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width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/4647/#Item_17" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <blockquote> <p>This entry is about the notion in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> in the sense of <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a> (<a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical</a>/<a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>). For the different notion of the same name in <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> see at <em><a class="existingWikiWord" href="/nlab/show/field">field</a></em>.</p> </blockquote> <hr /> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="physics">Physics</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a></p> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </p> </div></div></div> <h4 id="fields_and_quanta">Fields and quanta</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> and <a class="existingWikiWord" href="/nlab/show/fundamental+particle">particles</a> in <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></strong></p> <p><strong>and in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></strong>:</p> <p><strong><a class="existingWikiWord" href="/nlab/show/force">force</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge</a> <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/photon">photon</a> - <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> (<a class="existingWikiWord" href="/nlab/show/abelian+group">abelian</a> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/W-boson">W</a>, <a class="existingWikiWord" href="/nlab/show/Z-boson">Z</a>, <a class="existingWikiWord" href="/nlab/show/B-boson">B-boson</a> - <a class="existingWikiWord" href="/nlab/show/electroweak+field">electroweak field</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gluon">gluon</a> - <a class="existingWikiWord" href="/nlab/show/strong+nuclear+force">strong nuclear force</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graviton">graviton</a> - <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infraparticle">infraparticle</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/scalar+field">scalar</a> <a class="existingWikiWord" href="/nlab/show/bosons">bosons</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Higgs+boson">Higgs boson</a>, <a class="existingWikiWord" href="/nlab/show/inflaton">inflaton</a> (<a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a>)</li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/matter">matter</a> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a></strong> (<a class="existingWikiWord" href="/nlab/show/spinors">spinors</a>, <a class="existingWikiWord" href="/nlab/show/Dirac+fields">Dirac fields</a>)</p> <div> <table><thead><tr><th><strong><a class="existingWikiWord" href="/nlab/show/flavor+%28particle+physics%29">flavors</a> of <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental</a> <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> in the</strong> <br /> <strong><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>:</strong></th><th></th><th></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/generation+of+fermions">generation of fermions</a></td><td style="text-align: left;">1st generation</td><td style="text-align: left;">2nd generation</td><td style="text-align: left;">3d generation</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/quarks">quarks</a></strong> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>)</td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">up-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/up+quark">up quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/charm+quark">charm quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/top+quark">top quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;">down-type</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/down+quark">down quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strange+quark">strange quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom quark</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/leptons">leptons</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;">charged</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron">electron</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon">muon</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tauon">tauon</a></td></tr> <tr><td style="text-align: left;">neutral</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/electron+neutrino">electron neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/muon+neutrino">muon neutrino</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tau+neutrino">tau neutrino</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a>:</strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/mesons">mesons</a></strong></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/light+mesons">light mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/pion">pion</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>d</mi></mrow><annotation encoding="application/x-tex">u d</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a> <br /> <a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/strange+quark">strange</a>-mesons</strong>: <br /> <a class="existingWikiWord" href="/nlab/show/%CF%95-meson">ϕ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mover><mi>s</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">s \bar s</annotation></semantics></math>), <br /> <a class="existingWikiWord" href="/nlab/show/kaon">kaon</a>, <a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u s</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">d s</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/eta-meson">eta-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>u</mi><mo>+</mo><mi>d</mi><mi>d</mi><mo>+</mo><mi>s</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">u u + d d + s s</annotation></semantics></math>) <br /> <br /> <strong><a class="existingWikiWord" href="/nlab/show/charm+quark">charmed</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mi>c</mi></mrow><annotation encoding="application/x-tex"> u c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">d c</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">s c</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/J%2F%CF%88-meson">J/ψ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mover><mi>c</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">c \bar c</annotation></semantics></math>)</td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/bottom+quark">bottom</a> <a class="existingWikiWord" href="/nlab/show/heavy+mesons">heavy mesons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">q b</annotation></semantics></math>) <br /> <a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mover><mi>b</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">b \bar b</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/baryons">baryons</a></strong></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/nucleons">nucleons</a></strong>: <br /> <a class="existingWikiWord" href="/nlab/show/proton">proton</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>u</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u u d)</annotation></semantics></math> <br /> <a class="existingWikiWord" href="/nlab/show/neutron">neutron</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>u</mi><mi>d</mi><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(u d d)</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <p>(also: <a class="existingWikiWord" href="/nlab/show/antiparticles">antiparticles</a>)</p> <p><strong><a class="existingWikiWord" href="/nlab/show/effective+field+theory">effective particles</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Goldstone+bosons">Goldstone bosons</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/hadrons">hadrons</a></strong> (<a class="existingWikiWord" href="/nlab/show/bound+states">bound states</a> of the above <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/meson">meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/scalar+meson">scalar meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%83-meson">σ-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pion">pion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/kaon">kaon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-meson">D-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B-meson">B-meson</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+meson">vector meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%89-meson">ω-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%81-meson">ρ-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/f1-meson">f1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/a1-meson">a1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/b1-meson">b1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/h1-meson">h1-meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%2A-meson">K*-meson</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+meson">tensor meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quarkonium">quarkonium</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/charmonium">charmonium</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%CF%92-meson">ϒ-meson</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+meson">exotic meson</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/XYZ+meson">XYZ meson</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tetraquark">tetraquark</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/baryon">baryon</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/nucleon">nucleon</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/proton">proton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/neutron">neutron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chemical+element">chemical element</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/carbon">carbon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nitrogen">nitrogen</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lambda+baryon">Lambda baryon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pentaquark">pentaquark</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/solitons">solitons</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Skyrmion">Skyrmion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/caloron">caloron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> </ul> <p><strong>in <a class="existingWikiWord" href="/nlab/show/grand+unified+theory">grand unified theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/leptoquark">leptoquark</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Z%27-boson">Z'-boson</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/MSSM">minimally extended supersymmetric standard model</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/superpartners">superpartners</a></strong></p> <p>bosinos:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a> - <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> of <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> (<a class="existingWikiWord" href="/nlab/show/Rarita-Schwinger+field">Rarita-Schwinger field</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gaugino">gaugino</a> - <a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gluino">gluino</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/sfermions">sfermions</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/squark">squark</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/dark+matter">dark matter</a> candidates</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WIMP">WIMP</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/axion">axion</a></p> </li> </ul> <p><strong>Exotica</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/preon">preon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graviphoton">graviphoton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dilaton">dilaton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monopole">monopole</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dual+graviton">dual graviton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/giant+graviton">giant graviton</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/auxiliary+fields">auxiliary fields</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ghost+field">ghost field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/antifield">antifield</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/antighost+field">antighost field</a>, <a class="existingWikiWord" href="/nlab/show/Nakanishi-Lautrup+field">Nakanishi-Lautrup field</a></p> </li> </ul> </div></div> <h4 id="higher_geometry">Higher geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></strong> / <strong><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></strong></p> <p><strong>Ingredients</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></p> </li> </ul> <p><strong>Concepts</strong></p> <ul> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">little</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+scheme">generalized scheme</a></p> </li> </ul> </li> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">big</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-stacks">geometric ∞-stacks</a></li> </ul> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> / <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%28%E2%88%9E%2C1%29-site">étale (∞,1)-site</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-scheme">dg-scheme</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schematic+homotopy+type">schematic homotopy type</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></li> </ul> </li> <li> <p>derived smooth geometry</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Jones' theorem</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Deligne-Kontsevich conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+geometric+stacks">Tannaka duality for geometric stacks</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#AFirstIdeaOfQuantumFields'>A first idea of quantum fields</a></li> <ul> <li><a href='#spacetime'>Spacetime</a></li> <li><a href='#fields'>Fields</a></li> <li><a href='#field_variations'>Field variations</a></li> <li><a href='#equations_of_motion'>Equations of motion</a></li> </ul> <li><a href='#IdeaOfFieldBundlesAndItsProblems'>The traditional idea of field bundles and its problems</a></li> <ul> <li><a href='#IdeaLargeGaugeTransformations'>Large gauge transformations</a></li> <li><a href='#IdeaLocality'>Locality</a></li> <li><a href='#IdeaSpinStructures'>Spin-structures and other <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structures</a></li> <li><a href='#BackgroundFields'>Background fields</a></li> <li><a href='#HigherGaugeFields'>Higher gauge fields</a></li> </ul> <li><a href='#the_solution_field_bundles_and_moduli_stacks_of_fields'>The solution: Field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles and moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks of fields</a></li> </ul> <li><a href='#Definition'>Definition</a></li> <ul> <li><a href='#DefinitionPhysicalField'>Physical fields</a></li> <ul> <li><a href='#context_2'>Context</a></li> </ul> <li><a href='#RestrictionAndPullback'>Restriction and pullback of physical fields</a></li> <li><a href='#BoundaryAndDefectFields'>Boundary and defect fields</a></li> </ul> <li><a href='#FieldsProperties'>Properties</a></li> <ul> <li><a href='#ModuliStacksOfFields'>Moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks of fields</a></li> <li><a href='#RelationOfFieldsToSections'>Relation to sections of fiber <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles</a></li> <ul> <li><a href='#BriefReviewOfPrincipalInfinityBundlesAsBackgroundFields'>Background fields as principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles</a></li> <li><a href='#FieldsAsSectionsOfAssociatedBundles'>Fields as sections of associated fiber <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles</a></li> </ul> <li><a href='#RelationToTwistedCohomology'>Relation to twisted cohomology</a></li> <li><a href='#RelationToRelativeCohomology'>Relation to relative cohomology</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#SigmaModelFields'><strong>0)</strong> Sigma-model fields</a></li> <ul> <li><a href='#UnchargedScalarField'>Uncharged scalar field</a></li> <li><a href='#ParticleTrajectory'>Particle trajectory</a></li> <li><a href='#BraneTrajectory'>Brane trajectory</a></li> <li><a href='#OpenStringSigmaModelFields'>Open string sigma-model</a></li> </ul> <li><a href='#ForceFields'><strong>I)</strong> Force fields</a></li> <ul> <li><a href='#FieldsOfGravityAndGeneralizedGeometry'><strong>I a)</strong> Fields of gravity, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Structure and generalized geometry</a></li> <ul> <li><a href='#GStructure'><strong>General</strong> – <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure: (twisted) lift of structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-group</a></li> <li><a href='#OrdinaryGravity'>Gravity</a></li> <li><a href='#SpinStructures'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math>-structures</a></li> <li><a href='#SpinStructures'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-structures</a></li> <li><a href='#TypeIIGravity'>Type II gravity, exceptional geometry</a></li> <li><a href='#HigherSpinStructures'>Higher spin structures</a></li> <li><a href='#HigherSpincstructures'>Higher <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-structures</a></li> </ul> <li><a href='#GaugeFields'><strong>I b)</strong> Gauge fields</a></li> <ul> <li><a href='#GaugeFieldsGeneral'><strong>General</strong> – Coefficients in differential cohomology: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-connections</a></li> <ul> <li><a href='#contents_2'>Contents:</a></li> </ul> <li><a href='#ElectromagneticField'>Electromagnetic field</a></li> <li><a href='#YangMillsField'>Yang-Mills field</a></li> <li><a href='#KalbRamondBField'>Kalb-Ramond <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-field</a></li> <li><a href='#KalbRamondBField'>Supergeometric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-field</a></li> <li><a href='#SupergravityCField'>Supergravity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-field</a></li> </ul> </ul> <li><a href='#MatterFields'><strong>II)</strong> Matter fields</a></li> <ul> <li><a href='#SectionsOfAssociatedBundles'><strong>General</strong> – sections of associated fiber <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles</a></li> <li><a href='#Fermions'>Fermions</a></li> <li><a href='#TensorFields'>Tensor fields</a></li> </ul> <li><a href='#FieldCombiningVariousProperties'>Fields combining these properties</a></li> <ul> <li><a href='#TwistedDifferentialcStructurs'><strong>General</strong> – Twisted differential-cocycles and -<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math>-structures</a></li> <li><a href='#NonabelianChargedParticle'>Nonabelian charged particle trajectories – Wilson line</a></li> <li><a href='#ChernSimonsWithWilsonLines'>3d Chern-Simons field with Wilson line</a></li> <li><a href='#ChanPatonGaugeFields'>Chan-Paton gauge fields on D-branes: twisted differential K-cocycles</a></li> <li><a href='#HeteroticStringBackgroundField'>Anomaly-free heterotic string background: differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>String</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">String^c</annotation></semantics></math>-structure</a></li> </ul> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#lecture_notes_and_expositions'>Lecture notes and expositions</a></li> <li><a href='#original_articles_on_the_general_notion'>Original articles on the general notion</a></li> <li><a href='#original_articles_on_special_cases'>Original articles on special cases</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>In fundamental <a class="existingWikiWord" href="/nlab/show/physics">physics</a> the basic entities that are being described are called <em>fields</em>, as they appear in the terms <em><a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a></em> and <em><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></em>.</p> <h3 id="general">General</h3> <p>The basic example that probably gives the whole concept its name is the <a class="existingWikiWord" href="/nlab/show/electric+field">electric field</a> and the <a class="existingWikiWord" href="/nlab/show/magnetic+field">magnetic field</a> in the <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a> of <a class="existingWikiWord" href="/nlab/show/electromagnetism">electromagnetism</a>: if we fix a <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a> of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, then the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> splits into the <a class="existingWikiWord" href="/nlab/show/electric+field">electric field</a> and the <a class="existingWikiWord" href="/nlab/show/magnetic+field">magnetic field</a> which are both modeled by a <em><a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a></em>, traditionally denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec E</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>B</mi><mo stretchy="false">→</mo></mover></mrow><annotation encoding="application/x-tex">\vec B</annotation></semantics></math>, respectively, on this coordinate chart. The value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>E</mi><mo stretchy="false">→</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\vec E(x)</annotation></semantics></math> of the vector field at a given point of spacetime is a <a class="existingWikiWord" href="/nlab/show/vector">vector</a> that expresses the magnitude and direction of the electric <a class="existingWikiWord" href="/nlab/show/force">force</a> that is exerted on an <a class="existingWikiWord" href="/nlab/show/electric+charge">electrically charged</a> <a class="existingWikiWord" href="/nlab/show/particle">particle</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>.</p> <p>In fact more fundamentally, if we do not specify a <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, then the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> is not in fact represented by two vector fields. Rather, its <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> is represented by a <a class="existingWikiWord" href="/nlab/show/differential+2-form">differential 2-form</a>, hence a <a class="existingWikiWord" href="/nlab/show/tensor+field">tensor field</a> of rank <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,2)</annotation></semantics></math>, but the the whole field as such is not a tensor field, but is a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> of degree-2 in <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>: a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle bundle with connection</a>.</p> <p>Or for instance the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> if modeled as a <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+metric">pseudo-Riemannian metric</a> is a <a class="existingWikiWord" href="/nlab/show/tensor+field">tensor field</a> of rank <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,0)</annotation></semantics></math> – but subject to the constraint that this be pointwise non-degenerate. More fundamentally the field of gravity is instead a <a class="existingWikiWord" href="/nlab/show/vielbein+field">vielbein field</a>.</p> <p>Similar statements hold for all <a class="existingWikiWord" href="/nlab/show/forces">forces</a> of nature, such as the force of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> and the <a class="existingWikiWord" href="/nlab/show/weak+nuclear+force">weak nuclear force</a> and <a class="existingWikiWord" href="/nlab/show/strong+nuclear+force">strong nuclear force</a>: a configuration of these is mathematically modeled by <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connections</a>. Their <a class="existingWikiWord" href="/nlab/show/field+strengths">field strengths</a> are rank <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,2)</annotation></semantics></math>-tensor fields.</p> <p>The <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> and the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> are the physical fields that historically gave rise to what is now called <em><a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a></em>. But it turns out that fundamentally, in <a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum physics</a>, also all <a class="existingWikiWord" href="/nlab/show/matter">matter</a> in physics is constituted by fields in a similar sense. Specifically, where <a class="existingWikiWord" href="/nlab/show/force">force</a> fields in physics are usually <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connections on a bundle</a>, matter fields are <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of <a class="existingWikiWord" href="/nlab/show/associated+bundles">associated bundles</a>.</p> <p>Field theory was originally discovered as a <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a> of fields on <em><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></em>. But also the <a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a> consisting of a single <a class="existingWikiWord" href="/nlab/show/particle">particle</a> propagating in a <em>fixed</em> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is described by a field theory. In this case the field is not defined on spacetime, but on the abstract <em><a class="existingWikiWord" href="/nlab/show/worldline">worldline</a></em> of the particle, say the <em><a class="existingWikiWord" href="/nlab/show/real+line">real line</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>. A configuration of the system, namely a <a class="existingWikiWord" href="/nlab/show/trajectory">trajectory</a> of the particle, is then a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ℝ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \;\colon\; \mathbb{R}\to X</annotation></semantics></math>. This function may be regarded as a <em>field</em> on the <a class="existingWikiWord" href="/nlab/show/worldline">worldline</a> and in then called a <em><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></em> field. The <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> of a single particle may be equivalently thought of as a <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> on the 1-dimensional <a class="existingWikiWord" href="/nlab/show/worldline">worldline</a> of the particle.</p> <p>This perspective generalizes. Next one can consider fields on 2-dimensional <a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_2</annotation></semantics></math> which again are given by maps into some <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. The corresponding 2-dimensional <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> is then said to describe not a particle but a <em><a class="existingWikiWord" href="/nlab/show/string">string</a></em> propagating in spacetime, defined on the <em><a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_2</annotation></semantics></math>, replacing the <a class="existingWikiWord" href="/nlab/show/worldline">worldline</a> of the particle. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> of dimension 3 one accordingly speaks of the <em><a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a></em> of a <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> and then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> of general dimension here one speaks of the <em><a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a></em> of a <em><a class="existingWikiWord" href="/nlab/show/brane">brane</a></em>.</p> <p>But there is no fundamental distinction between physical fields on spaces that are interpreted as <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a> and those that are interpreted as <a class="existingWikiWord" href="/nlab/show/worldvolumes">worldvolumes</a> of objects propagating <em>in</em> a fixed spacetime. In general these notions mix. For instance the full description of <a class="existingWikiWord" href="/nlab/show/relativistic+particles">relativistic particles</a> and <a class="existingWikiWord" href="/nlab/show/relativistic+strings">relativistic strings</a> involves a field that is really a field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> on the <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>. Conversely, <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theories</a> on <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a> that arise by <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+compactification">Kaluza-Klein compactification</a> of higher dimensional theories typically have “<a class="existingWikiWord" href="/nlab/show/scalar+field">scalar moduli fields</a>” that used to be components of the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> in higher dimensions but now after compactifications become maps into some auxiliary <a class="existingWikiWord" href="/nlab/show/target+space">target space</a>, hence again <em><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></em> fields.</p> <h3 id="AFirstIdeaOfQuantumFields">A first idea of quantum fields</h3> <p>We introduce here the basic concepts of <em><a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a></em>, first for <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a> and then for its <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> to <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>.</p> <p>In full beauty these concepts are extremely general; but in this section the aim is to give a first good idea of the subject, and therefore we present for the moment only a restricted setup, notably assuming that <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> is <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>, that the <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> (see below) is an ordinary and <a class="existingWikiWord" href="/nlab/show/trivial+bundle">trivial</a> <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> and that all fields are <a class="existingWikiWord" href="/nlab/show/boson">bosonic</a>.</p> <p>This does subsume what is considered in most traditional texts on the subject. In subsequent sections we will eventually discuss more general situations, notably we will eventually allow spacetime to be any <a class="existingWikiWord" href="/nlab/show/globally+hyperbolic+Lorentzian+manifold">globally hyperbolic Lorentzian manifold</a> and the <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> to be an super <a class="existingWikiWord" href="/nlab/show/infinity-Lie+algebroid">infinity-Lie algebroid</a>. This is sufficient generality to capture the established perturbative <a class="existingWikiWord" href="/nlab/show/BV+formalism">BRST-BV quantization</a> of <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> coupled to <a class="existingWikiWord" href="/nlab/show/gauge+fields">gauge fields</a> <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetime">on curved spacetimes</a>.</p> <p>Throughout we use the case of the real <a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a> as an illustrative running example, which we develop alongside with the theory. The discussion of other <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> species that are of more genuine interest in applications is postponed to their dedicated sections below.</p> <h4 id="spacetime">Spacetime</h4> <p>Thoughout, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex"> p \in \mathbb{N} </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a> and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>≔</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mo>≔</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Sigma \coloneqq \mathbb{R}^{p,1} \coloneqq (\mathbb{R}^{p+1}, \eta) </annotation></semantics></math></div> <p>for <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p+1</annotation></semantics></math>, hence for the <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> which is the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{p+1}</annotation></semantics></math> of dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p+1</annotation></semantics></math> equipped with the <a class="existingWikiWord" href="/nlab/show/left+invariant+differential+form">constant</a> <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+metric">pseudo-Riemannian metric</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> which at the origin is given by the standard <a class="existingWikiWord" href="/nlab/show/quadratic+form">quadratic form</a> of <a class="existingWikiWord" href="/nlab/show/signature+of+a+quadratic+form">signature</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo>,</mo><mi>⋯</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (-, +, \cdots, +) </annotation></semantics></math></div> <p>in terms of the canonical <a class="existingWikiWord" href="/nlab/show/coordinate+functions">coordinate functions</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>k</mi></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>⟶</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> x^k \;\colon\; \mathbb{R}^{p+1} \longrightarrow \mathbb{R} </annotation></semantics></math></div> <p>which we index starting at zero: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>k</mi></msup><msubsup><mo stretchy="false">)</mo> <mrow><mi>k</mi><mo>=</mo><mn>0</mn></mrow> <mi>p</mi></msubsup></mrow><annotation encoding="application/x-tex">(x^k)_{k = 0}^p</annotation></semantics></math>.</p> <p>We write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>dvol</mi> <mi>Σ</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="mediummathspace" rspace="mediummathspace">∶−</mo><mspace width="thickmathspace"></mspace><mi>d</mi><msup><mi>x</mi> <mn>0</mn></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mn>1</mn></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mi>p</mi></msup><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> dvol_\Sigma \;\coloneq\; d x^0 \wedge d x^1 \wedge \cdots \wedge d x^p \in \Omega^{p+1}(\mathbb{R}^{p,1}) </annotation></semantics></math></div> <p>for the induced <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a>, and we call</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><msup><mi>x</mi> <mn>0</mn></msup><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d x^0 \in \Omega^1(\Sigma) </annotation></semantics></math></div> <p>the canonical representative of the canonical <a class="existingWikiWord" href="/nlab/show/time+orientation">time orientation</a> on Minkowski spacetime.</p> <h4 id="fields">Fields</h4> <p>A <em><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> configuration</em> on a given <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> is meant to be some kind of <a class="existingWikiWord" href="/nlab/show/quantity">quantity</a> assigned to each point of spacetime (each <a class="existingWikiWord" href="/nlab/show/event">event</a>), such that this assignment varies smoothly with spacetime points. For instance an <em><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a></em> configuration is at each point of spacetime a collection of <a class="existingWikiWord" href="/nlab/show/vectors">vectors</a> that encode the direction in which a <a class="existingWikiWord" href="/nlab/show/charged+particle">charged particle</a> passing through that point will feel a <a class="existingWikiWord" href="/nlab/show/force">force</a> (the <a class="existingWikiWord" href="/nlab/show/Lorentz+force">Lorentz force</a>).</p> <p>This is readily formalized: If</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><mi>SmthMfd</mi></mrow><annotation encoding="application/x-tex"> F \in SmthMfd </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of “values” that the the given kind of field may take at any spacetime point, then a field configuration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> is modeled as a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> from spacetime to this space of values:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Σ</mi><mo>⟶</mo><mi>F</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Phi \;\colon\; \Sigma \longrightarrow F \,. </annotation></semantics></math></div> <p>It will be useful to unify spacetime and the space of field values into a single space, the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>Σ</mi><mo>×</mo><mi>F</mi></mrow><annotation encoding="application/x-tex"> E \;\coloneqq\; \Sigma \times F </annotation></semantics></math></div> <p>and to think of this equipped with the <a class="existingWikiWord" href="/nlab/show/projection">projection</a> map onto the first factor as a <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> of spaces of field values over spacetime</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>E</mi></mtd> <mtd><mo>≔</mo></mtd> <mtd><mi>Σ</mi><mo>×</mo><mi>F</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>fb</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>pr</mi> <mn>1</mn></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd><mi>Σ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ E &\coloneqq& \Sigma \times F \\ {}^{\mathllap{fb}}\downarrow & \swarrow_{\mathrlap{pr_1}} \\ \Sigma } \,. </annotation></semantics></math></div> <p>This is then called the <em><a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a></em>, which specifies the kind of values that the given field species may take at any point of spacetime. Since the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> of field values is the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> of this <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a>, it is sometimes also called the <em><a class="existingWikiWord" href="/nlab/show/field+fiber">field fiber</a></em>.</p> <div class="num_defn"> <h6 id="definition">Definition</h6> <p><strong>(fields)</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>fb</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Σ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{E \\ \downarrow^{\mathrlap{fb}} \\ \Sigma }</annotation></semantics></math>, then a <em><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> configuration</em> (of type specified by this field bundle) is a <em>smooth <a class="existingWikiWord" href="/nlab/show/section">section</a></em> of this <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a>, namely a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mo lspace="verythinmathspace">:</mo><mi>Σ</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">\Phi \colon \Sigma \longrightarrow E</annotation></semantics></math> such that composed with the <a class="existingWikiWord" href="/nlab/show/projection">projection</a> map it is the <a class="existingWikiWord" href="/nlab/show/identity+function">identity function</a>, i.e. such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>fb</mi><mo>∘</mo><mi>Φ</mi><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">fb \circ \Phi = id</annotation></semantics></math>, or, diagrammatically, such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>Φ</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>fb</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Σ</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>Σ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && E \\ & {}^{\mathllap{\Phi}}\nearrow & \downarrow^{\mathrlap{fb}} \\ \Sigma & = & \Sigma } \,. </annotation></semantics></math></div> <p>The <em>field <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a></em> is the <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth</a> <a class="existingWikiWord" href="/nlab/show/space+of+sections">space of all these</a>, to be denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>SmoothSet</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma_\Sigma(E) \in SmoothSet \,. </annotation></semantics></math></div> <p>This is the <a class="existingWikiWord" href="/nlab/show/set">set</a> of all field configurations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> as above, and it is equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a> by declaring that a <em>smooth family</em> of field configurations, parameterized over any <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> is a smooth function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>U</mi><mo>×</mo><mi>Σ</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>Φ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>u</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>Φ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ U \times \Sigma &\overset{\Phi_{(-)}(-)}{\longrightarrow}& E \\ (u,x) &\mapsto& \Phi_u(x) } </annotation></semantics></math></div> <p>such that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">u \in U</annotation></semantics></math> we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><msub><mi>Φ</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>id</mi> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex">p \circ \Phi_{u}(-) = id_\Sigma</annotation></semantics></math>, i.e.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>Φ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>fb</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>U</mi><mo>×</mo><mi>Σ</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>pr</mi> <mn>2</mn></msub></mrow></munder></mtd> <mtd><mi>Σ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && E \\ & {}^{\mathllap{\Phi_{(-)}(-)}}\nearrow & \downarrow^{\mathrlap{fb}} \\ U \times \Sigma &\underset{pr_2}{\longrightarrow}& \Sigma } \,. </annotation></semantics></math></div></div> <div class="num_example" id="TrivialVectorBundleAsAFieldBundle"> <h6 id="example">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a> as a <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a>)</strong></p> <p>In applications the <a class="existingWikiWord" href="/nlab/show/field+fiber">field fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is often a <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> and equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>. In this case the trivial field bundle with fiber <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is of course a <em><a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial</a> <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a></em>.</p> <p>Choosing any <a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>ϕ</mi> <mi>a</mi></msup><msubsup><mo stretchy="false">)</mo> <mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow> <mi>s</mi></msubsup></mrow><annotation encoding="application/x-tex">(\phi^a)_{a = 1}^s</annotation></semantics></math> of the field fiber, then over <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> we have canonical <a class="existingWikiWord" href="/nlab/show/coordinates">coordinates</a> on the total space of the field bundle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>μ</mi></msup><msubsup><mo stretchy="false">)</mo> <mrow><mi>μ</mi><mo>=</mo><mn>0</mn></mrow> <mi>p</mi></msubsup><mo>,</mo><mo stretchy="false">(</mo><msup><mi>ϕ</mi> <mi>a</mi></msup><msubsup><mo stretchy="false">)</mo> <mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow> <mi>s</mi></msubsup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ( (x^\mu)_{\mu = 0}^p, ( \phi^a )_{a = 1}^s ) \,. </annotation></semantics></math></div></div> <div class="num_example" id="RealScalarFieldBundle"> <h6 id="example_2">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/real+scalar+field">real scalar field</a>)</strong></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> and if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>≔</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> F \coloneqq \mathbb{R} </annotation></semantics></math></div> <p>is simply the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, then the corresponding trivial <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Σ</mi><mo>×</mo><mi>ℝ</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>pr</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>Σ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Sigma \times \mathbb{R} \\ {}^{\mathllap{pr_1}}\downarrow \\ \Sigma } </annotation></semantics></math></div> <p>is the <em><a class="existingWikiWord" href="/nlab/show/trivial+fiber+bundle">trivial</a> <a class="existingWikiWord" href="/nlab/show/real+line+bundle">real line bundle</a></em> (a special case of example <a class="maruku-ref" href="#TrivialVectorBundleAsAFieldBundle"></a>) and the corresponding <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> is called the <em><a class="existingWikiWord" href="/nlab/show/real+scalar+field">real scalar field</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>. A configuration of this field is simply a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> with values in the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo>×</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma_\Sigma(\Sigma \times \mathbb{R}) \;\simeq\; C^\infty(\Sigma) \,. </annotation></semantics></math></div></div> <h4 id="field_variations">Field variations</h4> <p>Given a <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> as above, we know what type of quantities the corresponding fields assign to a given spacetime point. Among all consistent such field configurations, some are to qualify as those that “may occur in reality” if we think of the field theory as a means to describe parts of the <a class="existingWikiWord" href="/nlab/show/observable+universe">observable universe</a>. Moreover, if the reality to be described does not exhibit “action at a distance” then admissibility of its field configurations should be determined over arbitrary small spacetime regions, in fact over the <a class="existingWikiWord" href="/nlab/show/infinitesimal+neighbourhood">infinitesimal neighbourhood</a> of any point. This means equivalently that the realized field configurations should be those that satisfy a specific <em><a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a></em>, hence an <a class="existingWikiWord" href="/nlab/show/equation">equation</a> between the value of its <a class="existingWikiWord" href="/nlab/show/derivatives">derivatives</a> at any spacetime point.</p> <p>In order to formalize this, it is useful to first collect all the possible derivatives that a field may have at any given point into one big space of “field derivatives at spacetime points”. This collection is called the <em><a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a></em> of the <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a>, given as def. <a class="maruku-ref" href="#JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime"></a> below.</p> <p>Moving around in this space means to change the possible value of fields and their derivatives, hence to <em>vary</em> the fields. Accordingly <em><a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a></em> is just <a class="existingWikiWord" href="/nlab/show/differential+calculus">differential calculus</a> on a <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a>, this we consider in def. <a class="maruku-ref" href="#VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime"></a> below.</p> <div class="num_defn" id="JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> of a <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a> over <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a>)</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/field+fiber">field fiber</a> <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>=</mo><msup><mi>ℝ</mi> <mi>s</mi></msup></mrow><annotation encoding="application/x-tex">F = \mathbb{R}^s</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>ϕ</mi> <mi>a</mi></msup><msubsup><mo stretchy="false">)</mo> <mrow><mi>a</mi><mo>=</mo><mn>1</mn></mrow> <mi>s</mi></msubsup></mrow><annotation encoding="application/x-tex">(\phi^a)_{a = 1}^s</annotation></semantics></math>, then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k \in \mathbb{N}</annotation></semantics></math> a natural number, the <em>order-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msubsup><mi>J</mi> <mi>Σ</mi> <mi>k</mi></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>jb</mi> <mi>k</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Σ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ J^k_{\Sigma}( E ) \\ \downarrow^{\mathrlap{jb_k}} \\ \Sigma } </annotation></semantics></math></div> <p>over <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>≔</mo><mi>Σ</mi><mo>×</mo><mi>F</mi></mrow><annotation encoding="application/x-tex"> E \coloneqq \Sigma \times F </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> which is spanned by coordinate functions to be denoted as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><msup><mi>ϕ</mi> <mi>a</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><msub><mi>μ</mi> <mn>1</mn></msub><msub><mi>μ</mi> <mn>2</mn></msub></mrow> <mi>a</mi></msubsup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><msub><mi>μ</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>μ</mi> <mi>k</mi></msub></mrow> <mi>a</mi></msubsup><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( (x^\mu) \,,\, (\phi^a ) \,,\, ( \phi^a_{,\mu} ) \,,\, ( \phi^a_{,\mu_1\mu_2} ) \,,\, \cdots \,,\, ( \phi^a_{,\mu_1 \cdots \mu_k} ) \right) </annotation></semantics></math></div> <p>where the indices <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>,</mo><msub><mi>μ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>μ</mi> <mn>2</mn></msub><mo>,</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex">\mu, \mu_1, \mu_2, \cdots</annotation></semantics></math> range from 0 to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, while the index <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> ranges from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>. In terms of these coordinates the <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> <a class="existingWikiWord" href="/nlab/show/projection">projection</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>jb</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">jb_k</annotation></semantics></math> is just the one that remembers the spacetime coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>μ</mi></msup></mrow><annotation encoding="application/x-tex">x^\mu</annotation></semantics></math> and forgets the values of the field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow><annotation encoding="application/x-tex">\phi^a</annotation></semantics></math> and its derivatives <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>μ</mi></msub></mrow><annotation encoding="application/x-tex">\phi_{\mu}</annotation></semantics></math>. Similarly there are intermediate projection maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋯</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>jb</mi> <mrow><mn>3</mn><mo>,</mo><mn>2</mn></mrow></msub></mrow></mover></mtd> <mtd><msubsup><mi>J</mi> <mi>Σ</mi> <mn>2</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>jb</mi> <mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msub></mrow></mover></mtd> <mtd><msubsup><mi>J</mi> <mi>Σ</mi> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>jb</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>jb</mi> <mn>2</mn></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>jb</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>fb</mi></mpadded></msub></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Σ</mi></mtd> <mtd></mtd> <mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \cdots &\overset{jb_{3,2}}{\longrightarrow}& J^{2}_\Sigma(E) &\overset{jb_{2,1}}{\longrightarrow}& J^1_\Sigma(E) &\overset{jb_1}{\longrightarrow}& E \\ && &{}_{\mathllap{jb_2}}\searrow& {}^{\mathllap{jb_1}}\downarrow &\swarrow_{\mathrlap{fb}}& \\ && && \Sigma && } </annotation></semantics></math></div> <p>given by forgetting coordinates with more indices.</p> <p>The <em>infinite-order <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex"> J^\infty_\Sigma(E) \in SmoothSet </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a> defined so that a smooth function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mover><mo>⟶</mo><mi>f</mi></mover><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> U \overset{f}{\longrightarrow} J^\infty_\Sigma(E) </annotation></semantics></math></div> <p>from some <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> is equivalently a system of ordinary smooth functions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>(</mo><mi>U</mi><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mi>k</mi></msub></mrow></mover><msubsup><mi>J</mi> <mi>Σ</mi> <mi>k</mi></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>)</mo></mrow> <mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \left( U \overset{f_k}{\longrightarrow} J^k_\Sigma(E) \right)_{k \in \mathbb{N}} </annotation></semantics></math></div> <p>into all the finite-order jet bundles, such that this is compatible with the above projection maps, i.e. such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo>∀</mo><mrow><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><mrow><mo>(</mo><msub><mi>jb</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow></msub><mo>∘</mo><msub><mi>f</mi> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>f</mi> <mi>k</mi></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{k \in \mathbb{N}}{\forall} \left( jb_{k+1,k} \circ f_{k+1} = f_k \right) \,. </annotation></semantics></math></div> <p>Finally <em><a class="existingWikiWord" href="/nlab/show/jet+prolongation">jet prolongation</a></em> is that function from the <a class="existingWikiWord" href="/nlab/show/space+of+sections">space of sections</a> of the original bundle to the space of sections of the jet bundle which records the field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> and all its spacetimes <a class="existingWikiWord" href="/nlab/show/derivatives">derivatives</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mi>j</mi> <mn>∞</mn></msup></mrow></mover></mtd> <mtd><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><msup><mi>Φ</mi> <mi>a</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><mrow><mo>(</mo><mo stretchy="false">(</mo><msup><mi>Φ</mi> <mi>a</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mfrac><mrow><mo>∂</mo><msup><mi>Φ</mi> <mi>a</mi></msup></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msup><mi>Φ</mi> <mi>a</mi></msup></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mrow><msub><mi>μ</mi> <mn>1</mn></msub></mrow></msup><mo>∂</mo><msup><mi>x</mi> <mrow><msub><mi>μ</mi> <mn>2</mn></msub></mrow></msup></mrow></mfrac><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mi>⋯</mi><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \Gamma_\Sigma(E) &\overset{j^\infty}{\longrightarrow}& \Gamma_\Sigma(J^\infty_\Sigma(E)) \\ (\Phi^a) &\mapsto& \left( (\Phi^a) \,,\, ( \frac{\partial \Phi^a}{\partial x^\mu} ) \,,\, ( \frac{\partial^2 \Phi^a}{\partial x^{\mu_1} \partial x^{\mu_2}} ) \,,\, \cdots \right) } \,. </annotation></semantics></math></div></div> <p>Smooth functions on jet bundles turn out to <em>locally</em> depend on only finitely many of the jet coordinates:</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J^\infty_\Sigma(E)</annotation></semantics></math> as in def. <a class="maruku-ref" href="#JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime"></a>, then a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> out of it</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> J^\infty_\Sigma(E) \longrightarrow X </annotation></semantics></math></div> <p>is such that around each point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J^\infty_\Sigma(E)</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \subset J^\infty_\Sigma(E)</annotation></semantics></math> on which it is given by a function on a smooth function on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>Σ</mi> <mi>k</mi></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J^k_\Sigma(E)</annotation></semantics></math> for some finite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> </div> <div class="num_defn" id="VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime"> <h6 id="definition_3">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a>)</strong></p> <p>On the <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J^\infty_\Sigma(E)</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/trivial+vector+bundle">trivial vector bundle</a> over <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> as in def. <a class="maruku-ref" href="#JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime"></a> we may consider its <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> of <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a>; we write its <a class="existingWikiWord" href="/nlab/show/de+Rham+differential">de Rham differential</a> in boldface:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>Ω</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{d} \;\colon\; \Omega^\bullet(J^\infty_\Sigma(E)) \longrightarrow \Omega^{\bullet+1}(J^\infty_\Sigma(E)) \,. </annotation></semantics></math></div> <p>Since the jet bundle unified spacetime with field values, we want to decompose this differential into a contribution coming from forming the <a class="existingWikiWord" href="/nlab/show/total+derivatives">total derivatives</a> of fields along spacetime (“<a class="existingWikiWord" href="/nlab/show/horizontal+derivatives">horizontal derivatives</a>”), and actual <em>variation</em> of fields at a fixed spacetime point (“<a class="existingWikiWord" href="/nlab/show/vertical+derivatives">vertical derivatives</a>”):</p> <p>The <em><a class="existingWikiWord" href="/nlab/show/total+derivative">total spacetime derivative</a></em> or <em><a class="existingWikiWord" href="/nlab/show/horizontal+derivative">horizontal derivative</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J^\infty_\Sigma(E)</annotation></semantics></math> is the map on <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> on the jet bundle of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>Ω</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d \;\colon\; \Omega^\bullet( J^\infty_\Sigma(E) ) \longrightarrow \Omega^{\bullet+1}( J^\infty_\Sigma(E) ) </annotation></semantics></math></div> <p>which on functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f \colon J^\infty_\Sigma(E) \to \mathbb{R}</annotation></semantics></math> (i.e. on 0-forms) is defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>d</mi><mi>f</mi></mtd> <mtd><mo>≔</mo><mfrac><mrow><mi>d</mi><mi>f</mi></mrow><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>μ</mi></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≔</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>μ</mi><mo>=</mo><mn>0</mn></mrow><mi>p</mi></munderover><mrow><mo>(</mo><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow></mfrac><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>f</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi><mi>μ</mi></mrow> <mi>a</mi></msubsup><mo>+</mo><mi>⋯</mi><mo>)</mo></mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>μ</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} d f & \coloneqq \frac{d f}{d x^\mu} \mathbf{d} x^\mu \\ & \coloneqq \underoverset{\mu = 0}{p}{\sum} \left( \frac{\partial f}{\partial x^\mu} + \frac{\partial f}{\partial \phi^a} \phi^a_{,\mu} + \frac{ \partial f }{ \partial \phi^a_{,\nu}} \phi^a_{,\nu \mu } + \cdots \right) \mathbf{d} x^\mu \end{aligned} </annotation></semantics></math></div> <p>and extended to all forms by the graded <a class="existingWikiWord" href="/nlab/show/Leibniz+rule">Leibniz rule</a>, hence as a nilpotent <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> of degree +1.</p> <p>The <em><a class="existingWikiWord" href="/nlab/show/vertical+derivative">vertical derivative</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>Ω</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \delta \;\colon\; \Omega^\bullet( J^\infty_\Sigma(E) ) \longrightarrow \Omega^{\bullet+1}( J^\infty_\Sigma(E) ) </annotation></semantics></math></div> <p>is what remains of the full <a class="existingWikiWord" href="/nlab/show/de+Rham+differential">de Rham differential</a> when the total spacetime derivative (<a class="existingWikiWord" href="/nlab/show/horizontal+derivative">horizontal derivative</a>) is subtracted:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo>≔</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo>−</mo><mi>d</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \delta \coloneqq \mathbf{d} - d \,. </annotation></semantics></math></div> <p>This defines a bigrading on the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J^\infty_\Sigma(E)</annotation></semantics></math>, into horizontal degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> and vertical degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mrow><mo>(</mo><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><munder><mo>⊕</mo><mrow><mi>r</mi><mo>,</mo><mi>s</mi></mrow></munder><msup><mi>Ω</mi> <mrow><mi>r</mi><mo>,</mo><mi>s</mi></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega^\bullet\left( J^\infty_\Sigma(E) \right) \;\coloneqq\; \underset{r,s}{\oplus} \Omega^{r,s}(E) </annotation></semantics></math></div> <p>such that the horizontal and vertical derivative increase horizontal or vertical degree, respectively:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo></mtd> <mtd><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>d</mi></mover></mtd> <mtd><msup><mi>Ω</mi> <mrow><mn>1</mn><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>d</mi></mover></mtd> <mtd><msup><mi>Ω</mi> <mrow><mn>2</mn><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>d</mi></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>δ</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>δ</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>δ</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>d</mi></mover></mtd> <mtd><msup><mi>Ω</mi> <mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>d</mi></mover></mtd> <mtd><msup><mi>Ω</mi> <mrow><mn>2</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>d</mi></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>δ</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>δ</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>δ</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>d</mi></mover></mtd> <mtd><msup><mi>Ω</mi> <mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>d</mi></mover></mtd> <mtd><msup><mi>Ω</mi> <mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>d</mi></mover></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>δ</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>δ</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>δ</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ C^\infty(J^\infty_\Sigma(E)) = & \Omega^{0,0}(E) &\overset{d}{\longrightarrow}& \Omega^{1,0}(E) &\overset{d}{\longrightarrow}& \Omega^{2,0}(E) &\overset{d}{\longrightarrow}& \cdots \\ & \downarrow^{\mathrlap{\delta}} && \downarrow^{\mathrlap{\delta}} && \downarrow^{\mathrlap{\delta}} \\ & \Omega^{0,1}(E) &\overset{d}{\longrightarrow}& \Omega^{1,1}(E) &\overset{d}{\longrightarrow}& \Omega^{2,1}(E) &\overset{d}{\longrightarrow}& \cdots \\ & \downarrow^{\mathrlap{\delta}} && \downarrow^{\mathrlap{\delta}} && \downarrow^{\mathrlap{\delta}} \\ & \Omega^{0,2}(E) &\overset{d}{\longrightarrow}& \Omega^{1,2}(E) &\overset{d}{\longrightarrow}& \Omega^{2,2}(E) &\overset{d}{\longrightarrow}& \cdots \\ & \downarrow^{\mathrlap{\delta}} && \downarrow^{\mathrlap{\delta}} && \downarrow^{\mathrlap{\delta}} \\ & \vdots && \vdots && \vdots } \,. </annotation></semantics></math></div> <p>This is called the <em><a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a></em>.</p> </div> <p><strong>derivatives on <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a></strong></p> <table><thead><tr><th>symbols</th><th>name</th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{d}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/de+Rham+differential">de Rham differential</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>≔</mo><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">d \coloneqq d x^\mu \frac{d}{d x^\mu}</annotation></semantics></math></td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/total+derivative">total</a>) <a class="existingWikiWord" href="/nlab/show/horizontal+derivative">horizontal derivative</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mo>≔</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mo>+</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup><mfrac><mo>∂</mo><mrow><mo>∂</mo><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow></mfrac><mo>+</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> \frac{d}{d x^\mu} \coloneqq \frac{\partial}{\partial x^\mu} + \phi^a_{,\mu} \frac{\partial}{\partial \phi^a} + \cdots </annotation></semantics></math></td><td style="text-align: left;">(<a class="existingWikiWord" href="/nlab/show/total+derivative">total</a>) <a class="existingWikiWord" href="/nlab/show/horizontal+derivative">horizontal derivative</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>μ</mi></msub></mrow><annotation encoding="application/x-tex">\partial_\mu</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo>≔</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mo>−</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">\delta \coloneqq \mathbf{d} - d</annotation></semantics></math></td><td style="text-align: left;">(variational) <a class="existingWikiWord" href="/nlab/show/vertical+derivative">vertical derivative</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mi>L</mi><mo>≔</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>L</mi><mo>+</mo><mi>d</mi><mi>Θ</mi></mrow><annotation encoding="application/x-tex">\delta_{EL} L \coloneqq \mathbf{d}L + d \Theta</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+variational+derivative">Euler-Lagrange variational derivative</a></td></tr> </tbody></table> <div class="num_example" id="BasicFactsAboutVarationalCalculusOnJetBundleOfTrivialVectorBundle"> <h6 id="example_3">Example</h6> <p><strong>(basic facts about <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a>)</strong></p> <p>Given the jet bundle of a <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> as in def. <a class="maruku-ref" href="#JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime"></a>, then in its <a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a> (def. <a class="maruku-ref" href="#VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime"></a>) we have the following:</p> <ul> <li> <p>The <a class="existingWikiWord" href="/nlab/show/horizontal+derivative">horizontal derivative</a> of a spacetime coordinate function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>μ</mi></msup></mrow><annotation encoding="application/x-tex">x^\mu</annotation></semantics></math> coincides with its ordinary de Rham differential</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup><mo>=</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>μ</mi></msup><mo>∈</mo><msup><mi>Ω</mi> <mrow><mn>1</mn><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d x^\mu = \mathbf{d} x^\mu \in \Omega^{1,0}(E) </annotation></semantics></math></div> <p>and hence this is a horizontal 1-form.</p> </li> <li> <p>Therefore the vertical derivative of a spacetime coordinate vanishes:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><msup><mi>x</mi> <mi>μ</mi></msup><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \delta x^\mu = 0 \,. </annotation></semantics></math></div></li> <li> <p>In particular the given <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> gives a horizontal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">p+1</annotation></semantics></math>-form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo>=</mo><mi>d</mi><msup><mi>x</mi> <mn>0</mn></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mn>1</mn></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mi>p</mi></msup><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>0</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> dvol_\Sigma = d x^0 \wedge d x^1 \wedge \cdots \wedge d x^p \in \Omega^{p+1,0} \,. </annotation></semantics></math></div></li> <li> <p>Generally any horizontal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-form is of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><msub><mi>μ</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>μ</mi> <mi>k</mi></msub></mrow></msub><mi>d</mi><msup><mi>x</mi> <mrow><msub><mi>μ</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mrow><msub><mi>μ</mi> <mi>k</mi></msub></mrow></msup><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msup><mi>Ω</mi> <mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f_{\mu_1 \cdots \mu_k} d x^{\mu_1} \wedge \cdots \wedge d x^{\mu_k} \;\in\; \Omega^{k,0}(E) </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><msub><mi>μ</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>μ</mi> <mi>k</mi></msub></mrow></msub><mo>=</mo><msub><mi>f</mi> <mrow><msub><mi>μ</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>μ</mi> <mi>k</mi></msub></mrow></msub><mrow><mo>(</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msup><mi>ϕ</mi> <mi>a</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup><mo stretchy="false">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">f_{\mu_1 \cdots \mu_k} = f_{\mu_1 \cdots \mu_k}\left((x^\mu), (\phi^a), (\phi^a_{,\mu})\right)</annotation></semantics></math> any smooth function of the spacetime coordinates and the field coordinates.</p> </li> <li> <p>The horizontal differential of the vertical differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\delta \phi</annotation></semantics></math> of a field variable is the differential 2-form of horizontal degree 1 and vertical degree 2 given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>d</mi><mo stretchy="false">(</mo><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>δ</mi><mo stretchy="false">(</mo><mi>d</mi><msub><mi>ϕ</mi> <mi>a</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>δ</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup><mo stretchy="false">)</mo><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>μ</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} d (\delta \phi^a) & = - \delta (d \phi_a) \\ & = - (\delta \phi^a_{,\mu}) \wedge \mathbf{d} x^\mu \end{aligned} \,. </annotation></semantics></math></div> <p>In words this says that “the spacetime derivative of the variation of the field is the variation of its spacetime derivative”.</p> </li> </ul> </div> <div class="num_defn" id="LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime"> <h6 id="definition_4">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/local+Lagrangian+density">local Lagrangian density</a>)</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> over a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p+1)</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/Minkowski+spacetime">Minkowski spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> as in example <a class="maruku-ref" href="#TrivialVectorBundleAsAFieldBundle"></a>, then a <em><a class="existingWikiWord" href="/nlab/show/local+Lagrangian+density">local Lagrangian density</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{L}</annotation></semantics></math> (for the field species thus defined) is a <a class="existingWikiWord" href="/nlab/show/horizontal+differential+form">horizontal differential form</a> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p+1)</annotation></semantics></math> on the corresponding <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> (def. <a class="maruku-ref" href="#VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>L</mi></mstyle><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{L} \;\in \; \Omega^{p+1,0}(E) \,. </annotation></semantics></math></div> <p>By example <a class="maruku-ref" href="#BasicFactsAboutVarationalCalculusOnJetBundleOfTrivialVectorBundle"></a> any such Lagrangian density may uniquely be written as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>=</mo><mi>L</mi><msub><mi>dvol</mi> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex"> \mathbf{L} = L dvol_\Sigma </annotation></semantics></math></div> <p>with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>=</mo><mi>L</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msup><mi>ϕ</mi> <mi>a</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup><mo stretchy="false">)</mo><mo>,</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L = L((x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots ) </annotation></semantics></math> a smooth function on the jet bundle.</p> </div> <div class="num_prop" id="EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+operator">Euler-Lagrange operator</a>)</strong></p> <p>If a <a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{L}</annotation></semantics></math> as in def. <a class="maruku-ref" href="#LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime"></a>, then its de Rham differential has a <em>unique</em> decomposition as a sum of two terms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>=</mo><msub><mi>δ</mi> <mi>EL</mi></msub><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>−</mo><mi>d</mi><mi>Θ</mi></mrow><annotation encoding="application/x-tex"> \mathbf{d} \mathbf{L} = \delta_{EL} \mathbf{L} - d \Theta </annotation></semantics></math></div> <p>such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>EL</mi></msub></mrow><annotation encoding="application/x-tex">\delta_{EL}</annotation></semantics></math> is a “<a class="existingWikiWord" href="/nlab/show/source+form">source form</a>”:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>∧</mo><mi>δ</mi><msup><mi>Ω</mi> <mrow><mn>0</mn><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⊂</mo><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \delta_{EL} \mathbf{L} \in \Omega^{p+1,0}(E) \wedge \delta \Omega^{0,0}(E) \; \subset \Omega^{p+1,1}(E) \,. </annotation></semantics></math></div> <p>The map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>⟶</mo><msubsup><mi>Ω</mi> <mi>s</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \delta_{EL} \;\colon\; \Omega^{p+1,0}(E) \longrightarrow \Omega^{p+1,1}_s(E) </annotation></semantics></math></div> <p>thus defined is called the <em><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+operator">Euler-Lagrange operator</a></em> and is explicitly given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>δ</mi> <mi>EL</mi></msub><mi>L</mi><mspace width="thinmathspace"></mspace><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd> <mtd><mo>≔</mo><mfrac><mrow><mi>δ</mi><mi>L</mi></mrow><mrow><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow></mfrac><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup><mo>∧</mo><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≔</mo><mrow><mo>(</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow></mfrac><mo>−</mo><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi>d</mi> <mn>2</mn></msup></mrow><mrow><mi>d</mi><msup><mi>x</mi> <mrow><msup><mi>μ</mi> <mn>1</mn></msup></mrow></msup><mi>d</mi><msup><mi>x</mi> <mrow><msup><mi>μ</mi> <mn>2</mn></msup></mrow></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><msub><mi>μ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>μ</mi> <mn>2</mn></msub></mrow> <mi>a</mi></msubsup></mrow></mfrac><mo>−</mo><mi>⋯</mi><mo>)</mo></mrow><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup><mo>∧</mo><msub><mi>dvol</mi> <mi>Σ</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \delta_{EL} L \, dvol_\Sigma & \coloneqq \frac{\delta L}{ \delta \phi^a} \delta \phi^a \wedge dvol_\Sigma \\ & \coloneqq \left( \frac{\partial L}{\partial \phi^a} - \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} + \frac{d^2}{d x^{\mu^1} d x^{\mu^2}} \frac{\partial L}{\partial \phi^a_{\mu_1, \mu_2}} - \cdots \right) \delta \phi^a \wedge dvol_\Sigma \,. \end{aligned} </annotation></semantics></math></div> <p>The remaining term <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>Θ</mi></mrow><annotation encoding="application/x-tex">d \Theta</annotation></semantics></math> is unique, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Θ</mi><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Theta \in \Omega^{p,1}(E)</annotation></semantics></math> is unique only up to terms in the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>. One possible choice is</p> <div class="maruku-equation" id="eq:StandardThetaForTrivialVectorFieldBundleOnMinkowskiSpacetime"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>Θ</mi></mtd> <mtd><mo>≔</mo><mphantom><mo lspace="verythinmathspace" rspace="0em">+</mo></mphantom><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup><mspace width="thickmathspace"></mspace><mo>∧</mo><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>μ</mi></msub></mrow></msub><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><mrow><mo>(</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi></mrow> <mi>a</mi></msubsup><mo>−</mo><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi><mi>ν</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup><mo>)</mo></mrow><mo>∧</mo><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>μ</mi></msub></mrow></msub><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mphantom><mo>=</mo></mphantom><mo>+</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \Theta & \coloneqq \phantom{+} \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a \; \wedge \iota_{\partial_\mu} dvol_\Sigma \\ & \phantom{=} + \left( \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu} - \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a_{,\mu} \right) \wedge \iota_{\partial_\mu} dvol_\Sigma \\ & \phantom{=} + \cdots \,, \end{aligned} </annotation></semantics></math></div> <p>where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>μ</mi></msub></mrow></msub><msub><mi>dvol</mi> <mi>Σ</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>μ</mi></msup><mi>d</mi><msup><mi>x</mi> <mn>0</mn></msup><mo>∧</mo><mi>⋯</mi><mi>d</mi><msup><mi>x</mi> <mrow><mi>μ</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mrow><mi>μ</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mi>d</mi><msup><mi>x</mi> <mi>p</mi></msup></mrow><annotation encoding="application/x-tex"> \iota_{\partial_{\mu}} dvol_\Sigma \;\coloneqq\; (-1)^{\mu} d x^0 \wedge \cdots d x^{\mu-1} \wedge d x^{\mu+1} \wedge \cdots \wedge d x^p </annotation></semantics></math></div> <p>denotes the contraction of the volume form with the <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>μ</mi></msub></mrow><annotation encoding="application/x-tex">\partial_\mu</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>=</mo><mi>L</mi><msub><mi>dvol</mi> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{L} = L dvol_\Sigma</annotation></semantics></math> and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d \mathbf{L} = 0</annotation></semantics></math> by degree reasons, we find</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mstyle mathvariant="bold"><mi>d</mi></mstyle><mstyle mathvariant="bold"><mi>L</mi></mstyle></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow></mfrac><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><msub><mi>μ</mi> <mn>1</mn></msub><msub><mi>μ</mi> <mn>2</mn></msub></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><msub><mi>μ</mi> <mn>1</mn></msub><msub><mi>μ</mi> <mn>2</mn></msub></mrow> <mi>a</mi></msubsup><mo>+</mo><mi>⋯</mi><mo>)</mo></mrow><mo>∧</mo><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbf{d}\mathbf{L} & = \left( \frac{\partial L}{\partial \phi^a} \delta \phi^a + \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu} + \frac{\partial L}{\partial \phi^a_{,\mu_1 \mu_2}} \delta \phi^a_{,\mu_1 \mu_2} + \cdots \right) \wedge dvol_{\Sigma} \end{aligned} \,. </annotation></semantics></math></div> <p>The idea now is to have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>Θ</mi></mrow><annotation encoding="application/x-tex">d \Theta</annotation></semantics></math> pick up those terms that would appear as <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> terms under the <a class="existingWikiWord" href="/nlab/show/integral">integral</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mi>Σ</mi></msub><msup><mi>j</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>Φ</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>L</mi></mrow><annotation encoding="application/x-tex">\int_\Sigma j^\infty(\Phi)^\ast \mathbf{d}L</annotation></semantics></math> if we were to consider <a class="existingWikiWord" href="/nlab/show/integration+by+parts">integration by parts</a> to remove spacetime derivatives of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow><annotation encoding="application/x-tex">\delta \phi^a</annotation></semantics></math>.</p> <p>We compute, using example <a class="maruku-ref" href="#BasicFactsAboutVarationalCalculusOnJetBundleOfTrivialVectorBundle"></a>, the total horizontal derivative of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Θ</mi></mrow><annotation encoding="application/x-tex">\Theta</annotation></semantics></math> from <a class="maruku-eqref" href="#eq:StandardThetaForTrivialVectorFieldBundleOnMinkowskiSpacetime">(1)</a> as follows:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>d</mi><mi>Θ</mi></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mi>d</mi><mrow><mo>(</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup><mo>)</mo></mrow><mo>+</mo><mi>d</mi><mrow><mo>(</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi></mrow> <mi>a</mi></msubsup><mo>−</mo><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mi>μ</mi><mi>ν</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup><mo>)</mo></mrow><mo>+</mo><mi>⋯</mi><mo>)</mo></mrow><mo>∧</mo><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>μ</mi></msub></mrow></msub><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mrow><mo>(</mo><mrow><mo>(</mo><mi>d</mi><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mo>)</mo></mrow><mo>∧</mo><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup><mo>−</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><mi>d</mi><msup><mi>ϕ</mi> <mi>a</mi></msup><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mrow><mo>(</mo><mi>d</mi><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mo>)</mo></mrow><mo>∧</mo><mi>δ</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi></mrow> <mi>a</mi></msubsup><mo>−</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><mi>d</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi></mrow> <mi>a</mi></msubsup><mo>−</mo><mrow><mo>(</mo><mi>d</mi><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi><mi>ν</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mo>)</mo></mrow><mo>∧</mo><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup><mo>+</mo><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi><mi>ν</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><mi>d</mi><msup><mi>ϕ</mi> <mi>a</mi></msup><mo>)</mo></mrow><mo>+</mo><mi>⋯</mi><mo>)</mo></mrow><mo>∧</mo><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>μ</mi></msub></mrow></msub><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mrow><mo>(</mo><mrow><mo>(</mo><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi></mrow> <mi>a</mi></msubsup><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi><mi>μ</mi></mrow> <mi>a</mi></msubsup><mo>−</mo><mfrac><mrow><msup><mi>d</mi> <mn>2</mn></msup></mrow><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup><mi>d</mi><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi><mi>ν</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup><mo>−</mo><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi><mi>ν</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup><mo>)</mo></mrow><mo>+</mo><mi>⋯</mi><mo>)</mo></mrow><mo>∧</mo><msub><mi>dvol</mi> <mi>Σ</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} d \Theta & = \left( d \left( \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a \right) + d \left( \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu} - \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{\mu \nu}} \delta \phi^a \right) + \cdots \right) \wedge \iota_{\partial_\mu} dvol_\Sigma \\ & = \left( \left( \left( d \frac{\partial L}{\partial \phi^a_{,\mu}} \right) \wedge \delta \phi^a - \frac{\partial L}{\partial \phi^a_{,\mu}} \delta d \phi^a \right) + \left( \left(d \frac{\partial L}{\partial \phi^a_{,\nu \mu}}\right) \wedge \delta \phi^a_{,\nu} - \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta d \phi^a_{,\nu} - \left( d \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \right) \wedge \delta \phi^a + \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta d \phi^a \right) + \cdots \right) \wedge \iota_{\partial_\mu} dvol_\Sigma \\ & = - \left( \left( \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a + \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu} \right) + \left( \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu} + \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu \mu} - \frac{d^2}{ d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a - \frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a_{,\mu} \right) + \cdots \right) \wedge dvol_\Sigma \,, \end{aligned} </annotation></semantics></math></div> <p>where in the last line we used that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><msup><mi>x</mi> <mrow><msub><mi>μ</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mrow><msub><mi>μ</mi> <mn>2</mn></msub></mrow></msub></mrow></msub><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>if</mtext><mspace width="thinmathspace"></mspace><msub><mi>μ</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>μ</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex"> d x^{\mu_1} \wedge \iota_{\partial_{\mu_2}} dvol_\Sigma = \left\{ \array{ dvol_\Sigma &\vert& \text{if}\, \mu_1 = \mu_2 \\ 0 &\vert& \text{otherwise} } \right. </annotation></semantics></math></div> <p>Here the two terms proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi><mi>ν</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow><annotation encoding="application/x-tex">\frac{d}{d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} \delta \phi^a_{,\mu}</annotation></semantics></math> cancel out, and we are left with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>Θ</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mo>−</mo><mrow><mo>(</mo><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mo>−</mo><mfrac><mrow><msup><mi>d</mi> <mn>2</mn></msup></mrow><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup><mi>d</mi><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi><mi>ν</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mo>+</mo><mi>⋯</mi><mo>)</mo></mrow><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup><mo>∧</mo><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo>−</mo><mrow><mo>(</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mi>δ</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>ν</mi><mi>μ</mi></mrow> <mi>a</mi></msubsup><mo>+</mo><mi>⋯</mi><mo>)</mo></mrow><mo>∧</mo><msub><mi>dvol</mi> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex"> d \Theta \;=\; - \left( \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} - \frac{d^2}{ d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu \nu}} + \cdots \right) \delta \phi^a \wedge dvol_\Sigma - \left( \frac{\partial L}{\partial \phi^a_{,\mu}} \delta \phi^a_{,\mu} + \frac{\partial L}{\partial \phi^a_{,\nu \mu}} \delta \phi^a_{,\nu \mu} + \cdots \right) \wedge dvol_\Sigma </annotation></semantics></math></div> <p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>d</mi><mi>Θ</mi></mrow><annotation encoding="application/x-tex">-d \Theta</annotation></semantics></math> shares with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{d} \mathbf{L}</annotation></semantics></math> the terms that are proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><msub><mi>μ</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>μ</mi> <mi>k</mi></msub></mrow> <mi>a</mi></msubsup></mrow><annotation encoding="application/x-tex">\delta \phi^a_{,\mu_1 \cdots \mu_k}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k \geq 1</annotation></semantics></math>, and so the remaining terms are proportional to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow><annotation encoding="application/x-tex">\delta \phi^a</annotation></semantics></math>, as claimed:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>L</mi><mo>+</mo><mi>d</mi><mi>Θ</mi><mo>=</mo><munder><munder><mrow><mrow><mo>(</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow></mfrac><mo>−</mo><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi>d</mi> <mn>2</mn></msup></mrow><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup><mi>d</mi><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi><mi>ν</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mo>+</mo><mi>⋯</mi><mo>)</mo></mrow><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup><mo>∧</mo><msub><mi>dvol</mi> <mi>Σ</mi></msub></mrow><mo>⏟</mo></munder><mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow></munder><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{d}L + d \Theta = \underset{ \delta_{EL}\mathbf{L} }{ \underbrace{ \left( \frac{\partial L}{\partial \phi^a} - \frac{d}{d x^\mu}\frac{\partial L}{\partial \phi^a_{,\mu}} + \frac{d^2}{d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu\nu}} + \cdots \right) \delta \phi^a \wedge dvol_\Sigma }} \,. </annotation></semantics></math></div></div> <h4 id="equations_of_motion">Equations of motion</h4> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange</a> <a class="existingWikiWord" href="/nlab/show/equation+of+motion">equation of motion</a>)</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> over <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> as in example <a class="maruku-ref" href="#TrivialVectorBundleAsAFieldBundle"></a> equipped with a <a class="existingWikiWord" href="/nlab/show/local+Lagrangian+density">local Lagrangian density</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>∈</mo><msup><mi>Ω</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{L} \in \Omega^{p+1,1}(E)</annotation></semantics></math> as in def. <a class="maruku-ref" href="#LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime"></a> then the corresponding <em><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange</a> <a class="existingWikiWord" href="/nlab/show/equation+of+motion">equation of motion</a></em> on fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mo>∈</mo><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Phi \in \Gamma_\Sigma(E)</annotation></semantics></math> is the equation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>j</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>Φ</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mrow><mo>(</mo><msub><mi>δ</mi> <mi>EL</mi></msub><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> j^\infty(\Phi)^\ast \left(\delta_{EL} \mathbf{L}\right) = 0 \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>j</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>Φ</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>Σ</mi><mo>→</mo><msup><mi>J</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">j^\infty(\Phi) \colon \Sigma \to J^\infty(E)</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/jet+prolongation">jet prolongation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> (def. <a class="maruku-ref" href="#JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime"></a>), <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>j</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>E</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">j^\infty(E)^\ast</annotation></semantics></math> the operation of <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a> along this function, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>EL</mi></msub></mrow><annotation encoding="application/x-tex">\delta_{EL}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+operator">Euler-Lagrange operator</a> from prop. <a class="maruku-ref" href="#EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime"></a>.</p> <p>By that same proposition this equation is equivalently the <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow></mfrac><mo>−</mo><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mi>d</mi> <mn>2</mn></msup></mrow><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup><mi>d</mi><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><mfrac><mrow><mo>∂</mo><mi>L</mi></mrow><mrow><mo>∂</mo><msubsup><mi>ϕ</mi> <mrow><mo>,</mo><mi>μ</mi><mi>ν</mi></mrow> <mi>a</mi></msubsup></mrow></mfrac><mo>−</mo><mi>⋯</mi><mo>)</mo></mrow><mrow><mo>(</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><msup><mi>Φ</mi> <mi>a</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mrow><mo>(</mo><mfrac><mrow><mo>∂</mo><msup><mi>Φ</mi> <mi>a</mi></msup></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mo>,</mo><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msup><mi>Φ</mi> <mi>a</mi></msup></mrow><mrow><mo>∂</mo><msup><mi>x</mi> <mi>μ</mi></msup><mo>∂</mo><msup><mi>x</mi> <mi>ν</mi></msup></mrow></mfrac><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \frac{\partial L}{\partial \phi^a} - \frac{d}{d x^\mu} \frac{\partial L}{\partial \phi^a_{,\mu}} + \frac{d^2}{d x^\mu d x^\nu} \frac{\partial L}{\partial \phi^a_{,\mu\nu}} - \cdots \right) \left( (x^\mu), (\Phi^a), \left( \frac{\partial \Phi^a}{\partial x^\mu}, \frac{\partial^2 \Phi^a}{\partial x^\mu \partial x^\nu} \right) \right) \;=\; 0 \,. </annotation></semantics></math></div> <p>We write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><msub><mo stretchy="false">)</mo> <mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mi>L</mi><mo>=</mo><mn>0</mn></mrow></msub><mo>↪</mo><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Gamma_\Sigma(E)_{\delta_{EL} L = 0} \hookrightarrow \Gamma_\Sigma(E) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/smooth+space">smooth</a> subspace of the space of all field configurations on those that solve this differential equation.</p> </div> <h3 id="IdeaOfFieldBundlesAndItsProblems">The traditional idea of field bundles and its problems</h3> <p>A traditional approach to formalizing the notion of <em>physical field</em> is to declare that the specification of a <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory in physics</a>/<a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a> comes with a <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math> over the <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (or better: naturally over all spacetimes, see at <em><a href="#IdeaLocality">Locality</a></em> below) called the <strong><em><a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a></em></strong> and that a field configuration of the system is a <a class="existingWikiWord" href="/nlab/show/section">section</a> of this field bundle. This is for instance the basis for the theory of the <a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a>, hence of <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> for expressing <a class="existingWikiWord" href="/nlab/show/covariant+phase+spaces">covariant phase spaces</a>, for standard <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a>, etc.</p> <p>While this goes in the right direction, it cannot be quite the final answer, as it misses crucial properties that are demanded of a general notion of field. We now discuss these problems:</p> <ol> <li> <p><a href="#IdeaLargeGaugeTransformations">Large gauge transformations</a></p> </li> <li> <p><a href="#IdeaLocality">Locality</a></p> </li> <li> <p><a href="#IdeaSpinStructures">Spin structures and other G-structures</a></p> </li> <li> <p><a href="#BackgroundFields">Background fields</a></p> </li> <li> <p><a href="#HigherGaugeFields">Higher gauge fields</a></p> </li> </ol> <p>In the course of discussing the problems we also motivate and indicate their solution by a more natural notion of field moduli in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>. This is then discussed in full detail in the <em><a href="#Definition">Definition</a></em>-section below.</p> <h4 id="IdeaLargeGaugeTransformations">Large gauge transformations</h4> <p>In <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> specifically but in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> generally, physical fields come equipped with a notion of which fields configurations, while nominally different, are <a class="existingWikiWord" href="/nlab/show/equivalence">equivalent</a>, called <em><a class="existingWikiWord" href="/nlab/show/gauge+equivalence">gauge equivalent</a></em> and it is crucial to retain the information of gauge equivalences and not pass to <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of gauge equivalent fields. This means that generically for any <a class="existingWikiWord" href="/nlab/show/physical+theory">physical theory</a>, even if all field configurations would be represented by a section of some <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a>, many such sections are in fact to be regarded as being equivalent. Or more precisely, there should be a <em><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></em> or <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></em> of field configurations of which the sections of the field bundle only form the space of <a class="existingWikiWord" href="/nlab/show/objects">objects</a>, while the <a class="existingWikiWord" href="/nlab/show/gauge+transformations">gauge transformations</a> form the <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> and the <a class="existingWikiWord" href="/nlab/show/higher+gauge+transformations">higher gauge transformations</a> of order <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> form the <a class="existingWikiWord" href="/nlab/show/n-morphisms">n-morphisms</a>.</p> <p>To some extent this is dealt with in traditional <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a>: after a choice of <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> on the space of field configurations, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> spits out a <a class="existingWikiWord" href="/nlab/show/derived+L-%E2%88%9E+algebroid">derived L-∞ algebroid</a> whose objects are field configurations, and whose 1-cells are infinitesimal invariances of the given <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>.</p> <p>This goes in the right direction– it is the <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a> of the more encompassing <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a> of fields and gauge transformations – but has several problems, the main one being that this does now know about the <a class="existingWikiWord" href="/nlab/show/large+gauge+transformations">large gauge transformations</a>, those which are not connected to the identity (because it only sees infinitesimal data). These are important in the full quantum theory.</p> <p>Famous examples of the importance of <a class="existingWikiWord" href="/nlab/show/large+gauge+transformations">large gauge transformations</a> appear in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/2d+CFT">2d CFT</a>, for which all standard theory would break down if the global <a class="existingWikiWord" href="/nlab/show/conformal+transformations">conformal transformations</a> were not considered as gauge transformations.</li> </ul> <h4 id="IdeaLocality">Locality</h4> <p>Fields defined as sections of field bundles cannot capture gauge phenomena in a <em>local</em> way, as is necessary for a manifestly local formulation such in <a class="existingWikiWord" href="/nlab/show/extended+prequantum+field+theory">extended prequantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/extended+quantum+field+theory">extended quantum field theory</a> (sometimes called the “multi-tiered” formulation).</p> <p>Specifically, in <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> for <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, a field configuration – a <em><a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a> configuration</em> – is a combination of an <a class="existingWikiWord" href="/nlab/show/instanton+sector">instanton sector</a> – modeled by the <a class="existingWikiWord" href="/nlab/show/equivalence+class">equivalence class</a> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> – and the “<a class="existingWikiWord" href="/nlab/show/gauge+potential">gauge potential</a>”, modeled by a <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on this bundle</a> (see below at <em><a href="#GaugeFields">Gauge fields</a></em> for details). There is a <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E(P) \to X</annotation></semantics></math> such that its <a class="existingWikiWord" href="/nlab/show/sections">sections</a> are precisely the <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connections</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math>, and so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>c</mi></msub><mi>E</mi><mo stretchy="false">(</mo><msub><mi>P</mi> <mi>c</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\coprod_{c} E(P_c) \to X</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> ranges over the instanton sectors, is a field bundle for Yang-Mills fields on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>But this construction is not local: if we consider this assignment of field bundles to all suitable manifolds <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \to X</annotation></semantics></math> is a cover of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, then we cannot in general obtain the field bundle on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by gluing the field bundle on the cover. This is because <em>locally</em> every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> has trivial class, so that locally there is always only a single (the trivial) instanton sector.</p> <p>This failure of locality is often not recognized in the literature, since many if not most descriptions of physics restrict to trivial spacetime topology and/or restrict to <a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a> only. A formulation accurate and encompassing enough to see this issue is <em><a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetimes">AQFT on curved spacetimes</a></em>. A reference that explicitly runs into this non-locality issue of the field bundle in gauge theory in this context is (<a href="field+bundle#BDS">Benini-Dappiaggi-Schenkel 13</a>, <a href="field+bundle#Schenkel14">Schenkel 14</a>): the authors define a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a> equipped with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> that assigns the <a class="existingWikiWord" href="/nlab/show/algebras+of+observables">algebras of observables</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/Yang-Mills+fields">Yang-Mills fields</a> built from the field bundle of connections on the given principal bundles; and they observe that the result fails to be a <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> in that the inclusion of observables of a smaller spacetime into a larger patch may fail the isotony axiom (<a href="field+bundle#BDS">BDS, remark 5.6</a>). The authors then try to circumvent this by restricting to trivial instanton sectors. The fix later appears in (<a href="#BeniniSchenkelSzabo15">Benini-Schenkel-Szabo 15</a>), where the authors then consider proper <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a> of fields.</p> <p>But notice that instanton sectors is a non-negligible phenomenon. For instance the very <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> is a <a class="existingWikiWord" href="/nlab/show/superposition">superposition</a> of all possible instanton sectors (see at <em><a class="existingWikiWord" href="/nlab/show/instanton+in+QCD">instanton in QCD</a></em> for more on this). And there are field theories where the fields consist entirely of “<a class="existingWikiWord" href="/nlab/show/instanton+sectors">instanton sectors</a>” and where there is no infinitesimal information about the gauge group at all: these are theories whose gauge group is a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a>, which includes notably <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> and its higher analogy such as the <a class="existingWikiWord" href="/nlab/show/Yetter+model">Yetter model</a>. This means that for these <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theories</a> a local field bundle formalism can see nothing of the actual fields and also traditional tools applied to a global field bundle (such as traditional <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a>) see nothing of the actual fields. All this is fixed by the formulation that we discuss <a href="#Definition">below</a>.</p> <p>But this example already points to the general nature of the problem with field bundles, and also to its solution: while the <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a>-component of <a class="existingWikiWord" href="/nlab/show/Yang-Mills+fields">Yang-Mills fields</a> are not section of a bundle, they famously are sections of a <em><a class="existingWikiWord" href="/nlab/show/stack">stack</a></em> – the “<a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-principal bundles”, an object in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>.</p> <p>The problem with the locality of the <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> for <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> is solved by passing from <a class="existingWikiWord" href="/nlab/show/fiber+bundles">fiber bundles</a> to <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundles">fiber ∞-bundles</a>: in the <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">higher differential geometry</a> there is an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}G_{conn}</annotation></semantics></math> – the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a> (being the <a class="existingWikiWord" href="/nlab/show/stackification">stackification</a> of the <a class="existingWikiWord" href="/nlab/show/groupoid+of+Lie+algebra-valued+forms">groupoid of Lie algebra-valued forms</a>) such that maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G_{conn}</annotation></semantics></math> are equivalent to Yang-Mills fields on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (even including their <a class="existingWikiWord" href="/nlab/show/gauge+transformations">gauge transformations</a>). This means that if we allow field bundles in higher geometry – <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundles">fiber ∞-bundles</a>, then that for Yang-Mills theory over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is even a trivial field bundle, namely the <a class="existingWikiWord" href="/nlab/show/projection">projection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \times \mathbf{B}G_{conn} \to X </annotation></semantics></math></div> <p>out of the <a class="existingWikiWord" href="/nlab/show/product">product</a> of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> with the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of fields.</p> <p>This is a differential refinement of what is called the trivial <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, which is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \times \mathbf{B}G \to X </annotation></semantics></math></div> <p>and hence the “field bundle for instanton sectors” of Yang-Mills fields.</p> <p>In summary: there cannot be a <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> such that its <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of <a class="existingWikiWord" href="/nlab/show/local+sections">local sections</a> is the sheaf of configurations of the Yang-Mills field. But there is a <a class="existingWikiWord" href="/nlab/show/fiber+infinity-bundle">fiber 2-bundle</a> whose <a class="existingWikiWord" href="/nlab/show/stack">stack</a> of sections is the stack of configurations of the Yang-Mills field.</p> <p>Judging from these examples one might be tempted to guess that the notion of field <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> should simply be replaced by that of field <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>. But in fact what the example rather suggests is that what matters directly is the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Fields}</annotation></semantics></math> of fields, which for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> is simply</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>=</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} = \mathbf{B}G_{conn} \,. </annotation></semantics></math></div> <p>This perspective, which we describe in detail <a href="#Definition">below</a> also has the pleasant effect that it drastically simplifies and unifies notions of quantum field theory, for this says equivalently that if only we allow spaces in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>, then <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> is a <em><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></em> quantum field theory: one whose fields are simply maps to a given <a class="existingWikiWord" href="/nlab/show/target+space">target space</a>, only that this target space here is a <a class="existingWikiWord" href="/nlab/show/stack">stack</a>.</p> <p>But there are more advantages, slightly less obvious. These we come to in the following points.</p> <h4 id="IdeaSpinStructures">Spin-structures and other <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structures</h4> <p>Some fields in physics are (or involve) choices of <em><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a></em> in the sense of <a class="existingWikiWord" href="/nlab/show/reduction+and+lift+of+structure+groups">reduction and lift of structure groups</a>. Well-known examples include the choice of <em><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a></em> and of <em><a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a></em> in field theories with <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a> fields (discussed in detail in <em><a href="#Fermions">Fermions</a></em> below). Often in the literature the choice of <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> and <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a> is treated as an external parameter, but detailed analysis at least in low-dimensional examples shows that the in the full theory this is really a field configuration. For instance in <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> for theories with fermions, part of the integral over all field configurations is a sum over Spin structures.</p> <p>Now, a spin structure <em>is</em> equivalently a section of something, but again not of a <a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, but of an analog in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>, a <a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>.</p> <p>To see how this works, first recall the case of orientations, whose description as sections of the <a class="existingWikiWord" href="/nlab/show/orientation+bundle">orientation bundle</a> is familiar.</p> <p>For a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> represented by a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tau_X \;\colon\; X \to \mathbf{B}GL(n) </annotation></semantics></math></div> <p>be the map that modulates its <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> (discussed at <a href="geometry+of+physics#TangentBundle"><em>geometry of physics - tangent bundle</em></a>). Consider then the following diagram, which shows lifts of this map to the <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a>/<a class="existingWikiWord" href="/nlab/show/moduli+stacks">moduli stacks</a> for various other groups (this is the <em><a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}O(n)</annotation></semantics></math>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mpadded width="0" lspace="-100%width"><mrow><msub><mi>s</mi> <mi>X</mi></msub></mrow></mpadded><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>o</mi> <mi>X</mi></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>e</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0"><mrow><msub><mi>τ</mi> <mi>X</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \vdots \\ && \downarrow \\ && \mathbf{B}Spin(n) &\stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1) \\ &\mathllap{s_X}\nearrow& \downarrow \\ && \mathbf{B}SO(n) &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 \\ &{}^{\mathllap{o_X}}\nearrow& \downarrow \\ X & \stackrel{e_X}{\to}& \mathbf{B}O(n) &\stackrel{\mathbf{w}_1}{\to}& \mathbf{B}\mathbb{Z}_2 \\ &{}_{\mathrlap{\tau_X}} \searrow& \downarrow \\ && \mathbf{B}GL(n) } </annotation></semantics></math></div> <p>A lift of the tangent bundle map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\tau_X</annotation></semantics></math> to a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_X \colon X \to \mathbf{B}O(n)</annotation></semantics></math> as indicated is a choice of <a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a> (a <a class="existingWikiWord" href="/nlab/show/vielbein+field">vielbein field</a>, discussed in detail below in <em><a href="#OrdinaryGravity">Ordinary gravity</a></em>). For the present discussion assume that this is given.</p> <p>The a further lift to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>o</mi> <mi>X</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">o_X : X \to \mathbf{B}SO(n)</annotation></semantics></math> is a choice of <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, and finally a lift to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>X</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s_X : X \to \mathbf{B}Spin(n)</annotation></semantics></math> is a choice of <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>.</p> <p>Now, every hook-shaped sub-<a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in the above of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}\hat G \\ \downarrow \\ \mathbf{B}G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^n A } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a>. By the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> this means that the “space” – really: <em><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></em> or just <em><a class="existingWikiWord" href="/nlab/show/type">type</a></em>, for short – of lifts of a given map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G</annotation></semantics></math> to a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}\hat G</annotation></semantics></math> is equivalently the type of trivializations of the <a class="existingWikiWord" href="/nlab/show/composition">composite</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mover><mo>→</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>A</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^n A</annotation></semantics></math>.</p> <p>Now if we have an <a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>X</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">e_X : X \to \mathbf{B}O(n)</annotation></semantics></math> given, then this composite map according to the above diagram is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{w}_1(e_X) \;\colon\; X \to \mathbf{B}\mathbb{Z}_2 \,. </annotation></semantics></math></div> <p>This represents the <a class="existingWikiWord" href="/nlab/show/first+Stiefel-Whitney+class">first Stiefel-Whitney class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>w</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>∈</mo><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[w_1(\tau_X)] \in H^1(X, \mathbb{Z}_2)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\tau_X</annotation></semantics></math>, and it classifies a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, hence a <a class="existingWikiWord" href="/nlab/show/double+cover">double cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\hat X \to X</annotation></semantics></math> and this is precisely the <a class="existingWikiWord" href="/nlab/show/orientation+bundle">orientation bundle</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Sections of this bundle are choices of <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, hence are “orientation-structure fields”.</p> <p>Assume then such orientation field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>o</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">o_X</annotation></semantics></math> is given. Then in the next step the relevant composite map is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>o</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{w}_2(o_X) \;\colon\; X \to \mathbf{B}^2 \mathbb{Z}_2 \,. </annotation></semantics></math></div> <p>This now represents the <a class="existingWikiWord" href="/nlab/show/second+Stiefel-Whitney+class">second Stiefel-Whitney class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>w</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>∈</mo><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[w_2(\tau_X)] \in H^2(X, \mathbb{Z}_2)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and classifies a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbf{B}\mathbb{Z}_2)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>P</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}\mathbb{Z}_2 &\to& P \\ && \downarrow \\ && X } \,. </annotation></semantics></math></div> <p>This is sometimes called the <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/lifting+bundle+gerbe">lifting bundle gerbe</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>o</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">o_X</annotation></semantics></math>. A choice of <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a> is a choice of section of this 2-bundle. Hence spin structures are parts of fields in physics which are not sections of a field 1-bundle. Again, this is faithfully captured only in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>.</p> <p>This is only the most famous phenomenon in a large class of similar structures of fields in field theory. Notably in higher dimensional <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> and in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> there are fields which are ever higher lifts through this <a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> – <a class="existingWikiWord" href="/nlab/show/higher+spin+structures">higher spin structures</a>, such as <a class="existingWikiWord" href="/nlab/show/String+structures">String structures</a> and <a class="existingWikiWord" href="/nlab/show/Fivebrane+structures">Fivebrane structures</a> in the next two steps. Accordingly, these are fields which are equivalently sections of <a class="existingWikiWord" href="/nlab/show/principal+3-bundles">principal 3-bundles</a> (the “<a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+3-bundle">Chern-Simons circle 3-bundle</a>”) and <a class="existingWikiWord" href="/nlab/show/principal+infinity-bundle">principal 7-bundles</a> (the “<a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+7-bundle">Chern-Simons circle 7-bundle</a>”).</p> <h4 id="BackgroundFields">Background fields</h4> <p>Comparison of the above discussions under <em><a href="#IdeaLocality">Locality</a></em> and <em><a href="#IdeaSpinStructures">Spin structures</a></em> shows that there we had a higher-geometric field bundle of <a class="existingWikiWord" href="/nlab/show/Yang-Mills+fields">Yang-Mills fields</a> which was hower “trivial” in the sense that it was a projection out of the <a class="existingWikiWord" href="/nlab/show/product">product</a> of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> with a <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a>, so that a field configuration was equivalently of <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>-type, namely simply a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\phi \colon : X \to \mathbf{B}G_{conn}</annotation></semantics></math>; whereas here the “spin-lifting 2-bundles” and its higher analogs are, in general, not of this product form, hence “Spin structure”-fields, at least superficially do not seem to be of <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>-type, even in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>.</p> <p>But a closer inspection shows that in fact both situations are entirely analogous – once we realize that here these Spin-structure fields are not really defined just on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, but on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> <em>equipped with its orientation</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>o</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">o_X</annotation></semantics></math>. Since, by the same logic as above, also the orientation is a “field”, we may call it a <strong><em><a class="existingWikiWord" href="/nlab/show/background+field">background field</a></em></strong>. It serves as “background” over which spin structure fields can be considered.</p> <p>In <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> incarnated naturally as <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">higher topos theory</a>, this state of affairs is naturally modeled and indeed yields again a <em>moduli stack of spin structure fields</em> and makes spin-structures be <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>-type fields, as follows:</p> <p>the natural way to regard both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as well as its orientation structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>o</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">o_X</annotation></semantics></math> as a single object is to regard the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mrow><msub><mi>o</mi> <mi>X</mi></msub></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \stackrel{o_X}{\to} \mathbf{B}SO(n)</annotation></semantics></math> as an object in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (2,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}SO(n)}</annotation></semantics></math>. In here an <a class="existingWikiWord" href="/nlab/show/object">object</a> is a map of <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}SO(n)</annotation></semantics></math>, and a morphism is map of the domains of these maps together with a <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> filling the evident triangle <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>. Notably a lift of the orientation structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>o</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">o_X</annotation></semantics></math> to a spin structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>s</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">s_X</annotation></semantics></math> as above, hence a diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>s</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>o</mi> <mi>X</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>SpinStruc</mi></mstyle> <mi>n</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &&\stackrel{s_X}{\to}&& \mathbf{B}Spin(n) \\ & {}_{\mathllap{o_X}}\searrow &\swArrow_\simeq& \swarrow_{\mathrlap{\mathbf{SpinStruc}_n}} \\ && \mathbf{B}SO(n) } </annotation></semantics></math></div> <p>is equivalently a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>o</mi> <mi>X</mi></msub><mo>→</mo><msub><mstyle mathvariant="bold"><mi>SpinStruc</mi></mstyle> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> o_X \to \mathbf{SpinStruc}_n </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}SO(n)}</annotation></semantics></math>. This is again of the same simple form of the Yang-Mills fields on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, which are maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> X \to \mathbf{B}G_{conn} \,, </annotation></semantics></math></div> <p>but in the collection of stacks <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> itself, not in a <a class="existingWikiWord" href="/nlab/show/slice+%28infinity%2C1%29-topos">slice</a>.</p> <p>The slice here encodes the presence of <a class="existingWikiWord" href="/nlab/show/background+fields">background fields</a> – namely orientations in this case – whose <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> in turn is, in this case, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}SO(n)</annotation></semantics></math>.</p> <p>Notice that also the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> has a background field in this precise sense: as metioned above, a gravitational field configuration is a lift of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\tau_X</annotation></semantics></math> through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>OrthStruc</mi></mstyle> <mi>n</mi></msub></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}O(n) \stackrel{\mathbf{OrthStruc}_n}{\to} \mathbf{B}GL(n)</annotation></semantics></math>, hence a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub><mo>→</mo><msub><mstyle mathvariant="bold"><mi>OrthStruc</mi></mstyle> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> \tau_X \to \mathbf{OrthStruc}_n </annotation></semantics></math></div> <p>in the slice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}GL(n)}</annotation></semantics></math>. (Discussed in detail in <em><a href="#OrdinaryGravity">Ordinary gravity</a></em> below.) Hence also <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> becomes a <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a>-type field theory in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>. Notice that here it is <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, as embodied in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\tau_X</annotation></semantics></math>, which is the background.</p> <p>Now, at least for the field of gravity one can of course emulate the fields also by sections of a field bundle (while already for the second next step in the Whitehead tower, Spin structures, this is no longer the case, as we have seen). But even so, the field bundle formalism clearly misses then the relation between fields and background fields.</p> <p>In particular for two reasons</p> <ol> <li> <p>Typically the presence of background fields indicates that in a more comprehensive discussion background fields are also fields that vary;</p> </li> <li> <p>Often background fields on one space affect fields on <em>another</em> space.</p> </li> </ol> <p>An archetypical example for both these effects combined is 3d <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> with a compact, simple and simply connected <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> in the presence of <a class="existingWikiWord" href="/nlab/show/Wilson+lines">Wilson lines</a>. This is a <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a> on 3-dimensional <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> whose fields are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/gauge+fields">gauge fields</a> as for <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> above, hence given by maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><mo lspace="verythinmathspace">:</mo><mi>Σ</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\Phi \colon \Sigma \to \mathbf{B}G_{conn}</annotation></semantics></math>. At the same time, this theory has a “coupling” to a 1-dimensional <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a> which describes <a class="existingWikiWord" href="/nlab/show/particles">particles</a> propagating around <a class="existingWikiWord" href="/nlab/show/knots">knots</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>:</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">C : S^1 \to \Sigma</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> for which the restriction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><msub><mo stretchy="false">|</mo> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\Phi|_C</annotation></semantics></math> serves as the <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a>. Specifically, a field configuration of this 1-dimensional theory is equivalently a map in the slice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}G_{conn}}</annotation></semantics></math> which in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is given by a diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mn>1</mn></msup></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>T</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>C</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>OrbitStruct</mi></mstyle></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^1 &&\to&& \Omega^1(-,\mathfrak{g}//T) \\ & {}_{\mathllap{C}}\searrow &\swArrow& \swarrow_{\mathrlap{\mathbf{OrbitStruct}}} \\ && \mathbf{B}G_{conn} } </annotation></semantics></math></div> <p>for some map on the right which we discuss in detail below in <em><a href="#ChernSimonsWithWilsonLines">Chern-Simons fields with Wilson line fields</a></em>.</p> <p>Here considering just these fields in the background of a fixed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><msub><mo stretchy="false">|</mo> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\Phi|_C</annotation></semantics></math> produced a 1-dimensional <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> whose <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> is that “<a class="existingWikiWord" href="/nlab/show/Wilson+loop">Wilson loop</a>” observable of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi><msub><mo stretchy="false">|</mo> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\Phi|_C</annotation></semantics></math>. But this is not considered in isolation. The whole point of the relation of <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> to the <a class="existingWikiWord" href="/nlab/show/Jones+polynomial">Jones polynomial</a> <a class="existingWikiWord" href="/nlab/show/knot+invariant">knot invariant</a> of the <a class="existingWikiWord" href="/nlab/show/knot">knot</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is that one consider also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Φ</mi></mrow><annotation encoding="application/x-tex">\Phi</annotation></semantics></math> as a dynamical field, not as a fixed background. Indeed, in the full theory of Chern-Simons with Wilson loops that includes both the fields on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> as well as those on the knot, a field configuration is the diagram as above but regarded as the square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mn>1</mn></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>T</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>C</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>OrbitStruc</mi></mstyle></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Σ</mi></mtd> <mtd><mover><mo>→</mo><mi>Φ</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ S^1 &\to& \Omega^1(-,\mathfrak{g})//T \\ {}^{\mathllap{C}}\downarrow &\swArrow& \downarrow^{\mathrlap{\mathbf{OrbitStruc}}} \\ \Sigma &\stackrel{\Phi}{\to}& \mathbf{B}G_{conn} } \,, </annotation></semantics></math></div> <p>hence, again, a single map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mstyle mathvariant="bold"><mi>OrbiStruc</mi></mstyle></mrow><annotation encoding="application/x-tex"> C \to \mathbf{OrbiStruc} </annotation></semantics></math></div> <p>but now in the <a class="existingWikiWord" href="/nlab/show/arrow+%28%E2%88%9E%2C1%29-topos">arrow (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^{(\Delta^1)}</annotation></semantics></math>.</p> <p>This subtle interplay of “bulk fields” and “<a class="existingWikiWord" href="/nlab/show/QFT+with+defects">defect fields</a>” which is here captured most naturally in terms of <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> cannot really be expressed accurately just in terms of field bundles.</p> <h4 id="HigherGaugeFields">Higher gauge fields</h4> <p>Above we have seen the generalization of field bundles to <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> already for traditional notions such as Yang-Mills fields and Spin-structures. But many <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theories</a> considered in in theoretical physics have fields that are more “explicitly” entities in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>.</p> <p>For instance the higher analog of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> which is called the <em><a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></em> or <em><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></em> is a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">2-connection</a> on a <a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>. There is no way to faithfully encode this as a section of any ordinary <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a>. It follows that for instance also the <a href="magnetic+charge#MagneticChargeAnomaly">magnetic charge anomaly</a> (as discussed there) has no accurate description in terms of field bundles. Next the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> is a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">3-connection</a> on a <a class="existingWikiWord" href="/nlab/show/principal+3-bundle">principal 3-bundle</a>, and so on.</p> <p>There is a wide variety of <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theories">higher dimensional Chern-Simons theories</a> whose fields are such <a class="existingWikiWord" href="/nlab/show/higher+gauge+fields">higher gauge fields</a>. In some traditional literature one sees parts of this theory be discussed by standard <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> applied to <a class="existingWikiWord" href="/nlab/show/field+bundles">field bundles</a>, namely by ignoring the non-trivial <a class="existingWikiWord" href="/nlab/show/instanton+sectors">instanton sectors</a> and pretending that a field configuration for these <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">∞-connections</a> are given by globally dedined <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a>. In some special cases (for instance for <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a>/<a class="existingWikiWord" href="/nlab/show/worldvolumes">worldvolumes</a> of very special topology or low <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a>) this can be sufficient to capture everything, but in general (for instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a> and its <a class="existingWikiWord" href="/nlab/show/holographic+principle">holographically dual</a> <a class="existingWikiWord" href="/nlab/show/self-dual+higher+gauge+theory">self-dual higher gauge theory</a>) it is not.</p> <h3 id="the_solution_field_bundles_and_moduli_stacks_of_fields">The solution: Field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles and moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks of fields</h3> <p>By the <a href="#IdeaOfFieldBundlesAndItsProblems">above</a>, defining a physical field to be a section of some bundle goes in the right direction, but misses crucial aspects of physical fields. These problems are fixed by passing to <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>.</p> <p>Below in <em><a href="#Definition">Definition</a></em> we discuss a natural unified formulation of the notion of physical field in terms of <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> (the central definition being def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> ) and then we spell out many <a href="#Examples">Examples</a>.</p> <p>This definition turns out to be equivalent, at least under mild conditions, to a formulation where fields are sections of an <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>, hence a “field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundle”. This we discuss in <em><a href="#RelationOfFieldsToSections">Properties – Relation of fields to sections of ∞-bundles</a></em>. But this is just one of several equivalent perspectives on physical fields, and not always the most transparent one. In fact, sections of higher associated bundles are best known in the literature on <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> and indeed one equivalent characterization of fields is as <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> in <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> in the general sense of <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>. This we discuss below in <em><a href="#RelationToTwistedCohomology">Relation to twisted cohomology</a></em>.</p> <p>In summary we find and discuss that</p> <table><thead><tr><th>fields</th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> <a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative</a> <a class="existingWikiWord" href="/nlab/show/cohesion">cohesive</a> <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a></th></tr></thead><tbody></tbody></table> <h2 id="Definition">Definition</h2> <p>We give a general abstract definition of physical fields in</p> <ul> <li><em><a href="#DefinitionPhysicalField">Physical fields</a></em></li> </ul> <p>Then we consider some general abstract operations on fields in</p> <ul> <li><em><a href="#RestrictionAndPullback">Restriction and pullback of fields</a></em></li> </ul> <h3 id="DefinitionPhysicalField">Physical fields</h3> <p>A notion of <em>field</em> in physics is part of a specification of <em><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">physical theory</a></em> or <em><a class="existingWikiWord" href="/nlab/show/model+%28physics%29">physical model</a></em>. We consider specifically the framework of <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a>. Here a <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a>/<a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model</a> is specified by (or at least comes with) an <em><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a></em>. The <em>field</em> content of the theory is part of the specification of the <a class="existingWikiWord" href="/nlab/show/domain">domain</a> of the action functional. Therefore in def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> below we define <em>action functionals</em> and the fields relative to this notion.</p> <p>We work in the following context.</p> <div class="num_defn"> <h6 id="context_2">Context</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a>. For many of the examples below it is furthermore assumed that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is equipped with <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a>. This implies in particular that there is a notion of <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <p>Fix <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} \in Grp(\mathbf{H})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, hence a cohesive <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>.</p> </div> <p>For the main definition below we need the following basic notation.</p> <div class="num_defn"> <h6 id="definition_7">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">B \in \mathbf{H}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/object">object</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>A</mi><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X,A \in \mathbf{H}_{/B}</annotation></semantics></math> two objects in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>A</mi><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mi>B</mi></munder><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> [X,A]_{\mathbf{H}} \coloneqq \underset{B}{\prod} [X,A] \in \mathbf{H} </annotation></semantics></math></div> <p>for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>-valued <a class="existingWikiWord" href="/nlab/show/hom+object">hom object</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>: the <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">B \to *</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">[X, A] \in \mathbf{H}_{/B}</annotation></semantics></math>.</p> </div> <p>The following defines the notion of <em><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a></em> and as part of the data it defines the notion of <em>physical field</em>.</p> <div class="num_defn" id="FieldsInAnActionFunctional"> <h6 id="definition_8">Definition</h6> <p>Given an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{BgFields} \in \mathbf{H}</annotation></semantics></math> and given two objects, to be denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>B</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\Phi_X, \mathbf{Fields} \in \mathbf{H}_{/B}</annotation></semantics></math>, in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{BgFields}</annotation></semantics></math>, then an <strong><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a></strong> in (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>,</mo><mi>𝔾</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}, \mathbb{G})</annotation></semantics></math> “on fields on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>” is a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>→</mo><mi>𝔾</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S \;\colon\; [\Phi_X, \mathbf{Fields}]_{\mathbf{H}} \to \mathbb{G} \,. </annotation></semantics></math></div> <p>In this context we say that</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mstyle mathvariant="bold"><mi>BgField</mi></mstyle></munder><msub><mi>Φ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">X \coloneqq \underset{\mathbf{BgField}}{\sum} \Phi_X</annotation></semantics></math> is the <strong><a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a></strong> or <strong><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></strong>;</p> </li> <li> <p>the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></mrow><annotation encoding="application/x-tex">\Phi_X \;\colon\; X \to \mathbf{BgFields}</annotation></semantics></math> is the <strong><a class="existingWikiWord" href="/nlab/show/background+field">background field</a></strong>;</p> </li> <li> <p>the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Fields}</annotation></semantics></math> is the <strong><a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of fields</strong>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/elements">elements</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub></mrow><annotation encoding="application/x-tex">[\Phi_X,\mathbf{Fields}]_{\mathbf{H}}</annotation></semantics></math>, hence (see prop. <a class="maruku-ref" href="#GlobalPointsOfModuliOfFields"></a> below) the morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Φ</mi> <mi>X</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; \Phi_X \to \mathbf{Fields} </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{BgFields}}</annotation></semantics></math>, hence the <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>ϕ</mi></mover></mtd> <mtd></mtd> <mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></munder><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>Φ</mi> <mi>X</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &&\stackrel{\phi}{\to}&& \underset{\mathbf{BgFields}}{\sum} \mathbf{Fields} \\ & {}_{\mathllap{\Phi_X}}\searrow &\swArrow& \swarrow_{\mathrlap{\mathbf{Fields}}} \\ && \mathbf{BgFields} } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, are the <strong>fields</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>a <strong><a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a></strong> is a <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub></mrow><annotation encoding="application/x-tex">[\Phi_X, \mathbf{Fields}]_{\mathbf{H}}</annotation></semantics></math>, hence a</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>⇒</mo><mi>ϕ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex"> \phi \Rightarrow \phi' </annotation></semantics></math></div></li> <li> <p>a <strong><a class="existingWikiWord" href="/nlab/show/higher+gauge+transformation">higher gauge transformation</a></strong> is a <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub></mrow><annotation encoding="application/x-tex">[\Phi_X, \mathbf{Fields}]_{\mathbf{H}}</annotation></semantics></math>.</p> </li> </ul> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Definition <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> provides a unified perspective on fields from several perspectives.</p> <p>On the one hand, it almost explicitly says that in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> all fields are “<a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> fields” (see below at <em><a href="#SclarModuliFields">Examples – Scalar and Sigma-model fields</a></em>): if we regard the <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Fields}</annotation></semantics></math> as the <em><a class="existingWikiWord" href="/nlab/show/target+space">target space</a></em> then fields are simply maps from their domain (when regarded as <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <em>and</em> <a class="existingWikiWord" href="/nlab/show/background+field">background field</a>) to this target space.</p> <p>On the other hand, we see below in <em><a href="#RelationOfFieldsToSections">Relation to sections of ∞-bundles</a></em> that from another perspective def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> says that all fields are <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of an <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a> to an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundle modulated by the background fields. This means that in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> all fields are “<a class="existingWikiWord" href="/nlab/show/matter">matter</a> fields” (see below at <em><a href="#MatterFields">Matter fields</a></em>) <a class="existingWikiWord" href="/nlab/show/charge+%28physics%29">charged</a> under the <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a>.</p> <p>Finally, we see below in <em><a href="#RelationToTwistedCohomology">Relation to twisted cohomology</a></em> that from yet another perspective def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> says that fields are equivalently <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> in general <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>. This perspective is traditionally known for certain examples (see <em><a href="#ChanPatonGaugeFields">Examples – Chan-Paton gauge fields</a></em> below), but we see below that it is useful in its full generality. For instance the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> is in a precise sense a 0-cocycle with coefficients in the <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)/O(n)</annotation></semantics></math> that is twisted by the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> of spacetime (which exhibits the background gauge structure for <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>: the <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a> of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>). An inkling of this perspective is certainly visible in the traditional literature, notably in the generalization to <a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a> and <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>, and here we see how this is a precise mechanism on the same conceptual footing with the twisted K-theory seen over D-branes.</p> </div> <h3 id="RestrictionAndPullback">Restriction and pullback of physical fields</h3> <p>It is familiar from basic examples that not every type of physical field on a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> can be pulled back (in the sense of pullback of functions) along any <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \;\colon\;Y \to X</annotation></semantics></math>. For instance the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, a <a class="existingWikiWord" href="/nlab/show/vielbein+field">vielbein field</a> or <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+metric">pseudo</a>-<a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a>, discussed below in <em><a href="#OrdinaryGravity">Ordinary graviry</a></em> may be pulled back only along <a class="existingWikiWord" href="/nlab/show/local+diffeomorphisms">local diffeomorphisms</a>. More generally, one needs other <a class="existingWikiWord" href="/nlab/show/properties">properties</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> to pull back a given field and in fact in general one needs <a class="existingWikiWord" href="/nlab/show/extra+structure">extra structure</a>.</p> <p>In view of def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> above this is immediate: by that definition a field on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> in general does not just depend on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, but in fact also on the background field structure denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_X</annotation></semantics></math>. Accordingly, it can be pulled back only along maps that also carry this background field structure along.</p> <div class="num_remark" id="PullbackAlongGeneralizedLocalDiffeomorphisms"> <h6 id="remark_2">Remark</h6> <p>Since by def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> a physical field is a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\phi \colon \Phi_X \to \mathbf{Fields}</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{BgFields}}</annotation></semantics></math>, it may be “pulled back” along maps of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \colon Y \to X</annotation></semantics></math> when these are extended to maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>Y</mi></msub><mo>→</mo><msub><mi>Φ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_Y \to \Phi_X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{BgFields}}</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/background+fields">background fields</a>, hence to <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Y</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>Φ</mi> <mi>Y</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>κ</mi></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>Phi</mi> <mi>X</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ Y &&\stackrel{f}{\to}&& X \\ & {}_{\mathllap{\Phi_Y}}\searrow &\swArrow_{\kappa}& \swarrow_{\mathrlap{Phi_X}} \\ && \mathbf{BgFields} } \,, </annotation></semantics></math></div> <p>hence maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> equipped with a choice of <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>κ</mi><mo>;</mo><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>Φ</mi> <mi>X</mi></msub><mo>≃</mo><msub><mi>Φ</mi> <mi>Y</mi></msub></mrow><annotation encoding="application/x-tex"> \kappa ;\colon\; f^*\Phi_X \simeq \Phi_Y </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/background+fields">background fields</a>.</p> </div> <p>In standard <em><a href="#Examples">Examples</a></em> discussed below we see that this is a familar fact. For instance applied to the field of gravity (see <em><a href="#OrdinaryGravity">Gravity</a></em> below) it says that the gravitational field can be pulled back precisely along <a class="existingWikiWord" href="/nlab/show/local+diffeomorphisms">local diffeomorphisms</a>, or that spin structures on oriented manifolds (see <em><a href="#SpinStructures">Spin structures</a></em> below) can be pulled back along orientation-preserving maps. Or : for <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+fields">Chan-Paton gauge fields</a> on <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> (see <em><a href="#ChanPatonGaugeFields">Chan-Paton gauge fields</a></em>) it reproduces the familiar gauge relation for the <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> on D-branes known in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>, which is already less trivial. But the statement applies in full generality.</p> <h3 id="BoundaryAndDefectFields">Boundary and defect fields</h3> <p>In def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> the <a class="existingWikiWord" href="/nlab/show/background+field">background field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_X</annotation></semantics></math> is a fixed datum of the domain (<a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>) on which the physical fields are defined. In some <a class="existingWikiWord" href="/nlab/show/model+%28in+theoretical+physics%29">models</a> and for some of the fields this is precisely what one needs, but in other models one may need to be able to also regard the background fields as dynamical fields and to be able to switch between these perspectives, for instance to pass to a setup where what used to be a configuration of some field is now taken to be a fixed background field for the remaining fields. We now discuss how this more general setup is naturally formulated as a generalization of def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo lspace="verythinmathspace">:</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></munder><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>→</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Fields} \colon \underset{\mathbf{BgFields}}{\sum} \mathbf{Fields} \to \mathbf{BgFields}</annotation></semantics></math> a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, we may consider this also as an object in the <a class="existingWikiWord" href="/nlab/show/arrow+%28%E2%88%9E%2C1%29-topos">arrow (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^{(\Delta^1)}</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor+%28%E2%88%9E%2C1%29-category">(∞,1)-functor (∞,1)-category</a> from the <a class="existingWikiWord" href="/nlab/show/interval+category">interval category</a>/1-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>).</p> <p>A generic object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^{(\Delta^1)}</annotation></semantics></math> here is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\iota_X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. When we think of this as a domain on which to define fields we will write this</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>X</mi> <mi>def</mi></msub><mo>→</mo><msub><mi>X</mi> <mi>bulk</mi></msub></mrow><annotation encoding="application/x-tex"> \iota_X \;\colon\; X_{def} \to X_{bulk} </annotation></semantics></math></div> <p>where the subscripts are for “bulk” and for “defect” (as in <em><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></em>). A field in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^{(\Delta^1)}</annotation></semantics></math> which is given by a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>ι</mi> <mi>X</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; \iota_X \to \mathbf{Fields} </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^{(\Delta^1)}</annotation></semantics></math> is equivalently a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>X</mi> <mi>def</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>def</mi></msub></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>def</mi></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mi>X</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mi>bulk</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>bulk</mi></msub></mrow></mover></mtd> <mtd><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>bulk</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X_{def} &\stackrel{\phi_{def}}{\to}& \mathbf{Fields}_{def} \\ {}^{\mathllap{\iota_X}}\downarrow &\swArrow& \downarrow^{\mathrlap{\mathbf{Fields}}} \\ X_{bulk} &\stackrel{\phi_{bulk}}{\to}& \mathbf{Fields}_{bulk} } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <p>This we interpret as a configuration consisting of</p> <ol> <li> <p>a bulk field configuration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>bulk</mi></msub></mrow><annotation encoding="application/x-tex">\phi_{bulk}</annotation></semantics></math></p> </li> <li> <p>a defect field configuration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>def</mi></msub></mrow><annotation encoding="application/x-tex">\phi_{def}</annotation></semantics></math>,</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> that relates the restriction (or more generally: pullback to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>def</mi></msub></mrow><annotation encoding="application/x-tex">X_{def}</annotation></semantics></math>) of the bulk field to the embedding (or more generally: push-forward) of the defect field into the bulk field configuration on the defect.</p> </li> </ol> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>bulk</mi></msub></mrow><annotation encoding="application/x-tex">\phi_{bulk}</annotation></semantics></math> is regarded as fixed, then this is equivalently a field configuration as in def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> defined on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>def</mi></msub></mrow><annotation encoding="application/x-tex">X_{def}</annotation></semantics></math> and for background field the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mrow><msub><mi>X</mi> <mi>def</mi></msub></mrow></msub><mo>≔</mo><msubsup><mi>ι</mi> <mi>X</mi> <mo>*</mo></msubsup><msub><mi>ϕ</mi> <mi>bulk</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>X</mi> <mi>def</mi></msub><mover><mo>→</mo><mrow><msub><mi>ι</mi> <mi>X</mi></msub></mrow></mover><mi>X</mi><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>bulk</mi></msub></mrow></mover><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>bulk</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Phi_{X_{def}} \coloneqq \iota_X^* \phi_{bulk} \;\colon\; X_{def} \stackrel{\iota_X}{\to} X \stackrel{\phi_{bulk}}{\to} \mathbf{Fields}_{bulk} \,. </annotation></semantics></math></div></div> <p>This “fixing of bulk fields to background fields for defect fields” we discuss in more detail below in <em><a href="#ModuliStacksOfFields">Properties – Moduli stacks of fields</a></em>.</p> <p>We formalize the moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stack of all bulk and boundary fields as follows</p> <div class="num_defn"> <h6 id="definition_9">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow></msup><mover><munder><mo>→</mo><mrow><msub><mi>Γ</mi> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub></mrow></munder><mover><mo>←</mo><mrow><msub><mi>Disc</mi> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub></mrow></mover></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> \mathbf{H}^{(\Delta^1)} \stackrel{\overset{Disc_{\mathbf{H}}}{\leftarrow}}{\underset{\Gamma_{\mathbf{H}}}{\to}} \mathbf{H} </annotation></semantics></math></div> <p>for the canonical <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a>.</p> </div> <div class="num_defn" id="ModuliOfBulkAndBoundaryFields"> <h6 id="definition_10">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>X</mi></msub><mo>;</mo><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>X</mi> <mi>def</mi></msub><mo>→</mo><msub><mi>X</mi> <mi>bulk</mi></msub></mrow><annotation encoding="application/x-tex">\iota_X ;\colon\; X_{def} \to X_{bulk}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>def</mi></msub><mo>→</mo><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>bulk</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{Fields} \;\colon\; \mathbf{Fields}_{def} \to \mathbf{Fields}_{bulk}</annotation></semantics></math> morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, we say that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>ι</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>≔</mo><msub><mi>Γ</mi> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">[</mo><msub><mi>ι</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> [\iota_X, \mathbf{Fields}]_{\mathbf{H}} \coloneqq \Gamma_{\mathbf{H}}[\iota_X, \mathbf{Fields}] \;\; \in \mathbf{H} </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of bulk and boundary fields on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\iota_X</annotation></semantics></math>.</p> </div> <p>Several examples of this are discussed below.</p> <h2 id="FieldsProperties">Properties</h2> <p>The definition of fields in def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> is in fact the central part of a general theory of <em><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></em> and <em><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a></em> in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>/<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a> and various insights into <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a> follow by making this perspective explicit. This is what we do here</p> <ol> <li> <p><a href="#ModuliStacksOfFields">Moduli ∞-stacks of fields</a></p> </li> <li> <p><a href="#RelationOfFieldsToSections">Relation of fields to sections of ∞-bundles</a></p> </li> <li> <p><a href="#RelationToTwistedCohomology">Relation of fields to twisted cohomology</a></p> </li> <li> <p><a href="#RelationToRelativeCohomology">Relation of fields to relative cohomology</a></p> </li> </ol> <p>The central results that underlie these identifications are in (<a href="#NSS">NSS</a>), also <a href="#dcct">dcct, section 3.6.10, 3.6.11, 3.6.12, 3.6.15</a>.</p> <h3 id="ModuliStacksOfFields">Moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks of fields</h3> <p>The object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">[\Phi_X, \mathbf{Fields}]_{\mathbf{H}} \in \mathbf{H}</annotation></semantics></math> of def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> we may call the <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of fields. Here we discuss various properties of this object.</p> <div class="num_prop" id="GlobalPointsOfModuliOfFields"> <h6 id="proposition_3">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> of global <a class="existingWikiWord" href="/nlab/show/elements">elements</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub></mrow><annotation encoding="application/x-tex">[X, \mathbf{Fields}]_{\mathbf{H}}</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mo stretchy="false">[</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Γ</mi><mo stretchy="false">[</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></mrow></msub><mo stretchy="false">(</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(*,[\Phi_X, \mathbf{Fields}_{\mathbf{H}}]) \simeq \Gamma [\Phi_X, \mathbf{Fields}]_{\mathbf{H}} \simeq \mathbf{H}_{/\mathbf{BgFields}}(\Phi_X, \mathbf{Fields}) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Using the <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> <a class="existingWikiWord" href="/nlab/show/equivalences">equivalences</a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mo stretchy="false">[</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></munder><mo stretchy="false">[</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></mrow></msub><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle> <mo>*</mo></msup><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">[</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></mrow></msub><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mo stretchy="false">[</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></mrow></msub><mo stretchy="false">(</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbf{H}(*, [\Phi_X, \mathbf{Fields}]_{\mathbf{H}}) & \simeq \mathbf{H}(*, \underset{\mathbf{BgFields}}{\prod} [\Phi_X, \mathbf{Fields}]) \\ & \simeq \mathbf{H}_{/\mathbf{BgFields}}(\mathbf{BgFields}^*(*), [\Phi_X, \mathbf{Fields}]) \\ & \simeq \mathbf{H}_{/\mathbf{BgFields}}(*, [\Phi_X, \mathbf{Fields}]) \\ & \simeq \mathbf{H}_{/\mathbf{BgFields}}(\Phi_X, \mathbf{Fields}) \end{aligned} </annotation></semantics></math></div> <p>where the first line is the definition, the second is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle> <mo>*</mo></msup><mo>⊣</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></munder><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbf{BgFields}^* \dashv \underset{\mathbf{BgFields}}{\prod})</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a>-<a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a>, the third is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></munder><mo>⊣</mo><msup><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\underset{\mathbf{BgFields}}{\sum} \dashv \mathbf{BgFields}^*)</annotation></semantics></math>-adjunction implying that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathbf{BgFields}^*</annotation></semantics></math> preserves the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>, and finally the last line is the defining <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a>-<a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a>.</p> </div> <div class="num_prop" id="ModuliStackOfFieldsAsHomotopyFiber"> <h6 id="proposition_4">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub></mrow><annotation encoding="application/x-tex">[\Phi_X, \mathbf{BgFields}]_{\mathbf{H}}</annotation></semantics></math> sits in a <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">[</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></munder><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>κ</mi></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo>⊢</mo><msub><mi>Φ</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ [\Phi_X, \mathbf{Fields}]_{\mathbf{H}} &\stackrel{}{\to}& [X, \underset{\mathbf{BgFields}}{\sum}\mathbf{Fields}] \\ \downarrow &\swArrow_{\kappa}& \downarrow \\ {*} &\stackrel{\vdash \Phi_X}{\to}& [X, \mathbf{BgFields}] } \,. </annotation></semantics></math></div></div> <p>These relations are discussed at <em><a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a></em>, and (<a href="#dcct">dcct, section 3.6.1</a>).</p> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>Proposition <a class="maruku-ref" href="#ModuliStackOfFieldsAsHomotopyFiber"></a> makes precise the heuristic idea that a field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>∈</mo><mo stretchy="false">[</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub></mrow><annotation encoding="application/x-tex">\phi \in [\Phi_X, \mathbf{Fields}]_{\mathbf{H}}</annotation></semantics></math> is</p> <ol> <li> <p>a configuration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></munder><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\phi_X \;\colon\; X \to \underset{\mathbf{BgFields}}{\sum}\mathbf{Fields}</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>together with a <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>κ</mi> <mi>ϕ</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Φ</mi> <mi>X</mi></msub><mover><mo>→</mo><mo>≃</mo></mover><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>∘</mo><msub><mi>ϕ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\kappa_{\phi} \;\colon\; \Phi_X \stackrel{\simeq}{\to} \mathbf{Fields}\circ \phi_X</annotation></semantics></math> between the fixed <a class="existingWikiWord" href="/nlab/show/background+field">background field</a> and the background field induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\phi_X</annotation></semantics></math>.</p> </li> </ol> </div> <p>More generally, the moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks of combined bulk/boundary-defect fields as in def. <a class="maruku-ref" href="#ModuliOfBulkAndBoundaryFields"></a> is characterized as follows.</p> <div class="num_prop" id="ModuliOfBulkAndDefectFieldsAsPullback"> <h6 id="proposition_5">Proposition</h6> <p>The moduli stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>ι</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub></mrow><annotation encoding="application/x-tex">[\iota_X, \mathbf{Fields}]_{\mathbf{H}}</annotation></semantics></math> of bulk and defect fields in def. <a class="maruku-ref" href="#ModuliOfBulkAndBoundaryFields"></a> sits in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">[</mo><msub><mi>ι</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">[</mo><msub><mi>X</mi> <mi>def</mi></msub><mo>,</mo><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>def</mi></msub><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msub><mi>X</mi> <mi>bulk</mi></msub><mo>,</mo><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>bulk</mi></msub><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mo stretchy="false">[</mo><msub><mi>X</mi> <mi>def</mi></msub><mo>,</mo><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>bulk</mi></msub><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ [\iota_X, \mathbf{Fields}]_{\mathbf{H}} &\to& [X_{def}, \mathbf{Fields}_{def}] \\ \downarrow && \downarrow^{\mathrlap{}} \\ [X_{bulk}, \mathbf{Fields}_{bulk}] &\stackrel{}{\to}& [X_{def}, \mathbf{Fields}_{bulk}] } \,. </annotation></semantics></math></div></div> <p>The following proposition expresses that fixing a bulk field gives rise to a background field for the remaining defect fields</p> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>bulk</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>X</mi> <mi>bulk</mi></msub><mo>→</mo><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>bulk</mi></msub></mrow><annotation encoding="application/x-tex">\phi_{bulk} \;\colon\; X_{bulk} \to \mathbf{Fields}_{bulk}</annotation></semantics></math> a given bulk field, there is a <a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural</a> <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msubsup><mi>ι</mi> <mi>X</mi> <mo>*</mo></msubsup><msub><mi>ϕ</mi> <mi>bulk</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>≃</mo><mo stretchy="false">[</mo><msub><mi>ι</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><munder><mo>×</mo><mrow><mo stretchy="false">[</mo><msub><mi>X</mi> <mi>bulk</mi></msub><mo>,</mo><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>bulk</mi></msub><mo stretchy="false">]</mo></mrow></munder><mo stretchy="false">{</mo><msub><mi>ϕ</mi> <mi>bulk</mi></msub><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\iota_X^* \phi_{bulk}, \mathbf{Fields}]_{\mathbf{H}} \simeq [\iota_X, \mathbf{Fields}]_{\mathbf{H}} \underset{[X_{bulk}, \mathbf{Fields}_{bulk}]}{\times} \{\phi_{bulk}\} \,. </annotation></semantics></math></div></div> <h3 id="RelationOfFieldsToSections">Relation to sections of fiber <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles</h3> <p>We discuss here how def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> equivalently says that fields are <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of a <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a> which is which is <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated</a> to a (<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundle modulated by the <a class="existingWikiWord" href="/nlab/show/background+field">background field</a>.</p> <p>For simplicity of the discussion first consider the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{BgFields} \simeq \mathbf{B}G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in Grp(\mathbf{H})</annotation></semantics></math>. (In typical applications in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> we instead have the differential refinement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}G_{conn}</annotation></semantics></math> (see below at <em><a href="#GaugeFields">Examples – Gauge fields</a></em>) and the following discussion is directly generalized to that case. )</p> <ol> <li> <p><a href="#BriefReviewOfPrincipalInfinityBundlesAsBackgroundFields">Background fields as principal ∞-bundles</a></p> </li> <li> <p><a href="#FieldsAsSectionsOfAssociatedBundles">Fields as sections of associated fiber ∞-bundles</a></p> </li> </ol> <h4 id="BriefReviewOfPrincipalInfinityBundlesAsBackgroundFields">Background fields as principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles</h4> <p>We briefly recall the central aspects of <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a> in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/object">object</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mo lspace="0em" rspace="thinmathspace">X</mo></mrow><annotation encoding="application/x-tex">x \colon * \to \X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/global+element">global element</a>, the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> (<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a>) of the point along itse is the based <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Ω</mi> <mi>x</mi></msub><mi>X</mi><mo>≔</mo><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><munder><mo>×</mo><mi>X</mi></munder><mo stretchy="false">{</mo><mi>x</mi><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega_x X \coloneqq \{x\} \underset{X}{\times} \{x\} \,. </annotation></semantics></math></div> <p>Via composition of loops, this canonically has the structure of an <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a> object such that the <a class="existingWikiWord" href="/nlab/show/truncated+object+in+an+%28%E2%88%9E%2C1%29-category">0-truncation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mn>0</mn></msub><msub><mi>Ω</mi> <mi>x</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">\tau_0 \Omega_x X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a> which is a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a>. Such <em>groupal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-algebra objects</em> are <em><a class="existingWikiWord" href="/nlab/show/group+objects+in+an+%28%E2%88%9E%2C1%29-category">group objects</a></em> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a>.</p> <p>In fact every <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> is of this form. Moreover, restricted to <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> and <a class="existingWikiWord" href="/nlab/show/connected+object+in+an+%28%E2%88%9E%2C1%29-topos">connected objects</a>, the <a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi></mrow><annotation encoding="application/x-tex">\Omega</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mover><munderover><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>≃</mo></munderover><mover><mo>←</mo><mi>Ω</mi></mover></mover><msubsup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo>≥</mo><mn>1</mn></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Grp(\mathbf{H}) \stackrel{\overset{\Omega}{\leftarrow}}{\underoverset{\mathbf{B}}{\simeq}{\to}} \mathbf{H}^{*/}_{\geq 1} \,. </annotation></semantics></math></div> <p>Its inverse <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{B}</annotation></semantics></math> we call the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> operation. Notice that this means that we can conveniently discuss all aspects of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a> without ever explicitly considering <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a> structure by instead working with the pointed connected objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>.</p> <p>Notably one finds that a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is equivalently the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">g \;\colon\; X \to \mathbf{B}G</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>fib</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>≃</mo></mtd> <mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd> <mtd><mo>≃</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ fib(g) \simeq & P &\to& * & \simeq \mathbf{E}G \\ & \downarrow && \downarrow \\ & X &\stackrel{g}{\to}& \mathbf{B}G } \,. </annotation></semantics></math></div> <p>This construction yields an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%E2%88%9E-groupoids">equivalence of ∞-groupoids</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><munderover><mo>→</mo><mrow></mrow><mo>≃</mo></munderover><mover><mo>←</mo><mi>fib</mi></mover></mover><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> G Bund(X) \stackrel{\overset{fib}{\leftarrow}}{\underoverset{}{\simeq}{\to}} \mathbf{H}(X, \mathbf{B}G) </annotation></semantics></math></div> <p>between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a> with <a class="existingWikiWord" href="/nlab/show/gauge+transformations">gauge transformations</a> between them and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> with <a class="existingWikiWord" href="/nlab/show/coboundaries">coboundaries</a>. In particular this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a> are classified by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mo>∼</mo></mrow></msub><mo>≃</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G Bund(X)_{/\sim} \simeq \pi_0 G Bund(X) \stackrel{\simeq}{\to} \pi_0 \mathbf{H}(X, \mathbf{B}G) \simeq H^1(X,G) \,. </annotation></semantics></math></div> <p>Hence if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{BgFields} \simeq \mathbf{B}G</annotation></semantics></math> then a <a class="existingWikiWord" href="/nlab/show/background+field">background field</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is equivalently a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <h4 id="FieldsAsSectionsOfAssociatedBundles">Fields as sections of associated fiber <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles</h4> <div class="num_prop" id="EquivalenceOfActionCategoryWithSlice"> <h6 id="proposition_7">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in Grp(\mathbf{H})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> that sends an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">V \in \mathbf{H}</annotation></semantics></math> to the corresponding <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>G</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>G</mi><mi>Act</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\mathbf{E}G)\times_G(-) \;\colon\; G Act(\mathbf{H}) \stackrel{\simeq}{\to} \mathbf{H}_{/\mathbf{B}G} \,. </annotation></semantics></math></div> <p>Moreover there is a <a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural</a> <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi><mo>≃</mo><mo>*</mo><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi><mo>≃</mo><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> (\mathbf{E}G)\times_G V \simeq {*} \times_G V \simeq V//G </annotation></semantics></math></div> <p>with the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> (this is the general abstract and version and geometric refinement of the traditional <a class="existingWikiWord" href="/nlab/show/Borel+construction">Borel construction</a>).</p> </div> <div class="num_defn"> <h6 id="definition_11">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> of an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in Grp(\mathbf{H})</annotation></semantics></math> on some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">V \in \mathbf{H}</annotation></semantics></math> we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mover><mo>→</mo><mover><mi>ρ</mi><mo>¯</mo></mover></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> V//G \stackrel{\overline{\rho}}{\to} \mathbf{B}G </annotation></semantics></math></div> <p>for the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> under the equivalence of prop. <a class="maruku-ref" href="#EquivalenceOfActionCategoryWithSlice"></a>.</p> </div> <div class="num_prop"> <h6 id="proposition_8">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ρ</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{\rho}</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, hence we have a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>V</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mi>ρ</mi><mo>¯</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ V &\to& V//G \\ && \downarrow^{\mathrlap{\overline{\rho}}} \\ && \mathbf{B}G } </annotation></semantics></math></div> <p>identifying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ρ</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{\rho}</annotation></semantics></math> with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>. Moreover, this is the <em><a class="existingWikiWord" href="/nlab/show/universal+associated+%E2%88%9E-bundle">universal rho-associated ∞-bundle</a></em>: for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> with modulating map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">g_X \;\colon\; X \to \mathbf{B}G</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural</a> <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalence in an (∞,1)-category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi><mo>≃</mo><msubsup><mi>g</mi> <mi>X</mi> <mo>*</mo></msubsup><mo stretchy="false">(</mo><mover><mi>ρ</mi><mo>¯</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> P \times_G V \simeq g_X^*(\overline{\rho}) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/X}</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>In summary this means that an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> of an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on any object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">V \in \mathbf{H}</annotation></semantics></math> is equivalently a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>V</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mi>ρ</mi><mo>¯</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ V &\to& V//G \\ && \downarrow^{\mathrlap{\overline{\rho}}} \\ && \mathbf{B}G } \,. </annotation></semantics></math></div> <p>Moreover, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> modulated by a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>;</mo><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">g ;\colon\; X \to \mathbf{B}G</annotation></semantics></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, there is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mi>ρ</mi><mo>¯</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P \times_G V &\to& V//G \\ \downarrow && \downarrow^{\mathrlap{\overline{\rho}}} \\ X &\stackrel{g}{\to}& \mathbf{B}G } \,. </annotation></semantics></math></div></div> <p>Due to the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> this has the following consequence which, basic as it is, is fundamental for the interpretation of fields in def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a>.</p> <div class="num_prop"> <h6 id="proposition_9">Proposition</h6> <p>There is a canonical <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalence in an (∞,1)-category</a> between the <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>X</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">P \times_X G</annotation></semantics></math> and maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>X</mi></msub><mo>→</mo><mover><mi>ρ</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">g_X \to \overline{\rho}</annotation></semantics></math> in the slice, hence fields in the sense of def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> with background field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">g_X</annotation></semantics></math> and moduli stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ρ</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{\rho}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Γ</mi></mstyle> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><msub><mi>g</mi> <mi>X</mi></msub><mo>,</mo><mover><mi>ρ</mi><mo>¯</mo></mover><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\Gamma}_X(P \times_G V) \simeq [g_X, \overline{\rho}]_{\mathbf{H}} \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>This equivalence is constituted simply by forming the <a class="existingWikiWord" href="/nlab/show/pasting+diagram">pasting diagram</a> of the <a class="existingWikiWord" href="/nlab/show/section">section</a> with the above <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-pullback</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>σ</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mi>ρ</mi><mo>¯</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>id</mi></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>σ</mi></mover></mtd> <mtd></mtd> <mtd><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>g</mi></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mover><mi>ρ</mi><mo>¯</mo></mover></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && P \times_G V &\to& V//G \\ &{}^{\mathllap{\sigma}}\nearrow& \downarrow && \downarrow^{\mathrlap{\overline{\rho}}} \\ X &\stackrel{id}{\to}& X &\stackrel{g}{\to}& \mathbf{B}G } \;\;\;\;\;\; \simeq \;\;\;\;\;\; \array{ X &&\stackrel{\sigma}{\to}&& V//G \\ & {}_{\mathllap{g}}\searrow && \swarrow_{\overline{\rho}} \\ && \mathbf{B}G } \,. </annotation></semantics></math></div></div> <p>The following further central property leads in the <a href="#RelationToTwistedCohomology">following section</a> to an equivalent re-formulation of this in <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a>.</p> <div class="num_prop"> <h6 id="proposition_10">Proposition</h6> <p>Every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is <em>locally trivizable</em> in that there exists a <a class="existingWikiWord" href="/nlab/show/1-epimorphism">1-epimorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p \;\colon\; U \to X</annotation></semantics></math> and an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>U</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>P</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>U</mi></mtd> <mtd><mover><mo>→</mo><mi>p</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ U \times G &\to& P \\ \downarrow && \downarrow \\ U &\stackrel{p}{\to}& X } </annotation></semantics></math></div> <p>or equivalently a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>U</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ U &\to& * \\ \downarrow^{\mathrlap{p}} &\swArrow_{\simeq}& \downarrow \\ X & \stackrel{g}{\to} & \mathbf{B}G } \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>Together the above facts imply that for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>σ</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mi>ρ</mi><mo>¯</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>id</mi></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && P \times_G V &\to& V//G \\ & {}^{\mathllap{\sigma}}\nearrow & \downarrow && \downarrow^{\mathrlap{\overline{\rho}}} \\ X &\stackrel{id}{\to}& X &\stackrel{g}{\to}& \mathbf{B}G } </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/section">section</a> of the <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>, the restriction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub></mrow><annotation encoding="application/x-tex">\sigma|_U</annotation></semantics></math> of the section along the trivializing cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p \;\colon\; U \to X</annotation></semantics></math> factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>→</mo><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">V \to V//G</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>V</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>σ</mi><msub><mo stretchy="false">|</mo> <mi>U</mi></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mi>ρ</mi><mo>¯</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd><mi>U</mi></mtd> <mtd><mover><mo>→</mo><mi>p</mi></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && V &\to& V//G \\ & {}^{\mathllap{\sigma|_U}}\nearrow & && \downarrow^{\mathrlap{\overline{\rho}}} \\ U &\stackrel{p}{\to}& X &\stackrel{g}{\to}& \mathbf{B}G } \,. </annotation></semantics></math></div> <p>This means that a section of an <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a> is locally a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-valued map, hence a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>V</mi></mrow><annotation encoding="application/x-tex">\Omega V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>.</p> </div> <p>And so we say that <em>globally</em> it is a “twisted” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>V</mi></mrow><annotation encoding="application/x-tex">\Omega V</annotation></semantics></math>-cocycle. This leads to the following equivalent description.</p> <h3 id="RelationToTwistedCohomology">Relation to twisted cohomology</h3> <p>We discuss here how def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> of fields has an entirely equivalent expression in terms of <em><a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a></em> in general <em><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></em> and in fact, if the background field is nontrivial, in <em><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></em>. This follows by direct comparison with the corresponding notions in <em><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></em> and <em><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></em> as discussed there (<a href="#NSS">NSS</a>). The unification of notions seen this way gives a natural home for instance to the familiar observations such as that <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+fields">Chan-Paton gauge fields</a> on <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> are identified with <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> in (<a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential</a>) <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a> (see <em><a href="#ChanPatonGaugeFields">Chan-Paton gauge fields</a></em> below).</p> <p>First observe the following</p> <div class="num_remark"> <h6 id="remark_8">Remark</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Φ</mi> <mi>X</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\phi \;\colon\; \Phi_X \to \mathbf{Fields}</annotation></semantics></math> a field configuration as in def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> in the corresponding <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>ϕ</mi></mover></mtd> <mtd></mtd> <mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></munder><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>Φ</mi> <mi>X</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &&\stackrel{\phi}{\to}&& \underset{\mathbf{BgFields}}{\sum} \mathbf{Fields} \\ & {}_{\mathllap{\Phi_X}}\searrow &\swArrow& \swarrow_{\mathrlap{\mathbf{Fields}}} \\ && \mathbf{BgFields} } </annotation></semantics></math></div> <p>we have the following equivalently identifications when interpreting this as a <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>:</p> <ul> <li> <p>the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>∈</mo><mo lspace="0em" rspace="thinmathspace">matbbf</mo><msub><mi>H</mi> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{Fields} \in \matbbf{H}_{/\mathbf{BgFields}}</annotation></semantics></math> is the <strong><a class="existingWikiWord" href="/nlab/show/local+coefficient+%E2%88%9E-bundle">local coefficient ∞-bundle</a></strong>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>ϕ</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mrow></munder><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>→</mo><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle></mrow><annotation encoding="application/x-tex">\underset{\phi \in [X, \mathbf{Fields}]}{\sum}\mathbf{Fields} \to \mathbf{BgFields}</annotation></semantics></math> is the <strong>local <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a></strong> for a <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> theory;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/background+field">background field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_X</annotation></semantics></math> is the <strong><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twist</a></strong></p> </li> <li> <p>the field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\phi \colon \Phi_X \to \mathbf{Fields}</annotation></semantics></math> is</p> <ul> <li> <p>a <strong><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a></strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_X</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with local <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math></p> </li> <li> <p>equivalently: a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_X</annotation></semantics></math>-twisted <a class="existingWikiWord" href="/nlab/show/G-structure">V-structure</a></strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> </ul> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> is a <strong><a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a></strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_X</annotation></semantics></math>-twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-cohomology;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of fields are the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_X</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mrow><mo stretchy="false">[</mo><msub><mi>Φ</mi> <mi>X</mi></msub><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>τ</mi> <mn>0</mn></msub><mi>Γ</mi><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H_{[\Phi_X]}(X,V) \simeq \tau_0\Gamma[X,\mathbf{Fields}]_{\mathbf{H}} \,. </annotation></semantics></math></div></li> </ul> </div> <p>We see several illustrations of these identifications in the list of <em><a href="#Examples">Examples</a></em> below. More generally, there is a canonical identification of physical fields in the presence of background fields <em>and</em> boundary/defect with twisted and <em><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></em>. This we discuss below in <em><a href="#RelationToRelativeCohomology">Relation to relative cohomology</a></em>.</p> <h3 id="RelationToRelativeCohomology">Relation to relative cohomology</h3> <p>When we refine background fields to dynamical fields as discussed above in <em><a href="#BoundaryAndDefectFields">Boundary and defect fields</a></em> then the identification of fields with <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> in <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> as discussed above in <em><a href="#RelationToTwistedCohomology">Relation to twisted cohomology</a></em> accordingly generalized a bit: it becomes a combination of twisted cohomology and <em><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></em>.</p> <p>For consider the special case that the moduli of defect fields are trivial, hence that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo lspace="verythinmathspace">:</mo><mo>*</mo><mover><mo>→</mo><mrow><msub><mi>pt</mi> <mi>Q</mi></msub></mrow></mover><msub><mstyle mathvariant="bold"><mi>Fields</mi></mstyle> <mi>bulk</mi></msub></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \colon * \stackrel{pt_Q}{\to} \mathbf{Fields}_{bulk} </annotation></semantics></math></div> <p>is the global point inclusion into the bulk field moduli (the trivial bulk field). By prop. <a class="maruku-ref" href="#ModuliOfBulkAndDefectFieldsAsPullback"></a> it follows that</p> <div class="num_prop"> <h6 id="proposition_11">Proposition</h6> <p>There is a natural equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>ι</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>pt</mi> <mi>A</mi></msub><mo stretchy="false">]</mo><mo>≃</mo><mo stretchy="false">[</mo><msub><mi>X</mi> <mi>bulk</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><munder><mo>×</mo><mrow><mo stretchy="false">[</mo><msub><mi>X</mi> <mi>def</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo></mrow></munder><mo>*</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> [\iota_X, pt_A] \simeq [X_{bulk}, A] \underset{[X_{def}, A]}{\times} {*} \,, </annotation></semantics></math></div> <p>hence a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>ι</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>pt</mi> <mi>A</mi></msub><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><msub><mi>X</mi> <mi>bulk</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><msub><mi>X</mi> <mi>def</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [\iota_X, pt_A] \to [X_{bulk}, A] \to [X_{def}, A] </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_9">Remark</h6> <p>This identifies the equivalence classes of global points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>ι</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>pt</mi> <mi>A</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\iota_X, pt_A]</annotation></semantics></math> as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\iota_X</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative A-cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <p>In general, if the defect fields are not trivial, the fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>X</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\iota_X \to \mathbf{Fields}</annotation></semantics></math> (hence ordinary cocycles in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^{(\Delta^1)}</annotation></semantics></math>) are a kind of cocycles in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> that are a combination of relative and twisted cocycles: instead with their pullback to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>def</mi></msub></mrow><annotation encoding="application/x-tex">X_{def}</annotation></semantics></math> being equipped with a trivialization, it is equipped with a “twisted trivialization” in the sense of <a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structure">twisted c-structures</a>, discussed <a href="#TwistedDifferentialcStructurs">below</a>.</p> <h2 id="Examples">Examples</h2> <p>We distinguish four broad classes of examples of physical fields, according to def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a>:</p> <ol> <li> <p><a href="#SclarModuliFields">Scalar and moduli fields</a></p> <p>The simplest type of field is a (<a class="existingWikiWord" href="/nlab/show/smooth+function">smooth</a>) <a class="existingWikiWord" href="/nlab/show/function">function</a> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with values in the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> or <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>, called a <em><a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a></em>. Slightly more generally there are fields which are <a class="existingWikiWord" href="/nlab/show/functions">functions</a> into some other <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> or more general <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a>, often called <em>linear/non-linear <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></em> fields. Often that manifold is a space of parameters of some <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>, for instance of a compact space in <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+compactification">Kaluza-Klein compactification</a>, in which case these fields are often called <em>moduli fields</em>. Scalar fields may be <em>charged</em> under force fields, which we turn to next.</p> </li> <li> <p><a href="#ForceFields">Force fields</a></p> <ol> <li> <p><a href="#FieldsOfGravityAndGeneralizedGeometry">Fields of gravity, G-structure and generalized geometry</a></p> <p>In this case the <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a> is a <a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of fields is a <a class="existingWikiWord" href="/nlab/show/delooping">delooped</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\hat G \stackrel{\mathbf{Fields}}{\to} \mathbf{B}G</annotation></semantics></math> and a field is a generalized <a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a> exhibiting a <a class="existingWikiWord" href="/nlab/show/reduction+and+lift+of+structure+groups">reduction/lift of structure group</a>.</p> </li> <li> <p><a href="#GaugeFields">Gauge fields</a></p> <p>In this case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Fields}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}G_{conn}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/connections+on+an+%E2%88%9E-bundle">∞-connections</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a> for some <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mi>Grpd</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in Grpd(\mathbf{H})</annotation></semantics></math>.</p> </li> </ol> </li> <li> <p><a href="#MatterFields">Matter fields</a></p> <p>In this case the <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> for some <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in Grp(\mathbf{H})</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Fields}</annotation></semantics></math> is the univeral <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mover><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">V//G \stackrel{\mathbf{Fields}}{\to} \mathbf{B}G</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">V \in \mathbf{H}</annotation></semantics></math>. A field is <a class="existingWikiWord" href="/nlab/show/section">section</a> of the <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>.</p> </li> </ol> <p>Not all examples fall squarely into one of these types, some are mixtures of these. Relevant examples we discuss in</p> <ul> <li><a href="#FieldCombiningVariousProperties">Fields combining various of these properties</a></li> </ul> <p>In particular the moduli stacks <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> here are typically all differentially refined to moduli stacks <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}G_{conn}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">∞-connections</a> so that for instance every <a class="existingWikiWord" href="/nlab/show/reduction+and+lift+of+structure+groups">reduction and lift of structure groups</a> goes along with a corresponding data of the reduction of an <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">∞-connection</a>. The archetypical example for this are <a class="existingWikiWord" href="/nlab/show/spin+connections">spin connections</a>, see the example <em><a href="#OrdinaryGravity">Ordinary gravity</a></em> below.</p> <div class="un_remark" style="border:solid #000000;background: #E6DF13;border-width:2px 1px;padding:0 1em;margin:0 1em;"> <h6 id="examples_2">Examples</h6> <ol> <li> <p><a href="#SigmaModelFields">Sigma-model fields</a></p> <ol> <li> <p><a href="#UnchargedScalarField">Uncharged scalar particle</a></p> </li> <li> <p><a href="#ParticleTrajectory">Particle trajectors</a></p> </li> <li> <p><a href="#BraneTrajectory">Brane trajectory</a></p> </li> <li> <p><a href="#OpenStringSigmaModelFields">Open string sigma-model</a></p> </li> </ol> </li> <li> <p><a href="#ForceFields">Force fields</a></p> <ol> <li> <p><a href="#FieldsOfGravityAndGeneralizedGeometry">Fields of gravity, G-structure and generalized geometry</a></p> </li> <li> <p><a href="##GStructure">General – G-structure: twisted lift of structure group</a></p> </li> <li> <p><a href="#OrdinaryGravity">Gravity</a></p> </li> <li> <p><a href="#SpinStructures">Spin structures</a></p> </li> <li> <p><a href="#SpinStructures">Spin-c structures</a></p> </li> <li> <p><a href="##TypeIIGravity">Type II gravity, exceptional geometry</a></p> </li> <li> <p><a href="#HigherSpinStructures">Higher spin structures</a></p> </li> <li> <p><a href="#HigherSpincstructures">Higher spin-c structures</a></p> </li> </ol> </li> <li> <p><a href="#GaugeFields">Gauge fields</a></p> <ol> <li> <p><a href="#GaugeFieldsGeneral">General – Coefficients in differential cohomology: connections</a></p> </li> <li> <p><a href="#ElectromagneticField">Electromagnetic field</a></p> </li> <li> <p><a href="#YangMillsField">Yang-Mills field</a></p> </li> <li> <p><a href="#KalbRamondBField">Kalb-Ramond B-field</a></p> </li> <li> <p><a href="#SupergravityCField">Supergravity C-field</a></p> </li> </ol> </li> <li> <p><a href="#MatterFields">Matter fields</a></p> <ol> <li> <p><a href="#SectionsOfAssociatedBundles">General – Sections of associated bundles</a></p> </li> <li> <p><a href="#Fermions">Fermions</a></p> </li> <li> <p><a href="#TensorFields">Tensor fields</a></p> </li> </ol> </li> <li> <p><a href="##FieldCombiningVariousProperties">Fields combining these properties</a></p> <ol> <li> <p><a href="#TwistedDifferentialcStructurs">General – Twisted differential cocycles and c-structures</a></p> </li> <li> <p><a href="#NonabelianChargedParticle">Nonabelian charged particle trajectories – Wilson lines</a></p> </li> <li> <p><a href="#ChernSimonsWithWilsonLines">3d Chern-Simons field with Wilson line</a></p> </li> <li> <p><a href="#ChanPatonGaugeFields">Chan-Paton gauge bundles on D-branes: twisted differential K-cocycles</a></p> </li> <li> <p><a href="#HeteroticStringBackgroundField">Anomaly-free heterotic string background: differential String-c structures</a></p> </li> </ol> </li> </ol> </div> <h3 id="SigmaModelFields"><strong>0)</strong> Sigma-model fields</h3> <p>Traditionally a <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> field is a type of fields given simply by (<a class="existingWikiWord" href="/nlab/show/smooth+function">smooth</a>) <a class="existingWikiWord" href="/nlab/show/functions">functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math> from a <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> to a <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Now, by the general unified definition def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> and as shown in the following sections, in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> <em>every</em> type of field is of this form if we allow <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to be a general <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> and functions to be maps in a suitable <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a>. Nevertheless, here we start with briefly indicating those examples that are <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> fields also in the traditional restrictive sense of the term.</p> <h4 id="UnchargedScalarField">Uncharged scalar field</h4> <p>A <em><a class="existingWikiWord" href="/nlab/show/scalar+field">scalar field</a></em> is given simply by a <a class="existingWikiWord" href="/nlab/show/function">function</a> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>, typically with values in the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> (“real scalar field”) or the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> (“complex scalar field”). Hence this is the example of def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> with trivial <a class="existingWikiWord" href="/nlab/show/background+fields">background fields</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle><mo>≔</mo><mo>*</mo></mrow><annotation encoding="application/x-tex"> \mathbf{BgFields} \coloneqq * </annotation></semantics></math></div> <p>(the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>) and with the field moduli stack being the map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ℝ</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; \mathbb{R} \to * </annotation></semantics></math></div> <p>from (the <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> underlying) the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a> to the point, or else</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ℂ</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; \mathbb{C} \to * </annotation></semantics></math></div> <p>regarded as an object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mo>*</mo></mrow></msub><mo>≃</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/*} \simeq \mathbf{H}</annotation></semantics></math>.</p> <p><a class="existingWikiWord" href="/nlab/show/scalar+field">Scalar fields</a> are, due to their simplicity, prominent in toy examples used to discuss general properties of <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>. The only fundamental scalar particle observed in nature to date is the <a class="existingWikiWord" href="/nlab/show/Higgs+particle">Higgs particle</a> (or presumably so, in <a class="existingWikiWord" href="/nlab/show/technicolor">technicolor</a> models it is not actually fundamental but a composite of <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a> particles, discussed <a href="#Fermions">below</a>), but the Higgs field is, crucially, charged under the <a class="existingWikiWord" href="/nlab/show/electroweak+field">electroweak</a> <a class="existingWikiWord" href="/nlab/show/special+unitary+group">SU(2)</a>-<a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a>. A <a class="existingWikiWord" href="/nlab/show/model+%28in+theoretical+physics%29">model</a> of relevance in <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> which crucially features an uncharged scalar particle is <a class="existingWikiWord" href="/nlab/show/cosmic+inflation">cosmic inflation</a>. But the fundamental nature of the <a class="existingWikiWord" href="/nlab/show/inflaton+field">inflaton field</a> is hypothetical (if it exists at all), it might well be the <a class="existingWikiWord" href="/nlab/show/effective+QFT">effective</a> version of non-scalar fields.</p> <h4 id="ParticleTrajectory">Particle trajectory</h4> <p>The dynamics of a <a class="existingWikiWord" href="/nlab/show/particle">particle</a> propagating in a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is described by a field on the abstract <a class="existingWikiWord" href="/nlab/show/worldline">worldline</a> which is simply a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R} \to X</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>: a <a class="existingWikiWord" href="/nlab/show/trajectory">trajectory</a> of the particle. The <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> that describes the <a class="existingWikiWord" href="/nlab/show/dynamics">dynamics</a> of such a quantum particle is equivalently a 1-dimensional field theory on the <a class="existingWikiWord" href="/nlab/show/worldline">worldline</a>, with these maps as its physical fields.</p> <p>Therefore such wordline <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> fields are given by the special case of def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> with trivial <a class="existingWikiWord" href="/nlab/show/background+field">background field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle><mo>≃</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{BgFields} \simeq *</annotation></semantics></math> and with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{Fields} \;\colon\; X \to * </annotation></semantics></math> regarded as an object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mo>*</mo></mrow></msub><mo>≃</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/*} \simeq \mathbf{H}</annotation></semantics></math>.</p> <h4 id="BraneTrajectory">Brane trajectory</h4> <p>More generally, the <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> fields of any <a class="existingWikiWord" href="/nlab/show/brane">brane</a> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> with <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> are given as <a href="#ParticleTrajectory">above</a>.</p> <h4 id="OpenStringSigmaModelFields">Open string sigma-model</h4> <div class="num_remark"> <h6 id="remark_10">Remark</h6> <p>A <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mover><mo>→</mo><mrow><msub><mi>ι</mi> <mi>X</mi></msub></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">Q \stackrel{\iota_X}{\to} X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> for an <a class="existingWikiWord" href="/nlab/show/open+string">open string</a> with <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Σ</mi><mover><mo>↪</mo><mrow><msub><mi>ι</mi> <mi>Σ</mi></msub></mrow></mover><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\partial \Sigma \stackrel{\iota_\Sigma}{\hookrightarrow} \Sigma</annotation></semantics></math>: a <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a> configuration of the open string <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> is a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>ι</mi> <mi>Σ</mi></msub><mo>→</mo><msub><mi>ι</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; \iota_\Sigma \to \iota_X </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><msup><mi>Δ</mi> <mn>1</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^{\Delta^1}</annotation></semantics></math>, hence a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo>∂</mo><mi>Σ</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>bdr</mi></msub></mrow></mover></mtd> <mtd><mi>Q</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>ι</mi> <mi>Σ</mi></msub></mrow></mpadded></msup></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>ι</mi> <mi>X</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Σ</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>bulk</mi></msub></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \partial \Sigma &\stackrel{\phi_{bdr}}{\to}& Q \\ \downarrow^{\mathrlap{\iota_\Sigma}} &\swArrow& \downarrow^{\mathrlap{\iota_X}} \\ \Sigma &\stackrel{\phi_{bulk}}{\to}& X } \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> ordinary <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> just says that a field configuration is a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>bulk</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi_{bulk} \;\colon\; \Sigma \to X</annotation></semantics></math> subject to the constraint that it takes the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math>. This means that this is a <a class="existingWikiWord" href="/nlab/show/trajectory">trajectory</a> of an <a class="existingWikiWord" href="/nlab/show/open+string">open string</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> whose endpoints are constrained to sit on the <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Q \hookrightarrow X</annotation></semantics></math>.</p> <p>If however <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is more generally an <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a>, then the <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> filling the above diagram imposes this constraint only up to orbifold transformations, hence exhibits what in the physics literature are called “orbifold twisted sectors” of open string configurations.</p> </div> <div class="num_prop" id="TheTypeIIOpenStringSigmaModelModuliStackOfFields"> <h6 id="proposition_12">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>ι</mi> <mi>Σ</mi></msub><mo>,</mo><msub><mi>ι</mi> <mi>X</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\iota_\Sigma, \iota_X]</annotation></semantics></math> of such field configurations is the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">[</mo><msub><mi>ι</mi> <mi>Σ</mi></msub><mo>,</mo><msub><mi>ι</mi> <mi>X</mi></msub><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>Q</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ [\iota_{\Sigma}, \iota_X] &\to& [\Sigma, X] \\ \downarrow &\swArrow& \downarrow \\ [S^1, Q] &\to& [S^1, X] } \,. </annotation></semantics></math></div></div> <p>The <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a> for such a <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> is a <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+field">Chan-Paton gauge field</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Q \hookrightarrow X</annotation></semantics></math>.</p> <h3 id="ForceFields"><strong>I)</strong> Force fields</h3> <p>We discuss examples for the two classes of <a class="existingWikiWord" href="/nlab/show/force">force</a> fields, which are:</p> <ul> <li> <p><a href="#FieldsOfGravityAndGeneralizedGeometry">Fields of gravity, G-structure and generalized geometry</a>;</p> </li> <li> <p><a href="#GaugeFields">Gauge fields</a>.</p> </li> </ul> <h4 id="FieldsOfGravityAndGeneralizedGeometry"><strong>I a)</strong> Fields of gravity, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-Structure and generalized geometry</h4> <p>We discuss the example of the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, below in <em><a href="#OrdinaryGravity">Gravity</a></em>, and various closely related types of fields that all encode <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> in some sense, in fact all encode geometry in the sense of <em><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a></em>. This most general case we discuss in <em><a href="#GStructure">G-structure – (twisted lift of structure ∞-groups)</a></em>.</p> <h5 id="GStructure"><strong>General</strong> – <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structure: (twisted) lift of structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-group</h5> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mstyle mathvariant="bold"><mi>p</mi></mstyle></mover><mi>G</mi></mrow><annotation encoding="application/x-tex"> A \to \hat G \stackrel{\mathbf{p}}{\to} G </annotation></semantics></math></div> <p>be an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">g_X \colon X \to \mathbf{B}G</annotation></semantics></math> a given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, a <a class="existingWikiWord" href="/nlab/show/lift+of+the+structure+group">lift of the structure group</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math> is a field</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>g</mi> <mi>X</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mrow><mi>B</mi><mi>p</mi></mrow></mstyle><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi \; \colon\; g_X \to \mathbf{B p} \,. </annotation></semantics></math></div> <p>If the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a> is central in that it extends to a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mstyle mathvariant="bold"><mi>Bp</mi></mstyle></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mover><mo>→</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>A</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}A \to \mathbf{B}\hat G \stackrel{\mathbf{Bp}}{\to} \mathbf{B}G \stackrel{\mathbf{c}}{\to} \mathbf{B}^2 A </annotation></semantics></math></div> <p>then a <a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structure">twisted c-structure</a> is a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>g</mi> <mi>X</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; g_X \to \mathbf{c} </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}G}</annotation></semantics></math>. These kinds of fields are interpreted as fields of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> and its variants, as shown by the following examples.</p> <p>For recognizing traditional constructions in this formulation, the following basic fact is important.</p> <div class="num_prop" id="CosetIsHomotopyFiberOfDeloopedInclusion"> <h6 id="proposition_13">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo lspace="verythinmathspace">:</mo><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\iota \colon H \hookrightarrow G</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> inclusion of <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a>, we have a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><mi>H</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mover><mo>→</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>ι</mi></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> G/H \to \mathbf{B}H \stackrel{\mathbf{B}\iota}{\to} \mathbf{B}G </annotation></semantics></math></div> <p>with the <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> on the left.</p> </div> <h5 id="OrdinaryGravity">Gravity</h5> <p>We discuss the formulation of the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> as a special case of def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a>.</p> <div class="num_defn"> <h6 id="definition_12">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>OrthStruc</mi></mstyle> <mi>n</mi></msub></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; \mathbf{B}O(n) \stackrel{\mathbf{OrthStruc}_n}{\to} \mathbf{B}GL(n) </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of the <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie</a>-<a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> inclusion of the <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a> into the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a> in <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> (the inclusion of the <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a>).</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">X \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SmoothMfd">SmoothMfd</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>≔</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} \coloneqq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub><mo>≔</mo><msub><mi>τ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Phi_X \coloneqq \tau_X \;\colon\; X \to \mathbf{B}GL(n) </annotation></semantics></math></div> <p>for the canonical map modulating the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> to which the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> is canonically <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a>.</p> </div> <div class="num_example" id="RiemannianVielbein"> <h6 id="example_4">Example</h6> <p>Morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>τ</mi> <mi>X</mi></msub><mo>→</mo><msub><mstyle mathvariant="bold"><mi>OrthStruc</mi></mstyle> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; \tau_X \to \mathbf{OrthStruc}_n </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}GL(n)}</annotation></semantics></math> hence <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>e</mi></msub></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &&\stackrel{}{\to}&& \mathbf{B}O(n) \\ & \searrow &\swArrow_{e}& \swarrow \\ && \mathbf{B}GL(n) } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> are equivalently <a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a> fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> exhibiting the <a class="existingWikiWord" href="/nlab/show/reduction+of+the+structure+group">reduction of the structure group</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n)</annotation></semantics></math>. A <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> is a local frame transformation. Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo>,</mo><msub><mstyle mathvariant="bold"><mi>OrthStruc</mi></mstyle> <mi>n</mi></msub><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> [\tau_X, \mathbf{OrthStruc}_n]_{\mathbf{H}} \in \mathbf{H} </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of <a class="existingWikiWord" href="/nlab/show/Riemannian+metrics">Riemannian metrics</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Every such is a field configuration of ordinary <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. (But see example <a class="maruku-ref" href="#GenerallyCovariantFieldOfGravity"></a> below.)</p> <p>According to remark <a class="maruku-ref" href="#PullbackAlongGeneralizedLocalDiffeomorphisms"></a> such fields pull back along map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \colon Y \to X</annotation></semantics></math> that fit into a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Y</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>τ</mi> <mi>Y</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mi>τ</mi> <mi>X</mi></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Y &&\stackrel{f}{\to}&& X \\ & {}_{\mathllap{\tau_Y}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{\tau_X}} \\ && \mathbf{B}GL(n) } \,. </annotation></semantics></math></div> <p>These are precisely <a class="existingWikiWord" href="/nlab/show/local+diffeomorphisms">local diffeomorphisms</a> and indeed these are precisely the maps along which <a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a> fields / <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+metric">pseudo-Riemannian metric</a> fiedls pull back. (Pullback along <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> may not preserve the non-degeneracy of a metric tensor, but precisely the local diffeomorphisms do.)</p> </div> <div class="num_remark"> <h6 id="remark_11">Remark</h6> <p>The <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as embodied by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\tau_X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/background+field">background field</a> in example <a class="maruku-ref" href="#RiemannianVielbein"></a>.</p> </div> <div class="num_remark" id="OrthogonalStructureIsTrivialInPlainHomotopyTheory"> <h6 id="remark_12">Remark</h6> <p>Under <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+cohesive+infinity-groupoids">geometric realization</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>→</mo><mn>∞</mn><mi>Grpd</mi><mo>≃</mo><msub><mi>L</mi> <mi>whe</mi></msub><mi>Top</mi></mrow><annotation encoding="application/x-tex"> {\vert-\vert} \;\colon\; \mathbf{H} \to \infty Grpd \simeq L_{whe} Top </annotation></semantics></math></div> <p>the map of <a class="existingWikiWord" href="/nlab/show/moduli+stacks">moduli stacks</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}O(n) \to \mathbf{B}GL(n)</annotation></semantics></math> becomes an <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a> (a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>) of <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>B</mi><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B O(n) \stackrel{\simeq}{\to} B GL(n)</annotation></semantics></math>.</p> <p>This means that in plain <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, hence ignoring the <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>, a lift in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>τ</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd><mi>B</mi><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && B O(n) \\ & \nearrow & \downarrow \\ X &\stackrel{\tau_X}{\to}& B GL(n) } </annotation></semantics></math></div> <p>is no information, up to equivalence: it always exists and exists uniquely up to a contractible space of choices. In order for such a lift really to be equivalently an <a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a> it needs to be taken with geometry included, hence for <a class="existingWikiWord" href="/nlab/show/moduli+stacks">moduli stacks</a> (_<a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos">cohesive</a>_ <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a>) as above.</p> <p>The analog statement is true for every delooping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}H \to \mathbf{B}G</annotation></semantics></math> of the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">H \hookrightarrow G</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a> into a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>. More examples of this kind we discuss below in <em><a href="#TypeIIGravity">Type II Gravity and generalized geometry</a></em>.</p> </div> <div class="num_remark"> <h6 id="remark_13">Remark</h6> <p>The <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>OrthStruc</mi></mstyle> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{OrthStruc}_n</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/coset+space">coset space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)//O(n)</annotation></semantics></math>, hence we have a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> of <a class="existingWikiWord" href="/nlab/show/moduli+stacks">moduli stacks</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>OrthStruc</mi></mstyle> <mi>n</mi></msub></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> GL(n)//O(n) \to \mathbf{B}O(n) \stackrel{\mathbf{OrthStruc}_n}{\to} \mathbf{B}GL(n) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. This means that locally, over a <a class="existingWikiWord" href="/nlab/show/cover">cover</a> (<a class="existingWikiWord" href="/nlab/show/1-epimorphism">1-epimorphism</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p \colon U \to X</annotation></semantics></math> over which the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> has a trivialization <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub><msub><mo stretchy="false">|</mo> <mi>U</mi></msub><mo>≃</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\tau_X|_U \simeq *</annotation></semantics></math>, the space of fields is simply <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>U</mi><mo>,</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[U, GL(n)//O(n)]</annotation></semantics></math>.</p> </div> <p>The next example is the differential refinement of the previous one.</p> <div class="num_example"> <h6 id="example_5">Example</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mstyle mathvariant="bold"><mi>OrthStruc</mi></mstyle><mo>^</mo></mover> <mi>n</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \widehat \mathbf{OrthStruc}_n \;\colon\; \mathbf{B}O(n)_{conn} \to \mathbf{B}GL(n)_{conn} </annotation></semantics></math></div> <p>be the canonical differential refinement of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>OrthStruc</mi></mstyle> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{OrthStruc}_n</annotation></semantics></math>, where now the <a class="existingWikiWord" href="/nlab/show/moduli+stacks">moduli stacks</a> of <a class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a> appear. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a manifold as above, let then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>τ</mi><mo stretchy="false">^</mo></mover> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \hat \tau_X \;\colon\; X \to \mathbf{B}GL(n)_{conn} </annotation></semantics></math></div> <p>modulate an <a class="existingWikiWord" href="/nlab/show/affine+connection">affine connection</a> on the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>. A field</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mover><mi>τ</mi><mo stretchy="false">^</mo></mover> <mi>X</mi></msub><mo>→</mo><msub><mover><mstyle mathvariant="bold"><mi>OrthStruc</mi></mstyle><mo>^</mo></mover> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; \hat \tau_X \to \widehat \mathbf{OrthStruc}_n </annotation></semantics></math></div> <p>is now still equivalently just a <a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a>, but its component</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow></munder><msub><mover><mi>τ</mi><mo stretchy="false">^</mo></mover> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \underset{\mathbf{B}GL(n)_{conn}}{\sum} \hat \tau_X \;\colon\; X \to \mathbf{B}O(n)_{conn} </annotation></semantics></math></div> <p>now captures the orthognal connection which in the physics literature is often called (inaccurately) the <em><a class="existingWikiWord" href="/nlab/show/spin+connection">spin connection</a></em>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a> then exhibits the original <a class="existingWikiWord" href="/nlab/show/affine+connection">affine connection</a> as that whose components are the <a class="existingWikiWord" href="/nlab/show/Christoffel+symbols">Christoffel symbols</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a> defined as above, a relation that is familiar from the physics literature in the form of the <a class="existingWikiWord" href="/nlab/show/equation">equation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>Γ</mi><mi>e</mi><mo>+</mo><msup><mi>e</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>d</mi><mi>e</mi></mrow><annotation encoding="application/x-tex"> \omega = e^{-1}\Gamma e + e^{-1} d e </annotation></semantics></math></div> <p>between local connection 1-form components.</p> </div> <p>But both these examples do not fully accurately reflect the field content of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> yet. This is because the <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a> of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> is supposed to by <a class="existingWikiWord" href="/nlab/show/general+covariance">generally covariant</a>. This means that for two <a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a> field configurations as in example <a class="maruku-ref" href="#RiemannianVielbein"></a> such that one goes into the other under a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, there is a <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> between them.</p> <div class="num_example" id="GenerallyCovariantFieldOfGravity"> <h6 id="example_6">Example</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Diff</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Diff}(X) \coloneqq \mathbf{Aut}(X) \in Grp(\mathbf{H})</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/diffeomorphism+group">diffeomorphism group</a> of the <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. There is then the canonical <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> of the <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-group">automorphism ∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Diff</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Diff}(X)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> itself, exhibited by the universal <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>X</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>Diff</mi></mstyle><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mstyle mathvariant="bold"><mi>Diff</mi></mstyle><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mstyle mathvariant="bold"><mi>Diff</mi></mstyle><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left(X // \mathbf{Diff}\left(X\right) \to \mathbf{B}\mathbf{Diff}\left(X\right) \right) \in \mathbf{H}_{/\mathbf{B}\mathbf{Diff}\left(X\right)} \,. </annotation></semantics></math></div> <p>This induces also an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> also on the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>.</p> <p>Moreover the trivial bundle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>≔</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>Diff</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>OrthStruc</mi></mstyle><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mstyle mathvariant="bold"><mi>Diff</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mstyle mathvariant="bold"><mi>Diff</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \coloneqq (\mathbf{Diff}(X)\times \mathbf{OrthStruc} \to \mathbf{B}\mathbf{Diff}(X) ) \in \mathbf{H}_{/\mathbf{B}\mathbf{Diff}(X)} </annotation></semantics></math></div> <p>corresponds to the trivial action.</p> <p>Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Diff</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></munder><mo stretchy="false">[</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> \underset{\mathbf{B}Diff(n)}{\sum} \underset{\mathbf{B}(GL(n))}{\prod} [\tau_X, \mathbf{Fields}] \in \mathbf{H} </annotation></semantics></math></div> <p>is the accurate space of <a class="existingWikiWord" href="/nlab/show/general+covariance">generally covariant</a> fields of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>.</p> </div> <h5 id="SpinStructures"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spin</mi></mrow><annotation encoding="application/x-tex">Spin</annotation></semantics></math>-structures</h5> <p>In theories with <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> (discussed <a href="#Fermions">below</a>) the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> is more refined than just a <a class="existingWikiWord" href="/nlab/show/vielbein+field">vielbein field</a> as <a href="#OrdinaryGravity">above</a>, hence an <a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>: it also involves an <a class="existingWikiWord" href="/nlab/show/orientation+structure">orientation structure</a> and a <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>.</p> <p>To see that these structures are really all (fields) of the same kind, observe that they are the lifts through the first step of the <a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}GL</annotation></semantics></math>, as shown in the following table</p> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></th><th><a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stacks">moduli ∞-stacks</a></th><th><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>/<a class="existingWikiWord" href="/nlab/show/higher+spin+structure">higher spin structure</a></th><th><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋮</mi></mrow><annotation encoding="application/x-tex">\vdots</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ninebrane+10-group">ninebrane 10-group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Ninebrane</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}Ninebrane </annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ninebrane+structure">ninebrane structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/third+fractional+Pontryagin+class">third fractional Pontryagin class</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Fivebrane</mi><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mi>n</mi></mfrac></mstyle><msub><mi>p</mi> <mn>3</mn></msub></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>11</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}Fivebrane \stackrel{\tfrac{1}{n} p_3}{\to} \mathbf{B}^{11}U(1)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/second+fractional+Pontryagin+class">second fractional Pontryagin class</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>6</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}String \stackrel{\tfrac{1}{6}\mathbf{p}_2}{\to} \mathbf{B}^7 U(1)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/first+fractional+Pontryagin+class">first fractional Pontryagin class</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1)</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/second+Stiefel-Whitney+class">second Stiefel-Whitney class</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mover><mo>→</mo><mstyle mathvariant="bold"><mrow><msub><mi>w</mi> <mn>2</mn></msub></mrow></mstyle></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}SO \stackrel{\mathbf{w_2}}{\to} \mathbf{B}^2 \mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orientation+structure">orientation structure</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/first+Stiefel-Whitney+class">first Stiefel-Whitney class</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>1</mn></msub></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}O \stackrel{\mathbf{w}_1}{\to} \mathbf{B}\mathbb{Z}_2</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a>/<a class="existingWikiWord" href="/nlab/show/vielbein">vielbein</a>/<a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">↓</mo></mrow><annotation encoding="application/x-tex">\downarrow</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}GL</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></td><td style="text-align: left;"></td></tr> </tbody></table> <p>(all hooks are <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequences">homotopy fiber sequences</a>)</p> </div> <p>Notice, as in remark <a class="maruku-ref" href="#OrthogonalStructureIsTrivialInPlainHomotopyTheory"></a>, that for the interpretation of the first step here it is crucial to interpret this in <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stacks">moduli ∞-stacks</a> and not in <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a>.</p> <p>Hence if a background field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> is assumed, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{BgFields} \colon \mathbf{B}O(n)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub><mo>=</mo><msub><mi>e</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_X = e_X</annotation></semantics></math>, then the moduli stack for <a class="existingWikiWord" href="/nlab/show/orientation+structure">orientation structure</a>-fields is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \colon \mathbf{B} SO(n) \to \mathbf{B}O(n) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}O(n)}</annotation></semantics></math>. And if an orientation background field is assumed, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{BgFields} \colon \mathbf{B}SO(n)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub><mo>=</mo><msub><mi>o</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_X = o_X</annotation></semantics></math>, then the moduli stack for <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>-fields is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \colon \mathbf{B} Spin(n) \to \mathbf{B}SO(n) \,. </annotation></semantics></math></div> <p>Evidently there is then also a notion of <a class="existingWikiWord" href="/nlab/show/higher+spin+structure">higher spin structure</a>-fields. These appear as backgrounds when one passes from <a class="existingWikiWord" href="/nlab/show/spinning+particles">spinning particles</a> to <a class="existingWikiWord" href="/nlab/show/spinning+strings">spinning strings</a> and then to further “spinning” <a class="existingWikiWord" href="/nlab/show/branes">branes</a>. This we discuss below in <em><a href="#HigherSpinStructures">Higher spin structures</a></em>.</p> <h5 id="SpinStructures"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-structures</h5> <p>Some <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theories</a> involve not plain <a class="existingWikiWord" href="/nlab/show/spin+structures">spin structures</a> but <a class="existingWikiWord" href="/nlab/show/spin-c+structures">spin-c structures</a> on their <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> (for instance in <a class="existingWikiWord" href="/nlab/show/Seiberg-Witten+theory">Seiberg-Witten theory</a> or in the <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> over <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> in <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a>, see the example <em><a href="#ChanPatonGaugeFields">Chan-Paton gauge fields on D-branes</a></em> below). By the discussion there, the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}Spin^c(n)</annotation></semantics></math> for <a class="existingWikiWord" href="/nlab/show/spin%5Ec+group">spin^c group</a>-<a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a> sits in the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>Spin</mi> <mi>c</mi></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mstyle mathvariant="bold"><mi>c</mi></mstyle> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}Spin^c &\to& \mathbf{B}U(1) \\ \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 \,mod\, 2}} \\ \mathbf{B}SO(n) &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2\mathbb{Z}_2 } \,, </annotation></semantics></math></div> <p>where the right vertical map represents the universal <a class="existingWikiWord" href="/nlab/show/first+Chern+class">first Chern class</a> modulo 2. In other words this says that a <a class="existingWikiWord" href="/nlab/show/spin%5Ec-structure">spin^c-structure</a> on some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structure">twisted w2-structure</a> with twist the first Chern-class of a <a class="existingWikiWord" href="/nlab/show/circle+bundle">circle bundle</a>.</p> <p>So if that circle bundle is regarded as a <a class="existingWikiWord" href="/nlab/show/background+field">background field</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>X</mi></msub><mo>≔</mo><msub><mi>g</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mi>colon</mi><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \phi_X \coloneqq g_X \;colon\; X \to \mathbf{B}U(1) </annotation></semantics></math></div> <p>then a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-structure for that underlying circle bundle is a field</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Φ</mi> <mi>X</mi></msub><mo>→</mo><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; \Phi_X \to \mathbf{w}_2 </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}^2 \mathbb{Z}_2}</annotation></semantics></math>.</p> <p>Or rather, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> as before and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(n)</annotation></semantics></math>-principal bundle involved in the above is required to be a <a class="existingWikiWord" href="/nlab/show/reduction+of+the+structure+group">reduction</a> of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_X : X \to \mathbf{B}GL(n)</annotation></semantics></math> as before, then the <a class="existingWikiWord" href="/nlab/show/background+field">background field</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-structure fields is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>g</mi> <mi>X</mi></msub><mspace width="thinmathspace"></mspace><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn><mo stretchy="false">)</mo></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>×</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Phi_X \;\colon\; X \stackrel{(\tau_X, g_X \,mod\, 2)}{\to} \mathbf{B}GL(n) \times \mathbf{B}^2 \mathbb{Z}_2 \,, </annotation></semantics></math></div> <p>and the field moduli stack is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>SO</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><msub><mstyle mathvariant="bold"><mi>w</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; \mathbf{B}SO(n) \stackrel{(p, \mathbf{w}_2)}{\to} \mathbf{B}GL(n) \times \mathbf{B} \,. </annotation></semantics></math></div> <h5 id="TypeIIGravity">Type II gravity, exceptional geometry</h5> <p>By remark <a class="maruku-ref" href="#OrthogonalStructureIsTrivialInPlainHomotopyTheory"></a> above the ordinary field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> on some <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is equivalently a <a class="existingWikiWord" href="/nlab/show/reduction+of+the+structure+group">reduction of the structure group</a> from the <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a> to its <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a>, regarded as a field whose <a class="existingWikiWord" href="/nlab/show/background+field">background field</a> is the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> itself.</p> <p>If one instead considers a variant of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> as a <a class="existingWikiWord" href="/nlab/show/background+field">background field</a> – a <em><a class="existingWikiWord" href="/nlab/show/generalized+tangent+bundle">generalized tangent bundle</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>τ</mi> <mi>X</mi> <mi>gen</mi></msubsup><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\tau_X^{gen} : X \to \mathbf{B}G</annotation></semantics></math> – with some other structure <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, then a field with values in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; \mathbf{B}H \to \mathbf{B}G </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}G}</annotation></semantics></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>↪</mo></mrow><annotation encoding="application/x-tex">H \hookrightarrow</annotation></semantics></math> the inclusion of the <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a>, may be thought of as an accordingly <em>generized field of gravity</em> defining a <em>generalized</em> <a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a>.</p> <p>Such fields naturally appear in <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theories</a> of higher-dimensional <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, where related to the <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a> and more generally <a class="existingWikiWord" href="/nlab/show/U-duality">U-duality</a> structure of these <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theories</a>.</p> <p>Notably in manifestly <a class="existingWikiWord" href="/nlab/show/T-duality">T-duality</a>-equivariant <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a> the <a class="existingWikiWord" href="/nlab/show/generalized+tangent+bundle">generalized tangent bundle</a> on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(n,n)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>τ</mi> <mi>X</mi> <mi>gen</mi></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau^{gen}_X \;\colon\; X \to \mathbf{B}O(n,n)</annotation></semantics></math>. The corresponding <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a> inclusion yields the moduli stack of fields</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>×</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; \mathbf{B}(O(n)\times O(n)) \to \mathbf{B}O(n,n) </annotation></semantics></math></div> <p>and a field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>τ</mi> <mi>X</mi> <mi>gen</mi></msubsup><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\phi \;\colon\; \tau_X^{gen} \to \mathbf{Fields}</annotation></semantics></math> is equivalently the <a class="existingWikiWord" href="/nlab/show/generalized+vielbein">generalized vielbein</a> for the generalized Riemannian geometry called <em><a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a></em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>In fact also the generalized tangent bundle itself should be regarded as a field: Notice that the canonical diagonal inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> GL(n) \hookrightarrow O(n,n) </annotation></semantics></math></div> <p>does not have a <a class="existingWikiWord" href="/nlab/show/retraction">retraction</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>G</mi> <mi>geom</mi></msub><mo>↪</mo><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n) \hookrightarrow G_{geom} \hookrightarrow O(n,n)</annotation></semantics></math> for the maximal subgroup for which a retrection to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math> still exists, called the <strong>geometric subgroup</strong> in the context of <a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a>. Then a lift of the tangent bundle to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>geom</mi></msub></mrow><annotation encoding="application/x-tex">G_{geom}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msubsup><mi>τ</mi> <mi>X</mi> <mi>gen</mi></msubsup></mrow></mover></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>geom</mi></msub></mtd> <mtd><mo>↪</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>O</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mrow><mo lspace="0em" rspace="thinmathspace">athllap</mo><mrow><msub><mi>τ</mi> <mi>X</mi></msub></mrow></mrow></msub><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>p</mi></mstyle></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ X &&\stackrel{\tau_X^{gen}}{\to}&& \mathbf{B}G_{geom} & \hookrightarrow& \mathbf{B} O(n,n) \\ & {}_{\athllap{\tau_X}}\searrow &\swArrow& \swarrow_{\mathrlap{\mathbf{p}}} \\ && \mathbf{B}GL(n) } \,, </annotation></semantics></math></div> <p>hence a field</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>e</mi> <mi>X</mi> <mi>gen</mi></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>τ</mi> <mi>X</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>p</mi></mstyle></mrow><annotation encoding="application/x-tex"> e_X^{gen} \;\colon\; \tau_X \to \mathbf{p} </annotation></semantics></math></div> <p>is what is called a <em>geoemtric</em> <a class="existingWikiWord" href="/nlab/show/generalized+tangent+bundle">generalized tangent bundle</a>. These are exactly the generalized tangent bundles primarily considered in the literature (see at <em><a class="existingWikiWord" href="/nlab/show/type+II+geometry">type II geometry</a></em> for more).</p> <p>In the <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+compactification">Kaluza-Klein compactification</a> of <a class="existingWikiWord" href="/nlab/show/type+II+supergravity">type II supergravity</a> and of <a class="existingWikiWord" href="/nlab/show/11-dimensional+supergravity">11-dimensional supergravity</a> preserving some amount of global <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> this generalized type II geometry is further enhanced to various flavors of what is called <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a>.</p> <p>Here the <a class="existingWikiWord" href="/nlab/show/generalized+tangent+bundle">generalized tangent bundle</a> has as structure group an <a class="existingWikiWord" href="/nlab/show/exceptional+Lie+group">exceptional Lie group</a> from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>-series <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">E_{n(n)}</annotation></semantics></math> (for compactification on an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional compact space). The moduli stack of fields is then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>K</mi> <mi>n</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>E</mi> <mrow><mi>n</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; \mathbf{B} K_{n} \to \mathbf{B} E_{n(n)} </annotation></semantics></math></div> <p>and a field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>e</mi> <mi>X</mi> <mi>gen</mi></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>τ</mi> <mi>X</mi> <mi>gen</mi></msubsup><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex"> e_X^{gen} \;\colon\; \tau_X^{gen} \to \mathbf{Fields}</annotation></semantics></math> is equivalently a <a class="existingWikiWord" href="/nlab/show/generalized+vielbein">generalized vielbein</a> for <a class="existingWikiWord" href="/nlab/show/exceptional+generalized+geometry">exceptional generalized geometry</a>.</p> <h5 id="HigherSpinStructures">Higher spin structures</h5> <p>By the discussion of <a href="#SpinStructures">Spin structure fields</a> above there are evident higher analogs, obtained by climbing through the <a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> <a class="existingWikiWord" href="/nlab/show/higher+spin+structure+-+table">of BO</a>.</p> <p>In the next step we have <em><a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a></em>-fields which are maps to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; \mathbf{B}String \to \mathbf{B}Spin \,. </annotation></semantics></math></div> <p>These appear as fields in <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic supergravity</a> with <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a>-cancellation by the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a> and for trivial <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a>. In the presence of a non-trivial gauge field these are further refined to <em><a href="#HigherSpincstructures">Higher spin-c structrures</a></em> discussed below. This field content of <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic supergravity</a> is discussed in more detail below in <em><a href="#HeteroticStringBackgroundField">Anomaly-free heterotic supergravity fields – differential String-c structures</a></em>.</p> <p>Further up the <a class="existingWikiWord" href="/nlab/show/Whitehead+tower">Whitehead tower</a> <em><a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></em>-fields are maps to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Fivebrane</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; \mathbf{B}Fivebrane \to \mathbf{B}String </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}String}</annotation></semantics></math>. These, or their twisted variants, appear in <a class="existingWikiWord" href="/nlab/show/dual+heterotic+string+theory">dual heterotic string theory</a>.</p> <h5 id="HigherSpincstructures">Higher <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-structures</h5> <p>As discussed above, an ordinary <a class="existingWikiWord" href="/nlab/show/spin-c+structure">spin-c structure</a> is really a <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> which is <em><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structure">twisted</a></em> by the class of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>.</p> <p>Similarly the <a class="existingWikiWord" href="/nlab/show/higher+spin+structure">higher spin structure</a>-fields just discussed have further twistes by background unitary bundles. For</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{c} \;\colon\; \mathbf{B}G \to \mathbf{B}^3 U(1) </annotation></semantics></math></div> <p>some <a class="existingWikiWord" href="/nlab/show/universal+characteristic+map">universal characteristic map</a> on <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stacks">moduli ∞-stacks</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^3 U(1)</annotation></semantics></math> the moduli 3-stack of <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundles</a>, hence <a class="existingWikiWord" href="/nlab/show/circle+3-group">circle 3-group</a>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a> we say that the <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>String</mi> <mstyle mathvariant="bold"><mi>c</mi></mstyle></msup></mrow><annotation encoding="application/x-tex">String^{\mathbf{c}}</annotation></semantics></math> appearing in the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>String</mi> <mstyle mathvariant="bold"><mi>c</mi></mstyle></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>c</mi></mstyle></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mstyle mathvariant="bold"><mi>p</mi></mstyle></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}String^{\mathbf{c}} &\to& \mathbf{B} G \\ \downarrow && \downarrow^{\mathrlap{\mathbf{c}}} \\ \mathbf{B}Spin &\stackrel{\tfrac{1}{2}\mathbf{p}}{\to}& \mathbf{B}^3 U(1) } </annotation></semantics></math></div> <p>is a higher <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>-analog of <a class="existingWikiWord" href="/nlab/show/spin%5Ec">spin^c</a>.</p> <p>In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a compact, connected and simply connected <a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a> (such as the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a> or the <a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>) then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>BG</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> \pi_0\mathbf{H}(\mathbf{B}G, \mathbf{B}^3 U(1)) \simeq H^4(BG, \mathbb{Z}) \simeq \mathbb{Z} </annotation></semantics></math></div> <p>(the first equivalence is discussed at <em><a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></em> as a special case of a theorem discussed at <em><a class="existingWikiWord" href="/nlab/show/smooth+infinity-groupoid">smooth infinity-groupoid</a></em>). This means that there is an essentially unique map of higher moduli stacks</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{c} \;\colon\; \mathbf{B}G \to \mathbf{B}^3 U(1) </annotation></semantics></math></div> <p>which maps to the generator of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>BG</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^4(BG, \mathbb{Z})</annotation></semantics></math> under <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+cohesive+infinity-groupoids">geometric realization of cohesive infinity-groupoids</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>Spin</mi></mrow><annotation encoding="application/x-tex">G = Spin</annotation></semantics></math> this is the smooth refinement of the <a class="existingWikiWord" href="/nlab/show/first+fractional+Pontryagin+class">first fractional Pontryagin class</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\tfrac{1}{2}\mathbf{p}_1</annotation></semantics></math>, discussed further at <em><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a></em>.</p> <p>Of interest in <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic supergravity</a> here is the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">G = E_8 \times E_8</annotation></semantics></math> (the product of the <a class="existingWikiWord" href="/nlab/show/exceptional+Lie+group">exceptional Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">E_8</annotation></semantics></math> with itself) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math> is twice the canonical string class.</p> <p>If then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_X \;\colon\; X \to \mathbf{B}(E_8 \times E_8)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/instanton+sector">instanton sector</a> of the gauge field in <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic supergravity</a> regarded as a <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a> for the field of heterotic <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a>, then with</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; \mathbf{B}Spin \stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to} \mathbf{B}^3 U(1) </annotation></semantics></math></div> <p>a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>X</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex"> g_X \to \mathbf{Fields} </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}^3 U(1)}</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>String</mi></mstyle> <mstyle mathvariant="bold"><mi>c</mi></mstyle></msup></mrow><annotation encoding="application/x-tex">\mathbf{String}^{\mathbf{c}}</annotation></semantics></math>-structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> whose underlying gage bundle is the given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_X</annotation></semantics></math>.</p> <p>As before for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math>-structures, in applications one in addition demands that this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>String</mi></mstyle> <mstyle mathvariant="bold"><mi>c</mi></mstyle></msup></mrow><annotation encoding="application/x-tex">\mathbf{String}^{\mathbf{c}}</annotation></semantics></math>-structure is indeed a refinement of the field of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> of the theory, which means that one takes the <a class="existingWikiWord" href="/nlab/show/background+field">background field</a> to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo>,</mo><msub><mi>g</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Phi_X \;\colon\; X \stackrel{(\tau_X, g_X)}{\to} \mathbf{B}GL(n) \times \mathbf{B}(E_8 \times E_8) </annotation></semantics></math></div> <p>and the moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stack of fields to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>×</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; \mathbf{B}Spin(n) \stackrel{(p, \tfrac{1}{2}\mathbf{p}_1)}{\to} \mathbf{B}GL(n) \times \mathbf{B}^3 U(1) \,. </annotation></semantics></math></div> <p>These are precisely the <a class="existingWikiWord" href="/nlab/show/instanton+sectors">instanton sectors</a> of the fields of <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz anomaly free</a> <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic supergravity</a>, discussed further below in <em><a href="#HeteroticStringBackgroundField">Anomaly-free heterotic supergravity fields</a></em>. (<a href="#SSS">SSS</a>)</p> <h4 id="GaugeFields"><strong>I b)</strong> Gauge fields</h4> <h5 id="GaugeFieldsGeneral"><strong>General</strong> – Coefficients in differential cohomology: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-connections</h5> <p>The term <em>gauge field</em> in <em><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></em> with respect to a <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> refers to fields which are modeled by <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connections</a> either on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a> or on <a class="existingWikiWord" href="/nlab/show/associated+bundles">associated bundles</a> for these. The notion of <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a> between two such fields (hence between <a class="existingWikiWord" href="/nlab/show/connections+on+bundles">connections on bundles</a>) is the original meaning of the word <em><a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a></em>, even though that term is also used for equivalences between fields which are not modeled by connections.</p> <p>We discuss the general notion of gauge fields and then various special cases and variants. The following table gives an overview over the notions involved in the concept of gauge fields</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></strong>: models and components</p> <table><thead><tr><th><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong></th><th><strong><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></strong></th><th><strong><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></strong></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/instanton+sector">instanton</a>/<a class="existingWikiWord" href="/nlab/show/charge">charge</a> sector</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in underlying <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/gauge+potential">gauge potential</a></td><td style="text-align: left;">local connection <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a></td><td style="text-align: left;">local connection <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/curvature">curvature</a></td><td style="text-align: left;">underlying <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> in <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coboundary">coboundary</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/minimal+coupling">minimal coupling</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a> of <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a> of <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended Lagrangian</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universal+Chern-Simons+n-bundle">universal Chern-Simons n-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/universal+characteristic+map">universal characteristic map</a></td></tr> </tbody></table> </div> <div class="un_remark" style="border:solid #000000;background: #E6DF13;border-width:2px 1px;padding:0 1em;margin:0 1em;"> <h6 id="contents_2">Contents:</h6> <ol> <li> <p><a href="#ElectromagneticField">Electromagnetic field</a></p> </li> <li> <p><a href="#YangMillsField">Yang-Mills field</a></p> </li> <li> <p><a href="#KalbRamondBField">Kalb-Ramond B-field</a></p> </li> <li> <p><a href="#SupergeometricBField">Supergeometric B-field</a></p> </li> <li> <p><a href="#SupergravityCField">Supergravity C-field</a></p> </li> </ol> </div> <h5 id="ElectromagneticField">Electromagnetic field</h5> <p>A field configuratiotion of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> is a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle bundle with connection</a> and a <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> of the EM field is an equivalence of such connections.</p> <p>Let therefore</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>≔</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>DK</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>d</mi><mi>log</mi></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}U(1)_{conn} \coloneqq \Omega^1//U(1) \simeq DK(U(1) \stackrel{d log}{\to} \Omega^1) </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/quotient+stack">quotient stack</a> of the <a class="existingWikiWord" href="/nlab/show/action">action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of <a class="existingWikiWord" href="/nlab/show/differential+1-forms">differential 1-forms</a>, equivalently the image under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> of the sheaf of <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> given by the <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a> for degree-2 <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>, as indicated.</p> <p>This is the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a>. Hence setting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle><mo>≃</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{BgFields} \simeq *</annotation></semantics></math> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>≔</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \coloneqq \mathbf{B}U(1)_{conn} </annotation></semantics></math></div> <p>in def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> yields the type of the electromagnetic field.</p> <h5 id="YangMillsField">Yang-Mills field</h5> <p>More generally, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^1(-,\mathfrak{g})</annotation></semantics></math> the sheaf of <a class="existingWikiWord" href="/nlab/show/Lie+algebra+valued+1-forms">Lie algebra valued 1-forms</a></p> <div class="num_defn" id="ModuliStackOfGPrincipalConnection"> <h6 id="definition_13">Definition</h6> <p>The <strong>moduli stack of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a></strong> is the <a class="existingWikiWord" href="/nlab/show/quotient+stack">quotient stack</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo>≔</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}G_{conn} \coloneqq \Omega^1(-,\mathfrak{g})//G </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/action">action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on it <a class="existingWikiWord" href="/nlab/show/Lie+algebra+valued+differential+forms">Lie algebra valued differential forms</a> by <a class="existingWikiWord" href="/nlab/show/gauge+transformations">gauge transformations</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><msup><mi>A</mi> <mi>g</mi></msup><mo>≔</mo><mo>↦</mo><msup><mi>A</mi> <mi>g</mi></msup><msub><mi>Ad</mi> <mi>g</mi></msub><mi>A</mi><mo>+</mo><msup><mi>g</mi> <mo>*</mo></msup><mi>θ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> g \colon A^g \coloneqq \mapsto A^g Ad_g A + g^* \theta \,. </annotation></semantics></math></div></div> <p>This is the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>. Hence setting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle><mo>≃</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{BgFields} \simeq *</annotation></semantics></math> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>≔</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \coloneqq \mathbf{B}G_{conn} </annotation></semantics></math></div> <p>in def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> yields <a class="existingWikiWord" href="/nlab/show/Yang-Mills+fields">Yang-Mills fields</a>.</p> <h5 id="KalbRamondBField">Kalb-Ramond <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-field</h5> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mrow><msup><mi>B</mi> <mn>2</mn></msup></mrow></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>≔</mo><msup><mi>Ω</mi> <mrow><mn>2</mn><mo>≤</mo><mo>•</mo><mo>≤</mo><mn>1</mn></mrow></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>DK</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>d</mi><mi>log</mi></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mover><mo>→</mo><mi>d</mi></mover><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B^2}U(1)_{conn} \coloneqq \Omega^{2 \leq \bullet \leq 1} // \mathbf{B}U(1) \simeq DK(U(1) \stackrel{d log}{\to} \Omega^1 \stackrel{d}{\to} \Omega^2) </annotation></semantics></math></div> <p>be the image under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> of the <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a> for degree-3 <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>. This is the <a class="existingWikiWord" href="/nlab/show/moduli+infinity-stack">moduli 2-stack</a> for <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundle with connections</a>.</p> <p>Setting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle><mo>≃</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{BgFields} \simeq *</annotation></semantics></math> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>≔</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \coloneqq \mathbf{B}^2 U(1)_{conn} </annotation></semantics></math></div> <p>in def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> yields the 2-form gauge field known as the <em><a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></em> or the <em><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></em>.</p> <h5 id="KalbRamondBField">Supergeometric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>-field</h5> <p>The <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> appears both in <a class="existingWikiWord" href="/nlab/show/bosonic+string+theory">bosonic string theory</a> as well as in <a class="existingWikiWord" href="/nlab/show/type+II+superstring+theory">type II superstring theory</a>. In (<a href="#Precis">Distler-Freed-Moore</a>) it was pointed out that for the <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> the plain formulation <a href="#KalbRamondBField">above</a> needs to be refined. We discuss here the natural formulation of this observation in <a class="existingWikiWord" href="/nlab/show/smooth+super+infinity-groupoid">higher supergeometry</a> as in (<a href="#FiorenzaSatiSchreiberCSIntroAndSurvey">FSS CSIntroAndSurvey, section 4.3</a>).</p> <p>Generally instead of the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> one can consider the <a class="existingWikiWord" href="/nlab/show/multiplicative+group">multiplicative group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}^\times</annotation></semantics></math> of non-zero <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>. The canonical <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>↪</mo><msup><mi>ℂ</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">U(1) \hookrightarrow \mathbb{C}^\times</annotation></semantics></math> induces for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> a canonical morphism of <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stacks">moduli ∞-stacks</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msup><mi>ℂ</mi> <mo>×</mo></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}^n U(1) \to \mathbf{B}^n \mathbb{C}^\times \,. </annotation></semantics></math></div> <p>This is not quite an <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a> but it is a <a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a>-equivalence in that <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+cohesive+%E2%88%9E-groupoids">geometric realization of cohesive ∞-groupoids</a> sends it to an equivalence as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow><mo>≃</mo><mrow><mo stretchy="false">|</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msup><mi>ℂ</mi> <mo>×</mo></msup><mo stretchy="false">|</mo></mrow><mo>≃</mo><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> {\vert \mathbf{B}^n U(1)\vert} \simeq {\vert \mathbf{B}^n \mathbb{C}^\times\vert} \simeq K(\mathbb{Z}, n+1) \,. </annotation></semantics></math></div> <p>For instance smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}^\times</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a> are both classified by the universal <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^2(-, \mathbb{Z})</annotation></semantics></math> but the <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge</a> <a class="existingWikiWord" href="/nlab/show/automorphisms">automorphisms</a> of the trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-principal bundle form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(-,U(1))</annotation></semantics></math>, while that of the trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}^\times</annotation></semantics></math>-principal bundle form the larger <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mi>ℂ</mi> <mo>×</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(-,\mathbb{C}^\times)</annotation></semantics></math>.</p> <p>Both versions of the moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-stack of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-bundles have their use. The version <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msup><mi>ℂ</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">\mathbf{B}^n \mathbb{C}^\times</annotation></semantics></math> has the advantage that it is actually equivalent to the moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-stack of complex line <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-bundles.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><msup><mi>ℂ</mi> <mo>×</mo></msup><mo>≃</mo><mi>n</mi><msub><mi>Line</mi> <mi>ℂ</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}^n \mathbb{C}^\times \simeq n Line_\mathbb{C} \,. </annotation></semantics></math></div> <p>Specifically for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n = 2</annotation></semantics></math> it is equivalent to the moduli 2-stack of <a class="existingWikiWord" href="/nlab/show/line+2-bundles">line 2-bundles</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msup><mi>ℂ</mi> <mo>×</mo></msup><mo>≃</mo><mn>2</mn><msub><mi>Line</mi> <mi>ℂ</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}^2 \mathbb{C}^\times \simeq 2 Line_\mathbb{C} \,. </annotation></semantics></math></div> <p>But now in <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> <a href="http://ncatlab.org/nlab/show/line%202-bundle#SuperLine2BundlesAndTwistedK">complex super-line 2-bundle</a> have a richer classification that plain <a class="existingWikiWord" href="/nlab/show/line+2-bundle">line 2-bundle</a>. Explicitly, let now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo lspace="verythinmathspace">:</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} \colon </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SmoothSuper%E2%88%9EGrpd">SmoothSuper∞Grpd</a> be the ambient <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> for higher <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mstyle mathvariant="bold"><mi>sLine</mi></mstyle> <mi>ℂ</mi></msub><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> 2\mathbf{sLine}_{\mathbb{C}} \in \mathbf{H} </annotation></semantics></math></div> <p>for the 2-stack of <a href="http://ncatlab.org/nlab/show/line%202-bundle#SuperLine2BundlesAndTwistedK">complex super-line 2-bundle</a>.</p> <p>This has <a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos">homotopy sheaves</a></p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mn>2</mn><mstyle mathvariant="bold"><mi>sLine</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_k( 2 \mathbf{sLine})</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">3</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℂ</mi> <mo>×</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}^\times</annotation></semantics></math></td></tr> </tbody></table> <p>in the <a class="existingWikiWord" href="/nlab/show/topos">topos</a> over <a class="existingWikiWord" href="/nlab/show/supermanifolds">supermanifolds</a>. The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mn>2</mn><mstyle mathvariant="bold"><mi>sLine</mi></mstyle><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\pi_0(2 \mathbf{sLine}) \simeq \mathbb{Z}_2</annotation></semantics></math> says that locally there is not just one super line 2-bundle, namely the standard line 2-bundle, but also its odd “superpartner”.</p> <p>Under <a class="existingWikiWord" href="/nlab/show/geometric+realization+of+cohesive+infinity-groupoids">geometric realization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo></mrow><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>→</mo><mn>∞</mn><mi>Grpd</mi><mo>≃</mo><msub><mi>L</mi> <mi>whe</mi></msub><mi>Top</mi></mrow><annotation encoding="application/x-tex">{\vert-\vert} \;\colon\; \mathbf{H} \to \infty Grpd \simeq L_{whe} Top</annotation></semantics></math> we have the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mn>2</mn><mstyle mathvariant="bold"><mi>sLine</mi></mstyle><mo stretchy="false">|</mo></mrow></mrow><annotation encoding="application/x-tex">{\vert 2 \mathbf{sLine}\vert}</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a></p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mrow><mo stretchy="false">|</mo><mn>2</mn><mstyle mathvariant="bold"><mi>sLine</mi></mstyle><mo stretchy="false">|</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_k({\vert 2 \mathbf{sLine}\vert})</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;">0</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">3</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">4</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math></td></tr> </tbody></table> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mstyle mathvariant="bold"><mi>sLine</mi></mstyle></mrow><annotation encoding="application/x-tex">2 \mathbf{sLine}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Picard+3-group">Picard 3-group</a> of the monoidal 2-stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mstyle mathvariant="bold"><mi>sVect</mi></mstyle></mrow><annotation encoding="application/x-tex">2 \mathbf{sVect}</annotation></semantics></math> of super <a class="existingWikiWord" href="/nlab/show/2-vector+bundles">2-vector bundles</a> it is a supergeometric <a class="existingWikiWord" href="/nlab/show/3-group">3-group</a>. Therefore there is the further <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mn>2</mn><mstyle mathvariant="bold"><mi>sLine</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{B}2\mathbf{sLine}</annotation></semantics></math> and hence the differential refinement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><msub><mstyle mathvariant="bold"><mi>sLine</mi></mstyle> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">2\mathbf{sLine}_{conn}</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mn>2</mn><msub><mstyle mathvariant="bold"><mi>sLine</mi></mstyle> <mi>conn</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msubsup><mi>Ω</mi> <mi>cl</mi> <mn>3</mn></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>2</mn><mstyle mathvariant="bold"><mi>sLine</mi></mstyle></mtd> <mtd><mover><mo>→</mo><mi>curv</mi></mover></mtd> <mtd><msub><mo>♭</mo> <mi>dR</mi></msub><mstyle mathvariant="bold"><mi>B</mi></mstyle><mn>2</mn><mstyle mathvariant="bold"><mi>sLine</mi></mstyle></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ 2\mathbf{sLine}_{conn} &\to& \Omega^3_{cl} \\ \downarrow && \downarrow \\ 2 \mathbf{sLine} &\stackrel{curv}{\to}& \flat_{dR} \mathbf{B}2 \mathbf{sLine} } \,. </annotation></semantics></math></div> <p>This is now the moduli 2-stack</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>≔</mo><mn>2</mn><msub><mstyle mathvariant="bold"><mrow><mn>2</mn><mi>Line</mi></mrow></mstyle> <mi>conn</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>SmoothSuper</mi><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \coloneqq 2\mathbf{2Line}_{conn} \;\; \in SmoothSuper\infty Grpd </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> in <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a>.</p> <h5 id="SupergravityCField">Supergravity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-field</h5> <p>Proceeding in this fashion, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mrow><msup><mi>B</mi> <mn>3</mn></msup></mrow></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>≔</mo><msup><mi>Ω</mi> <mrow><mn>3</mn><mo>≤</mo><mo>•</mo><mo>≤</mo><mn>1</mn></mrow></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>DK</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>d</mi><mi>log</mi></mrow></mover><msup><mi>Ω</mi> <mn>1</mn></msup><mover><mo>→</mo><mi>d</mi></mover><msup><mi>Ω</mi> <mn>2</mn></msup><mover><mo>→</mo><mi>d</mi></mover><msup><mi>Ω</mi> <mn>3</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B^3}U(1)_{conn} \coloneqq \Omega^{3 \leq \bullet \leq 1} // \mathbf{B}^2 U(1) \simeq DK(U(1) \stackrel{d log}{\to} \Omega^1 \stackrel{d}{\to} \Omega^2 \stackrel{d}{\to}\Omega^3) </annotation></semantics></math></div> <p>be the image under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> of the <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a> for degree-4 <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>. This is the <a class="existingWikiWord" href="/nlab/show/moduli+infinity-stack">moduli 3-stack</a> for <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle with connections</a>.</p> <p>Fields whose moduli stack is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; \mathbf{B}^3 U(1)_{conn} \to * </annotation></semantics></math></div> <p>are one component of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a>.</p> <h3 id="MatterFields"><strong>II)</strong> Matter fields</h3> <p>Fundamental <a class="existingWikiWord" href="/nlab/show/matter">matter</a>-fields as they appear in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a>, hence fundamental <a class="existingWikiWord" href="/nlab/show/fermions">fermions</a> such as <a class="existingWikiWord" href="/nlab/show/electrons">electrons</a>, <a class="existingWikiWord" href="/nlab/show/quarks">quarks</a> and <a class="existingWikiWord" href="/nlab/show/neutrinos">neutrinos</a>, are <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math> on <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> of the <a class="existingWikiWord" href="/nlab/show/force">force</a> fields that interact with the matter fields and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a> to the <a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> underlying such a force <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a>, as discussed in <em><a href="#GaugeFields">Gauge fields</a></em> above, and, if we are talking indeed about fermions, to the <a class="existingWikiWord" href="/nlab/show/spin+bundle">spin bundle</a> given by the gravity-spin structure field discussed in <em><a href="#MatterFields">Gravity and generalized geometry</a></em> above.</p> <p>More precisely, fermioninc matter fields are sections of these bundles regarded in <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> with the fibers regarded as odd-graded. (…)</p> <h4 id="SectionsOfAssociatedBundles"><strong>General</strong> – sections of associated fiber <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-bundles</h4> <p>We start by discussing the general mechanism by which <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundles">associated ∞-bundles</a> are an example of the general definition <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mo lspace="0em" rspace="thinmathspace">Grp</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in \Grp(\mathbf{H})</annotation></semantics></math> a geometric <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>, there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mi>Act</mi><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> G Act \simeq \mathbf{H}_{/\mathbf{B}G} </annotation></semantics></math></div> <p>between that of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-actions">∞-actions</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> over the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. Under this equivalence an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">V \in \mathbf{H}</annotation></semantics></math> is equivalently a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>V</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mover><mi>ρ</mi><mo>¯</mo></mover></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ V &\to& V//G \\ && \downarrow^{\overline \rho} \\ && \mathbf{B}G } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. This is the <a class="existingWikiWord" href="/nlab/show/universal+associated+%E2%88%9E-bundle">universal rho-associated ∞-bundle</a>: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> with modulating map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">g_X \;\colon\; X \to \mathbf{B}G</annotation></semantics></math> the corresponding <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a> is naturally equivalent to the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ρ</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{\rho}</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi><mo>≃</mo><msubsup><mi>g</mi> <mi>X</mi> <mo>*</mo></msubsup><mo stretchy="false">(</mo><mover><mi>ρ</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> P \times_G V \simeq g_X^*(\overline{\rho}) \,. </annotation></semantics></math></div> <p>From this and the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> one finds that a <a class="existingWikiWord" href="/nlab/show/section">section</a> of the <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>V</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \times_G V \to X</annotation></semantics></math> is equivalently a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><msub><mi>g</mi> <mi>X</mi></msub><mo>→</mo><mover><mi>ρ</mi><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex"> \phi : g_X \to \overline{\rho} </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}G}</annotation></semantics></math>.</p> <p>This means that such sections are fields in the sense of def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BgFields</mi></mstyle><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{BgFields} \simeq \mathbf{B}G</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/background+field">background field</a> is the moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">g_X</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of fields is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>ρ</mi><mo>¯</mo></mover><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\overline{\rho} \in \mathbf{H}_{/\mathbf{B}G}</annotation></semantics></math>.</p> </li> </ul> <h4 id="Fermions">Fermions</h4> <p><a class="existingWikiWord" href="/nlab/show/fermion">fermion</a>:</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/spin+representation">spin representation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>S</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>Spin</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; S//Spin \to \mathbf{B}Spin </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> (…)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/electron">electron</a>, <a class="existingWikiWord" href="/nlab/show/quark">quark</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravitino">gravitino</a> (<a class="existingWikiWord" href="/nlab/show/Rarita-Schwinger+field">Rarita-Schwinger field</a>), <a class="existingWikiWord" href="/nlab/show/gluino">gluino</a></p> </li> </ul> <h4 id="TensorFields">Tensor fields</h4> <p>While the term <em>physical field</em> probably orignates from <em><a class="existingWikiWord" href="/nlab/show/tensor+field">tensor field</a></em>, few fields are fundamentally given by tensor fields. Nevertheless, tensor fields, being <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of a <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of copies of the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> and the <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a> are certainly examples of the general notion of field as in def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a>. Gere we spell this out.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, a <a class="existingWikiWord" href="/nlab/show/tensor+field">tensor field</a> of rank <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p,q)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/section">section</a> of the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> bundle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><msup><mo stretchy="false">)</mo> <mrow><msub><mo>⊗</mo> <mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mi>p</mi></mrow></msup><msub><mo>⊗</mo> <mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mi>Γ</mi><mo stretchy="false">(</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>X</mi><msup><mo stretchy="false">)</mo> <mrow><mo>⊗</mo><mi>q</mi></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma(T X)^{\otimes_{C^\infty(X)} p} \otimes_{C^\infty(X)} \Gamma(T^* X)^{\otimes q} \,. </annotation></semantics></math></div> <p>This is canonically the <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> which to which the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> is also canonically associated, by the given <a class="existingWikiWord" href="/nlab/show/representation">representation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL(n)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n(p+q)}</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/universal+associated+%E2%88%9E-bundle">universal associated ∞-bundle</a> of this representation is</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></msup><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Fields} \;\colon\; \mathbb{R}^{n (p+q)}//GL(n) \to \mathbf{B}GL(n) </annotation></semantics></math>.</p> <p>Hence for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_X \colon X \to \mathbf{B}GL(n)</annotation></semantics></math> the canonical map, the space of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p,q)</annotation></semantics></math>-tensor fields on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>τ</mi> <mi>X</mi></msub><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\tau_X, \mathbf{Fields}]_{\mathbf{H}} \in \mathbf{H} \,. </annotation></semantics></math></div> <p>By passing to <a class="existingWikiWord" href="/nlab/show/irreducible+representations">irreducible representations</a> in the above, we obtain specifically the corresponding subclasses of tensor fields, such as <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a>.</p> <h3 id="FieldCombiningVariousProperties">Fields combining these properties</h3> <p>The above distinction of types of physical fields into into <a href="SigmaModelFields">Sigma-model fields</a>, <a href="#FieldsOfGravityAndGeneralizedGeometry">Force fields of gravity</a>, <a href="#GaugeFields">Gauge force fields</a> and <a href="#MatterFields">Matter fields</a> can be helpful for reasoning about fields and types of <a class="existingWikiWord" href="/nlab/show/theories+%28physics%29">theories in physics</a>, but is not fundamental. The general definition <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> of which all these types are examples reflects that. In fact, one way to read that definition and the above list of examples is to say that it shows that in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> <em>all</em> types of fields are <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> fields: they are all given as maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\Phi_X \to \mathbf{Fields}</annotation></semantics></math> from a domain <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>+<a class="existingWikiWord" href="/nlab/show/background+field">background field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_X</annotation></semantics></math> to a generalized <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Fields}</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of fields in the given <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> characterized by the nature of the <a class="existingWikiWord" href="/nlab/show/background+fields">background fields</a>).</p> <p>Accordingly, a general physical field in a general <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a> does not fall squarely into one of the above categories, but combines aspects of all of these. Here we discuss such examples.</p> <h4 id="TwistedDifferentialcStructurs"><strong>General</strong> – Twisted differential-cocycles and -<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math>-structures</h4> <p>We have seen that the <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of a field of plain <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> or general geometry is a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{c} \colon \mathbf{B}\hat G \to \mathbf{B}G</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a>, regarded as an object in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}G}</annotation></semantics></math>, and that the <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of a plain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a> is that of <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connections">principal ∞-connections</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}G_{conn}</annotation></semantics></math>. These two concepts have an evident unification if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math> has a differential refinement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">at</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\at \mathbf{c}</annotation></semantics></math> to a map of differential moduli stacks</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mover><mi>G</mi><mo stretchy="false">^</mo></mover> <mi>conn</mi></msub></mtd> <mtd><mover><mo>→</mo><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mover><mo>→</mo><mstyle mathvariant="bold"><mi>c</mi></mstyle></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}\hat G_{conn} &\stackrel{\hat \mathbf{c}}{\to}& \mathbf{B}G_{conn} \\ \downarrow && \downarrow \\ \mathbf{B}\hat G &\stackrel{\mathbf{c}}{\to}& \mathbf{B}G } \,, </annotation></semantics></math></div> <p>regarded then as an object in the slice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}G_{conn}}</annotation></semantics></math>.</p> <p>Since, as discussed <a href="#GStructure">above</a>, a field with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math> is equivalently a <em>twisted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>c</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{c}</annotation></semantics></math>-structure</em>, we may calla field with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat \mathbf{c}</annotation></semantics></math> a <strong><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structure">twisted differential c-structure</a></strong>.</p> <p>Moreover, we have seen that <a href="#MatterFields">matter fields</a> have <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stacks">moduli ∞-stacks</a> coming not from a direct <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>≃</mo><mo>*</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \simeq *//G</annotation></semantics></math> of an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, but from the <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">V//G</annotation></semantics></math> of an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on some object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>. Combining this with differential refinement as above we consider fields whose <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stacks">moduli ∞-stacks</a> are maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mover><mo>→</mo><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover></mover><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (V//G)_{conn} \stackrel{\hat \mathbf{c}}{\to} (\mathbf{B}G)_{conn} \;\;\;\; \in \mathbf{H}_{/\mathbf{B}G_{conn}} \,. </annotation></semantics></math></div> <p>This is a differential refinement of fields which are <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> in <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> with local <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>, hence <strong>twisted differential cohomology</strong>. The above case of <a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a> is a special case of this for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex">V \simeq \mathbf{B}A</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> that <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">extended</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>G</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat G</annotation></semantics></math>.</p> <p>Some further slight variants of these combinations appear in the examples below.</p> <h4 id="NonabelianChargedParticle">Nonabelian charged particle trajectories – Wilson line</h4> <p>We describe here a variant of the particle propagating on a spacetime <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, where now the particle is charged under a <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a> for a nonabelian <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a> which is connected, simply connected and <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/irreducible+representation">irreducible</a> <a class="existingWikiWord" href="/nlab/show/unitary+representation">unitary representation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> under which the particle is to be <a class="existingWikiWord" href="/nlab/show/charge+%28physics%29">charged</a>. By the <a class="existingWikiWord" href="/nlab/show/Borel-Weil-Bott+theorem">Borel-Weil-Bott theorem</a> this corresponds equivalently to a <a class="existingWikiWord" href="/nlab/show/weight+%28in+representation+theory%29">weight</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>λ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo><mo>∈</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \langle \lambda, -\rangle \in \mathfrak{g}^* \,. </annotation></semantics></math></div> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>≃</mo><msub><mi>G</mi> <mi>λ</mi></msub><mo>↪</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">T \simeq G_\lambda \hookrightarrow G</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/maximal+torus">maximal torus</a> which is the <a class="existingWikiWord" href="/nlab/show/stabilizer+subgroup">stabilizer subgroup</a> that fixes this weight under the <a class="existingWikiWord" href="/nlab/show/coadjoint+action">coadjoint action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on its dual <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathfrak{g}^*</annotation></semantics></math>.</p> <p>Set then in def. <a class="maruku-ref" href="#FieldsInAnActionFunctional"></a> the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of <a class="existingWikiWord" href="/nlab/show/background+fields">background fields</a> to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BgField</mi></mstyle><mo lspace="verythinmathspace">:</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex"> \mathbf{BgField} \colon \Omega^1(-,\mathfrak{g})//G \simeq \mathbf{B}G_{conn} \;\;\; \mathbf{H} </annotation></semantics></math></div> <p>as in def. <a class="maruku-ref" href="#ModuliStackOfGPrincipalConnection"></a>. Moreover, let the moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stack of fields in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}G_{conn}}</annotation></semantics></math> be given by the canonical map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>T</mi><mo>→</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mo>≃</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; \Omega^1(-,\mathfrak{g})//T \to \Omega^1(-,\mathfrak{g})//G \simeq \mathbf{B}G_{conn} </annotation></semantics></math></div> <p>which is induced by the defining inclusion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>.</p> <div class="num_remark"> <h6 id="remark_14">Remark</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">U \in</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\hookrightarrow \mathbf{H}</annotation></semantics></math> a field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>→</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">\phi \colon U \to \Omega^1(-,\mathfrak{g})//T</annotation></semantics></math> is equivalently a <a class="existingWikiWord" href="/nlab/show/Lie+algebra+valued+form">Lie algebra valued form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in \Omega^1(U,\mathfrak{g})</annotation></semantics></math>, but a <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> of such a field is constrained to be a smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>↪</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">t \in C^\infty(U,T) \hookrightarrow C^\infty(U, G)</annotation></semantics></math> instead of an arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-valued function.</p> </div> <div class="num_remark"> <h6 id="remark_15">Remark</h6> <p>With these definitions we have for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> that</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/background+gauge+field">background gauge field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_X \;\colon\; X \to \mathbf{B}G_{conn}</annotation></semantics></math> is equivalently a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connection">principal connection</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> (which if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is assumed connected and simply connected and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> is of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> at most 3 is equivalently just a <a class="existingWikiWord" href="/nlab/show/Lie+algebra+valued+form">Lie algebra valued form</a>);</p> </li> <li> <p>a field configuration is a diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msup><mo>∇</mo> <mi>g</mi></msup></mrow></mover></mtd> <mtd></mtd> <mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>T</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mo>∇</mo></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>g</mi></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ X &&\stackrel{\nabla^g}{\to}&& \Omega^1(-,\mathfrak{g})//T \\ & {}_{\nabla}\searrow &\swArrow_{g}& \swarrow_{\mathrlap{\mathbf{Fields}}} \\ && \mathbf{B}G_{conn} } \,, </annotation></semantics></math></div> <p>which is equivalently a differentially refined <a class="existingWikiWord" href="/nlab/show/reduction+of+the+structure+group">reduction of the structure group</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> is given by a globally defined connection form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> then this is equivalently just a smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-valued function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> that takes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mi>g</mi></msup></mrow><annotation encoding="application/x-tex">A^g</annotation></semantics></math> as indicated.</p> </li> </ul> </div> <p>As a slight variant of prop. <a class="maruku-ref" href="#CosetIsHomotopyFiberOfDeloopedInclusion"></a> we have</p> <div class="num_prop"> <h6 id="proposition_14">Proposition</h6> <p>We have a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>λ</mi></msub><mo>≃</mo><mi>G</mi><mo stretchy="false">/</mo><mi>T</mi><mo>→</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>T</mi><mover><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{O}_\lambda \simeq G/T \to \Omega^1(-,\mathfrak{g})//T \stackrel{\mathbf{Fields}}{\to} \mathbf{B}G_{conn} \,, </annotation></semantics></math></div> <p>where on the left we have the <a class="existingWikiWord" href="/nlab/show/coadjoint+orbit">coadjoint orbit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>λ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle \lambda , -\rangle</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_16">Remark</h6> <p>This implies that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\nabla = 0</annotation></semantics></math> is the trivial background field, than fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\phi : X \to \mathbf{Fields}</annotation></semantics></math> are equivalently maps to the coadjoint orbit</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><msub><mo stretchy="false">|</mo> <mrow><msub><mi>Φ</mi> <mi>X</mi></msub><mo>=</mo><mn>0</mn></mrow></msub><mo>≃</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>𝒪</mi> <mi>λ</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [X, \mathbf{Fields}]_{\mathbf{H}}|_{\Phi_X = 0} \simeq C^\infty(X, \mathcal{O}_\lambda) \,. </annotation></semantics></math></div> <p>Hence in this sector we have simply a <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> field as in <a href="#SigmaModelFields">Sigma-models</a> above.</p> </div> <div class="num_remark"> <h6 id="remark_17">Remark</h6> <p>There is a canonical <a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended Lagrangian</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><msub><mo stretchy="false">]</mo> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub></mrow><annotation encoding="application/x-tex">[X, \mathbf{Fields}]_{\mathbf{H}}</annotation></semantics></math> with the above definitions, whose <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi>S</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">X = S^1</annotation></semantics></math> is the closed connected 1-dimensional manifold, is that of a <a class="existingWikiWord" href="/nlab/show/1d+Chern-Simons+theory">1d Chern-Simons theory</a>. The <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> is the <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a> map – the “<a class="existingWikiWord" href="/nlab/show/Wilson+line">Wilson line</a>” – on the background gauge field connection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>.</p> </div> <p>This is discussed further at <a href="#ActionFunctionalsForChernSimonsTypeGaugeTheories">geometry of physics – Prequantum gauge theory and gravity</a>.</p> <h4 id="ChernSimonsWithWilsonLines">3d Chern-Simons field with Wilson line</h4> <p>We discuss the field content of 3d <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> for a simple, simply connected compact Lie group with <a class="existingWikiWord" href="/nlab/show/Wilson+loops">Wilson loops</a>. This is an example of <em><a href="#BoundaryAndDefectFields">Bulk fields with defect fields</a></em>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>3</mn></msub><mo>∈</mo><mi>SmthMfd</mi><mo>↪</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\Sigma_3 \in SmthMfd \hookrightarrow \mathbf{H}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> 3, and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><msup><mi>Σ</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex"> C_X \;\colon\; S^1 \to \Sigma^3 </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/submanifold">embedding</a> of a <a class="existingWikiWord" href="/nlab/show/knot">knot</a>. Let the moduli of fields be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>:</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔠</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>T</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} : \Omega^1(-,\mathfrak{c})//T \to \mathbf{B}G_{conn} </annotation></semantics></math></div> <p>as defined in <em><a href="#NonabelianChargedParticle">Nonabelian charged particle trajectories – Wilson lines</a></em> above. Regarding the not inclusion as a defect in the 3-dimnensional manifold, a bulk-defect fiedl configuration according to def. <a class="maruku-ref" href="#ModuliOfBulkAndBoundaryFields"></a> is a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>C</mi> <mi>X</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; C_X \to \mathbf{Fields} </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^{(\Delta^1)}</annotation></semantics></math>. This is equivalently a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mrow><mi>A</mi><msubsup><mo stretchy="false">|</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow> <mi>g</mi></msubsup></mrow></mover></mtd> <mtd><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>T</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>C</mi></mpadded></msup></mtd> <mtd><msub><mo>⇙</mo> <mi>g</mi></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>Σ</mi> <mn>3</mn></msub></mtd> <mtd><mover><mo>→</mo><mi>A</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^1 &\stackrel{A|_{S^1}^g}{\to}& \Omega^1(-,\mathfrak{g})//T \\ \downarrow^{\mathrlap{C}} &\swArrow_{g}& \downarrow \\ \Sigma_3 &\stackrel{A}{\to}& \mathbf{B}G_{conn} } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. This in turn is equivalently</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/Lie+algebra+valued+form">Lie algebra valued form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>Σ</mi> <mn>3</mn></msub><mo>,</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in \Omega^1(\Sigma_3, \mathfrak{g})</annotation></semantics></math> (the bulk <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a>)</p> </li> <li> <p>a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math>-valued function on the circle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g \in C^\infty(S^1, G)</annotation></semantics></math></p> </li> </ol> <p>which determine a background gauge field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msubsup><mo stretchy="false">|</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow> <mi>g</mi></msubsup></mrow><annotation encoding="application/x-tex">A|_{S^1}^g</annotation></semantics></math> on the knot.</p> <p>Moreover a <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> between two such field configurations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ϕ</mi><mo>⇒</mo><mi>ϕ</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\kappa \;\colon\; \phi \Rightarrow \phi'</annotation></semantics></math> is equivalently a gauge transformaiton of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo stretchy="false">|</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub></mrow><annotation encoding="application/x-tex">A|_{S^1}</annotation></semantics></math> such that together they intertwine <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g'</annotation></semantics></math>. In particular if the bulk field is held fixed, then such a gauge transformation is a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo lspace="verythinmathspace">:</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">t \colon S^1 \to T</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>′</mo><mo>=</mo><mi>t</mi><mi>g</mi></mrow><annotation encoding="application/x-tex">g' = t g</annotation></semantics></math>. This means that the gauge equivalence classes of field confiurations for fixed background gauge field are labeled by maps to the <a class="existingWikiWord" href="/nlab/show/coadjoint+orbit">coadjoint orbit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>λ</mi></msub><mo>≃</mo><mi>G</mi><mo stretchy="false">/</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}_\lambda \simeq G/T</annotation></semantics></math> as above.</p> <p>This are the field confugurations for 3d <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> (see the discussion there) with Wilson lines (<a href="#FiorenzaSatiSchreiberCSIntroAndSurvey">FSS</a>).</p> <h4 id="ChanPatonGaugeFields">Chan-Paton gauge fields on D-branes: twisted differential K-cocycles</h4> <p>We discuss the <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+fields">Chan-Paton gauge fields</a> over <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> in <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/extension+of+groups">extension of groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1) \to U(n) \to PU(n)</annotation></semantics></math> sits in a long <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>PU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>PU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mstyle mathvariant="bold"><mi>dd</mi></mstyle> <mi>n</mi></msub></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> U(1) \to U(n) \to PU(n) \to \mathbf{B}U(1) \to \mathbf{B}U(n) \to \mathbf{B}PU(n) \stackrel{\mathbf{dd}_n}{\to} \mathbf{B}^2 U(1) </annotation></semantics></math></div> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \colon (\mathbf{B}U(n)//\mathbf{B}U(1))_{conn} \to \mathbf{B}^2 U(1)_{conn} </annotation></semantics></math></div> <p>be the differential refinement of that universal <a class="existingWikiWord" href="/nlab/show/Dixmier-Douady+class">Dixmier-Douady class</a>.</p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Q</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \iota_X \;\colon\; Q \hookrightarrow X </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/submanifold">submanifold</a>, to be thought of as a <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> in an ambient <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>Then a field configuration of the <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> together with a compatible rank-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> gauge field on the <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a> is a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mi>X</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex"> \iota_X \to \mathbf{Fields} </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{H}^{(\Delta^1)}</annotation></semantics></math>, hence a diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Q</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mrow><msub><mi>ι</mi> <mi>X</mi></msub></mrow></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mo>≃</mo></msub></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mover><mstyle mathvariant="bold"><mi>dd</mi></mstyle><mo stretchy="false">^</mo></mover> <mi>n</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mo>∇</mo> <mi>B</mi></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Q &\stackrel{}{\to}& (\mathbf{B}U(n)//\mathbf{B}U(1)) \\ {}^{\iota_X}\downarrow &\swArrow_{\simeq}& \downarrow^{\mathrlap{\hat \mathbf{dd}_n}} \\ X &\stackrel{\nabla_B}{\to}& \mathbf{B}^2 U(1)_{conn} } </annotation></semantics></math></div> <p>This identifies a <a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a> with connection on the D-brane whose twist is the class in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^3(X, \mathbb{Z})</annotation></semantics></math> of the bulk <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a>.</p> <p>This relation is the Kapustin-part of the <a class="existingWikiWord" href="/nlab/show/Freed-Witten-Kapustin+anomaly">Freed-Witten-Kapustin anomaly</a> cancellation for the <a class="existingWikiWord" href="/nlab/show/bosonic+string">bosonic string</a> or else for the <a class="existingWikiWord" href="/nlab/show/type+II+string">type II string</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Spin</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">Spin^c</annotation></semantics></math> D-branes. (<a href="#FiorenzaSatiSchreiberCSIntroAndSurvey">FSS</a>)</p> <div class="num_remark"> <h6 id="remark_18">Remark</h6> <p>If we regard the <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> as a <a class="existingWikiWord" href="/nlab/show/background+field">background field</a> for the <a class="existingWikiWord" href="/nlab/show/Chan-Paton+gauge+field">Chan-Paton gauge field</a>, then remark <a class="maruku-ref" href="#PullbackAlongGeneralizedLocalDiffeomorphisms"></a> determines along which maps of the B-field the Chan-Paton gauge field may be transformed.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Y &\stackrel{}{\to}& X &\stackrel{}{\to}& (\mathbf{B}U(n)//\mathbf{B}U(1))_{conn} \\ & \searrow & \downarrow & \swarrow \\ &&\mathbf{B}^2 U(1)_{conn} } \,. </annotation></semantics></math></div> <p>On the local connection forms this acts as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↦</mo><mi>A</mi><mo>+</mo><mi>α</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A \mapsto A + \alpha \,. </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>↦</mo><mi>B</mi><mo>+</mo><mi>d</mi><mi>α</mi></mrow><annotation encoding="application/x-tex"> B \mapsto B + d \alpha </annotation></semantics></math></div> <p>This is the famous gauge transformation law known from the string theory literature.</p> </div> <h4 id="HeteroticStringBackgroundField">Anomaly-free heterotic string background: differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>String</mi> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">String^c</annotation></semantics></math>-structure</h4> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>Spin</mi> <mi>conn</mi></msub><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; \mathbf{B}Spin_{conn} \stackrel{\tfrac{1}{2}\hat \mathbf{p}_1}{\to} \mathbf{B}^3 U(1)_{conn} </annotation></semantics></math></div> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>→</mo><mrow><msub><mi>g</mi> <mi>X</mi></msub></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>8</mn></msub><mo>×</mo><msub><mi>E</mi> <mn>8</mn></msub><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mover><mo>→</mo><mrow><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Phi_X \;\colon\; X \stackrel{g_X}{\to} \mathbf{B}(E_8 \times E_8)_{conn} \stackrel{\hat \mathbf{c}_2}{\to} \mathbf{B}^3 U(1) </annotation></semantics></math></div> <p>then fields are <a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structures">twisted differential string structures</a> or equivalently differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>String</mi></mstyle> <mstyle mathvariant="bold"><mi>c</mi></mstyle></msup></mrow><annotation encoding="application/x-tex">\mathbf{String}^{\mathbf{c}}</annotation></semantics></math>-structure with underlying gauge bundle give by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\Phi_X</annotation></semantics></math>, the differential refinement of the discussion in <em><a href="#HigherSpinStructures">Higher spin structure</a></em> above.</p> <p>As in the discussion there, we implement the constraint that the string structure is on the tangent bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="0em" rspace="thinmathspace">colol</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\tau_X \;\colol\; X \to \mathbf{B}GL(n)</annotation></semantics></math> of the manifold by settting</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>Spin</mi> <mi>conn</mi></msub><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>,</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; \mathbf{B}Spin_{conn} \stackrel{(p,\tfrac{1}{2}\hat \mathbf{p}_1)}{\to} \mathbf{B}GL(n)_conn \mathbf{B}^3 U(1)_{conn} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Φ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mo>∇</mo> <mi>X</mi></msub><mo>,</mo><msub><mover><mstyle mathvariant="bold"><mi>c</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>g</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>GL</mi><mo stretchy="false">(</mo><mi>n</mi><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>×</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Phi_X \;\colon\; X \stackrel{(\nabla_X, \hat \mathbf{c}_2(g_X))}{\to} \mathbf{B}GL(n)_{conn} \times \mathbf{B}^3 U(1)_{conn} \,. </annotation></semantics></math></div> <p>Then a field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Φ</mi> <mi>X</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\phi \;\colon\; \Phi_X \to \mathbf{Fields}</annotation></semantics></math> is the higher spin-connection version as discussed in <em><a href="#OrdinaryGravity">Gravity</a></em> above of a twisted differential string structure.</p> <p>The moduli stack of these fields is that of background fields that satisfy the <a class="existingWikiWord" href="/nlab/show/Green-Schwarz+anomaly+cancellation">Green-Schwarz anomaly cancellation</a> in <a class="existingWikiWord" href="/nlab/show/heterotic+supergravity">heterotic supergravity</a>. (<a href="#SSS">SSS</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+history">field history</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+observable">field observable</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a></p> <ul> <li> <p><strong>physical field</strong></p> <p><a class="existingWikiWord" href="/nlab/show/free+field">free field</a></p> <p><a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></p> <p><a class="existingWikiWord" href="/nlab/show/superfield">superfield</a></p> <p><a class="existingWikiWord" href="/nlab/show/auxiliary+field">auxiliary field</a></p> <p><a class="existingWikiWord" href="/nlab/show/fiber+bundles+in+physics">fiber bundles in physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/antifield">antifield</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ghost">ghost</a>, <a class="existingWikiWord" href="/nlab/show/antighost">antighost</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+Lagrangian">extended Lagrangian</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantum n-bundle</a>, <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/charge+%28physics%29">charge (physics)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbation+theory">perturbation theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a>, <a class="existingWikiWord" href="/nlab/show/tachyon">tachyon</a></li> </ul> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/bulk+field+theory">bulk field theory</a></th><th><a class="existingWikiWord" href="/nlab/show/boundary+field+theory">boundary field theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_09f5a372cf29c6ceb81738a718c5b5bf0f4bac5c_1"><semantics><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n+1</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_09f5a372cf29c6ceb81738a718c5b5bf0f4bac5c_2"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/source+field">source</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/correlation+function">correlation function</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/conformal+blocks">conformal blocks</a></td></tr> </tbody></table> </div> <ul> <li><a class="existingWikiWord" href="/nlab/show/concept+with+an+attitude">concept with an attitude</a></li> </ul> <h2 id="references">References</h2> <h3 id="lecture_notes_and_expositions">Lecture notes and expositions</h3> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+perturbative+quantum+field+theory">geometry of physics – perturbative quantum field theory</a></em>, chaper <em><a href="https://ncatlab.org/nlab/show/geometry+of+physics+--+perturbative+quantum+field+theory#Fields">3. Fields</a></em></li> </ul> <p>A survey of the main field species is given in</p> <ul> <li id="Kocic16"><a class="existingWikiWord" href="/nlab/show/Mikica+Kocic">Mikica Kocic</a>, <em>Overview of the fields in QFT</em>, 2016 (<a class="existingWikiWord" href="/nlab/files/FieldSpecies.pdf" title="pdf">pdf</a>)</li> </ul> <p>Most of the above material as of 2013 was written as part of a lecture series</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em>Higher Chern-Simons theory</em>, lecture series at <em><a href="http://www2.ims.nus.edu.sg/Programs/013wquantum/index.php">Workshop on Topological Aspects of Quantum Field Theories, Singapore (14 - 18 Jan 2013)</a></em> on the first sections at <em><a href="geometry%20of%20physics#PHYSICS">geometry of physics, II) Physics</a></em></li> </ul> <p>A exposition specifically for gauge fields is also in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a href="https://www.physicsforums.com/insights/examples-prequantum-field-theories-gauge-fields/">Examples for Prequantum field theories I: Gauge fields</a></em>.</li> </ul> <p>An exposition of the general formulation of fields in terms of <a class="existingWikiWord" href="/nlab/show/moduli+stacks">moduli stacks</a> in <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-toposes">slice (∞,1)-toposes</a> is in section 4 of</p> <ul id="FiorenzaSatiSchreiberCSIntroAndSurvey"> <li><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/A+higher+stacky+perspective+on+Chern-Simons+theory">A higher stacky perspective on Chern-Simons theory</a></em></li> </ul> <p>Lecture notes on fields as discussed here with applications in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> are in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/nlab/show/twisted+smooth+cohomology+in+string+theory">twisted smooth cohomology in string theory</a></em>, lecture notes at <em><a href="http://maths-old.anu.edu.au/esi/2012/">K-theory and Quantum Fields</a></em>, ESI program June 2012</li> </ul> <p>An introductory survey is also in section 1 of</p> <ul id="dcct"> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></em></li> </ul> <h3 id="original_articles_on_the_general_notion">Original articles on the general notion</h3> <p>For references on the tradtional formulation of physical fields by sections of <em><a class="existingWikiWord" href="/nlab/show/field+bundles">field bundles</a></em> as discussed <a href="#IdeaOfFieldBundlesAndItsProblems">above</a> see there references <a class="existingWikiWord" href="/nlab/show/field+bundle">there</a>.</p> <p>The formulation of physical fields as <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> in <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> as in the <em><a href="#Definition">Definition</a></em>-section above originates around</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Background+fields+in+twisted+differential+nonabelian+cohomology">Background fields in twisted differential nonabelian cohomology</a></em>, talk at <em><a class="existingWikiWord" href="/nlab/show/Oberwolfach+Workshop%2C+June+2009+--+Strings%2C+Fields%2C+Topology">Oberwolfach Workshop, June 2009 – Strings, Fields, Topology</a></em></li> </ul> <p>Further articles since then are listed at</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></em></li> </ul> <p>In particular the general notion of fields as <a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a> appears in</p> <ul id="SSS"> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em><a class="existingWikiWord" href="/schreiber/show/Twisted+Differential+String+and+Fivebrane+Structures">Twisted Differential String and Fivebrane Structures</a></em> (<a href="http://arxiv.org/abs/0910.4001">arXiv:0910.4001</a>)</li> </ul> <p>and the general theory of <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> and <a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a> with <a class="existingWikiWord" href="/nlab/show/local+coefficient+%E2%88%9E-bundles">local coefficient ∞-bundles</a> as referred to in <em><a href="#RelationToTwistedCohomology">Relation to twisted cohomology</a></em> above as well as the theory of <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundles">associated ∞-bundles</a> as in <em><a href="#SectionsOfAssociatedBundles">Sections of associated ∞-bundles</a></em> is laid out in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Thomas+Nikolaus">Thomas Nikolaus</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Danny+Stevenson">Danny Stevenson</a>, <em><a class="existingWikiWord" href="/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications">Principal ∞-bundles – theory, presentations and applications</a></em> (<a href="http://arxiv.org/abs/1207.0248">arXiv:1207.0248</a>)</li> </ul> <p>Some examples of fields in this sense are called “relative fields” in</p> <ul id="FreedTeleman"> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, <em>Relative quantum field theory</em> (<a href="http://arxiv.org/abs/1212.1692">arXiv:1212.1692</a>)</li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> with stacky fields and co-stacky <a class="existingWikiWord" href="/nlab/show/local+nets+of+observables">local nets of observables</a> is in</p> <ul> <li id="BeniniSchenkelSzabo15"><a class="existingWikiWord" href="/nlab/show/Marco+Benini">Marco Benini</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, <a class="existingWikiWord" href="/nlab/show/Richard+Szabo">Richard Szabo</a>, <em>Homotopy colimits and global observables in Abelian gauge theory</em> (<a href="http://arxiv.org/abs/1503.08839">arXiv:1503.08839</a>)</li> </ul> <h3 id="original_articles_on_special_cases">Original articles on special cases</h3> <p>The supergeometric nature of the <a class="existingWikiWord" href="/nlab/show/B-field">B-field</a> in <a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a> had been pointed out in</p> <ul id="Precis"> <li><a class="existingWikiWord" href="/nlab/show/Jacques+Distler">Jacques Distler</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <em>Orientifold Précis</em> in: <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a> (eds.) <em><a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a></em> Proceedings of Symposia in Pure Mathematics, AMS (2011) (<a href="http://arxiv.org/abs/0906.0795">arXiv:0906.0795</a>, <a href="http://www.ma.utexas.edu/users/dafr/bilbao.pdf">slides</a>)</li> </ul> <p>with a more detailed account in</p> <ul id="FreedLectures"> <li><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em>Lectures on twisted K-theory and orientifolds</em> (<a href="http://www.ma.utexas.edu/users/dafr/ESI.pdf">pdf</a>)</li> </ul> <p>The formulation of this in <a class="existingWikiWord" href="/nlab/show/smooth+super+infinity-groupoids">smooth super infinity-groupoids</a> is (<a href="#FiorenzaSatiSchreiberCSIntroAndSurvey">FSS, section 4.3</a>).</p> </body></html> </div> <div class="revisedby"> <p> Last revised on November 15, 2024 at 17:20:52. 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