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<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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#global_gauge_group'>Global gauge group</a></li> <li><a href='#local_gauge_groups'>Local gauge groups</a></li> </ul> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#NotARedundancy'>Not a redundancy</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#global_gauge_group_2'>Global gauge group</a></li> <li><a href='#local_gauge_group'>Local gauge group</a></li> <li><a href='#local_gauging_of_global_symmetries_in_quantum_gravity'>Local gauging of global symmetries in quantum gravity</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/physics">physics</a> there are (at least) two different concepts that go by the name <strong>gauge group</strong>:</p> <ul> <li> <p>a <em>local gauge group</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> is a structure group 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+bundle">principal bundle</a>s in the configuration space of a classical <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>: it acts by <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>s on the space of field configurations.</p> </li> <li> <p>a <em>global gauge group</em> is a group of <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a>s that acts on the (<a class="existingWikiWord" href="/nlab/show/local+net">local net</a> of) <a class="existingWikiWord" href="/nlab/show/observable">observable</a>s of a <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>.</p> </li> </ul> <p>Notably after <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> the gauge group in the first sense does <em>not</em> become a gauge group in the second sense. On the contrary, observables in quantum field theory are required <em>not</em> to depend on gauge transformations in the first sense, and part of what makes quantization of <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> nontrivial is to find among all potential candidate observables those that actually are invariant under gauge transformations, i.e. under isomorphisms of <a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>s with <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a> in the configuration space of the gauge theory.</p> <h3 id="global_gauge_group">Global gauge group</h3> <p>The concept of gauge groups is most prominent in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>, where the gauge group of a <a class="existingWikiWord" href="/nlab/show/physical+system">physical system</a> is the <a class="existingWikiWord" href="/nlab/show/group">group</a> of transformations of the mathematical model of the system that do not correspond to any measurable <a class="existingWikiWord" href="/nlab/show/physics">physical</a> effects. In this sense, nontrivial gauge groups arise from redundancies of the mathematical description. Gauge groups are a central ingredient of <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> gauge groups are introduced via a <a class="existingWikiWord" href="/nlab/show/net+of+C-star-systems">net of C-star-systems</a>.</p> <h3 id="local_gauge_groups">Local gauge groups</h3> <p>In <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> and other <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a> the <em>gauge groups</em> is the structure group <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> 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> on which the <a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a> is a <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection</a>.</p> <p>Local gauge groups are visible in the <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian</a> approach to <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>, where they act on the configuration space on which the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> is a function by <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>s. A large machinery has been developed to handle the (<a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a>) <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>s on such configuration spaces. See for instance <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a>.</p> <h2 id="Properties">Properties</h2> <h3 id="NotARedundancy">Not a redundancy</h3> <p>Sometimes one sees the statement being made that gauge symmetry in theories of <a class="existingWikiWord" href="/nlab/show/physics">physics</a> is just a sign of a <em>redundancy</em> in the theory, since, after all, gauge equivalent configurations are equivalent.</p> <p>But, instead, there is genuine information contained in the gauge group of a physical theory: it encodes the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of configurations; or in other words: the higher <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>s in the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> of <a class="existingWikiWord" href="/nlab/show/configuration+space">configurations</a> of the system. <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">Infinitesimally</a> this is given by the <a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a> of the system, and the nature of the gauge group controls its higher <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cohomology groups</a>. For instance the degree-1 cohomology of the BRST complex (meaning: “ghost degree-1”) of a system contains the possible gauge <a class="existingWikiWord" href="/nlab/show/quantum+anomalies">quantum anomalies</a> (as discussed there) of the system. This is clearly not redundant information.</p> <p>Abstractly speaking, the idea that symmetry is <em>just</em> a redundancy is a mistake of <a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a>: passing from a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> of <a class="existingWikiWord" href="/nlab/show/configuration+space">configurations</a> – where different configurations are related by <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s called <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>s – to the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> space of configurations modulo gauge transformations is the <a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a> of the groupoid. More technically speaking, it is the <em><a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncation</a></em> . It computes the 0-th <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a> and forgets all the higher homotopy groups.</p> <p>While there is indeed some information extracted by this process, that about the gauge equivalence classes of configurations, other information is lost.</p> <p>Physically speaking, notably the <em>locality</em> of field theory would break down if one insisted on always passing to gauge equivalence classes of configurations.</p> <p>For instance in a finite <a class="existingWikiWord" href="/nlab/show/QFT">QFT</a> such as <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> with gauge group <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>, the <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>/<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><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>. This space is <a class="existingWikiWord" href="/nlab/show/connected">connected</a>, reflecting the fact that <em>locally</em> every configuration of DW-theory is gauge equivalent to every other: namely the field configurations 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/principal+bundle">principal bundle</a>s and locally on a piece of space <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> these are all equivalent to the trivial principal bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">U \times G</annotation></semantics></math>. If one insisted that only gauge equivalence classes of configuration contain non-redundant information, then one would find that either DW theory is not a local QFT or else that it contains the trivial configuration.</p> <p>Entirely analogous comments apply to more interesting theories such as <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>, only that there the moduli space is richer: it is the <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential refinement</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><mo>:</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mi>P</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G_{conn} := [P_1(-),\mathbf{B}G]</annotation></semantics></math> 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>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> (see <em><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a></em> for details). Again, this cannot be reconstructed from its decategorification and if one insisted on passing to gauge equivalence classes of configurations then either Yang-Mills theory would no longer appear as a local QFT, or else it would have just the the globally trivial configurations.</p> <p>To say this more precisely, here instead of “<a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a>” we should be saying “<a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a>”. The mathematical theory of <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a> is the theory that deals with systems that are both <em>local</em> (in that global configurations are glued together from local data) as well as equipped with (gauge) symmetries. In that more refined language of <a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a> one would find and say that the <a class="existingWikiWord" href="/nlab/show/0-truncated">0-truncation</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><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}G_{conn}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> of gauge equivalence classes of <a class="existingWikiWord" href="/nlab/show/Lie+algebra+valued+differential+form">Lie algebra valued differential form</a>s. Insisting that this is the only relevant information amounts to insisting that either Yang-Mills theory is not a local theory, or else that all configurations of Yang-Mills theory are given by globally defined differential form data. This would exclude from the theory all configurations with nontrivial <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a>es, hence in particular it would exclude the <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> solutions from the theory, which are known to crucially encode information about the <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum theory</a>.</p> <p>That all said, one should note that in <em>higher</em> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> redundancies may appear after all. This is related to the fact that in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> the notion of <em><a class="existingWikiWord" href="/nlab/show/resolution">resolution</a>s</em> becomes relevant: there are higher <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a>s that look very rich on first sight but turn out to be <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28infinity%2C1%29-category">equivalent</a> to simpler moduli stacks.</p> <p>For instance if <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><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">U(1) \to \hat G \to G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/central+extension">central extension</a> of the gauge group <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 there is a <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U(1) \to \hat G)</annotation></semantics></math> (see <em><a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a></em>) which is equivalent to <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>. Therefore 2-gauge theory (such as, say, the <a class="existingWikiWord" href="/nlab/show/Courant+sigma-model">Courant sigma-model</a>) with gauge 2-group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U(1) \to \hat G)</annotation></semantics></math> is in fact equivalent to ordinary <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>-gauge theory: there is indeed a lot of redundancy in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U(1) \to \hat G)</annotation></semantics></math>. One can detect this by looking at the invariant information: the higher <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U(1) \to \hat G)</annotation></semantics></math> or else of its <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> (now a <a class="existingWikiWord" href="/nlab/show/infinity-stack">2-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><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}(U(1) \to \hat G)</annotation></semantics></math> are the same as that 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>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>.</p> <p>But even in such a situation, these redundant <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a>s have a good use, in general as well as in applications to physics. In the above example the resolution serves to support an evident morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mn>1</mn><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></mrow><annotation encoding="application/x-tex">(U(1) \to \hat G) \to (U(1) \to 1) = \mathbf{B}U(1)</annotation></semantics></math>. Together with the equivalence to <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> this constitutes an <a class="existingWikiWord" href="/nlab/show/infinity-anafunctor">2-anafunctor</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><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mover><mi>G</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><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></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><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}(U(1) \to \hat G) &\to& \mathbf{B}^2 U(1) \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B}G } \,. </annotation></semantics></math></div> <p>This structure exhibits a <a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</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>-gauge theory (namely the class that classifies the <a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>). The nontriviality of this class measures the failure 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>-gauge theory to lift 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>-gauge theory.</p> <p>A famous example of this in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> comes from the case where <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><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\hat G \to G</annotation></semantics></math> is the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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(n) \to PU(n)</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a> to the <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary group</a>. In this case a configuration of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PU(n)</annotation></semantics></math>-gauge theory is a <a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a> representing a class in <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a> as it appears on the worldvolume of <a class="existingWikiWord" href="/nlab/show/D-brane">D-brane</a>s. The corresponding higher <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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">\mathbf{B}U(1)</annotation></semantics></math>-gauge theory configuration is the <a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a>(<a class="existingWikiWord" href="/nlab/show/B-field">B-field</a>) restricted to the brane, which is the <em>twist</em> that twists the bundle and prevents it from being a configuration in genuine <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(n)</annotation></semantics></math>-gauge theory.</p> <h2 id="Examples">Examples</h2> <p>We list examples of local gauge groups and <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a> for various higher <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a>.</p> <ul> <li> <p>the gauge group 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/Yang-Mills+theory">Yang-Mills theory</a> is the given <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>; for the Yang-Mills theory appearing in the <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> this is the <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary 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>3</mn><mo stretchy="false">)</mo><mo>×</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>×</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(3) \times SU(2) \times U(1)</annotation></semantics></math>;</p> </li> <li> <p>the local gauge group of <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> on a <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 the <a class="existingWikiWord" href="/nlab/show/Poincare+group">Poincare group</a>;</p> </li> <li> <p>the gauge <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a> of the <a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a> is the <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle 2-group</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>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B} U(1) = (U(1) \to 1)</annotation></semantics></math>;</p> </li> <li> <p>the gauge <a class="existingWikiWord" href="/nlab/show/3-group">3-group</a> of the <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a> is the <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle 3-group</a> <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>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mn>1</mn><mo>→</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^2 U(1) = (U(1) \to 1 \to 1)</annotation></semantics></math>;</p> </li> <li> <p>the gauge group of abelian <a class="existingWikiWord" href="/nlab/show/higher+dimensional+Chern-Simons+theory">higher dimensional Chern-Simons theory</a> in dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn><mi>k</mi><mo>+</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">4 k+3</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle (2k+1)-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mn>2</mn><mi>k</mi></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^{2k} U(1)</annotation></semantics></math>;</p> </li> <li> <p>the 7-dimensional “<a class="existingWikiWord" href="/nlab/show/differential+fivebrane+structure">fivebrane Chern-Simons theory</a>” has as gauge 2-group the <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>;</p> </li> <li> <p>the symmetry group of <a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a> is some <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> that is not 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>-group for any finite <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>;</p> </li> <li> <p>an <a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a> has in general not only a gauge <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> but an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> of symmetries:</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Poisson+sigma-model">Poisson sigma-model</a> has as gauge groupoid the <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a> that is the <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the given <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Courant+sigma-model">Courant sigma-model</a> has as gauge <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a> the <a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic 2-groupoid</a> that integrates the given <a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a>;</p> </li> <li> <p>generally, the <a class="existingWikiWord" href="/nlab/show/AKSZ+sigma-model">AKSZ sigma-model</a> in grade <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> has as gauge <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>-groupoid a <a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic Lie n-groupoid</a>.</p> </li> </ul> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge">gauge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/structure+group">structure group</a>, <a class="existingWikiWord" href="/nlab/show/equivariance+group">equivariance group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</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/enhanced+gauge+symmetry">enhanced gauge symmetry</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li>Wikipedia: <a href="http://en.wikipedia.org/wiki/Gauge_group">gauge group</a></li> </ul> <h3 id="global_gauge_group_2">Global gauge group</h3> <p>Chapter IV of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Rudolf+Haag">Rudolf Haag</a>, <em><a class="existingWikiWord" href="/nlab/show/Local+Quantum+Physics">Local Quantum Physics</a></em></li> </ul> <h3 id="local_gauge_group">Local gauge group</h3> <p>(…)</p> <h3 id="local_gauging_of_global_symmetries_in_quantum_gravity">Local gauging of global symmetries in quantum gravity</h3> <p>It is being argued that after embedding into consistent <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a>, all <a class="existingWikiWord" href="/nlab/show/global+symmetries">global symmetries</a> must become <a class="existingWikiWord" href="/nlab/show/local+symmetries">local symmetries</a>.</p> <p>A substantiation of this argument via <a class="existingWikiWord" href="/nlab/show/AdS%2FCFT">AdS/CFT</a> is given in:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Harlow">Daniel Harlow</a>, <a class="existingWikiWord" href="/nlab/show/Hirosi+Ooguri">Hirosi Ooguri</a>, <em>Symmetries in quantum field theory and quantum gravity</em> (<a href="https://arxiv.org/abs/1810.05338">arXiv:1810.05338</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Harlow">Daniel Harlow</a>, <a class="existingWikiWord" href="/nlab/show/Hirosi+Ooguri">Hirosi Ooguri</a>, <em>Constraints on symmetry from holography</em>, Phys. Rev. Lett. 122, 191601 (2019) (<a href="https://arxiv.org/abs/1810.05337">arXiv:1810.05337</a>)</p> </li> </ul> <p>See also:</p> <ul> <li>Sylvain Fichet, Prashant Saraswat, <em>Approximate Symmetries and Gravity</em> (<a href="https://arxiv.org/abs/1909.02002">arXiv:1909.02002</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 16, 2024 at 16:56:34. See the <a href="/nlab/history/gauge+group" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/gauge+group" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/1530/#Item_20">Discuss</a><span class="backintime"><a href="/nlab/revision/gauge+group/22" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/gauge+group" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/gauge+group" accesskey="S" class="navlink" id="history" rel="nofollow">History (22 revisions)</a> <a href="/nlab/show/gauge+group/cite" style="color: black">Cite</a> <a href="/nlab/print/gauge+group" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/gauge+group" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>