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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> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="physics">Physics</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a> </li> <li> <a class="existingWikiWord" 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href="/nlab/show/moment+of+inertia">moment of inertia</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> </li> <li> <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> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> <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> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a> <ul> <li> Axiomatizations <ul> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> <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> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <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> Structural phenomena <ul> <li> <a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> </li> <li> <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> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> </li> <li> <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> </li> <li> examples <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> <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <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> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a> <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> <a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <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> </li> <li> <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> </li> <li> <a 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class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+object">group object</a>, <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object in an (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></li> <li><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+action">group action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></li> <li><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></li> <li><a class="existingWikiWord" href="/nlab/show/progroup">progroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a></li> </ul> Classical groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>. <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary group</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a> </li> </ul> Finite groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a>, <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/classification+of+finite+simple+groups">classification of finite simple groups</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/sporadic+finite+simple+groups">sporadic finite simple groups</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Monster+group">Monster group</a>, <a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a></li> </ul> </li> </ul> Group schemes <ul> <li> <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/abelian+variety">abelian variety</a> </li> </ul> Topological groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/compact+topological+group">compact topological group</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+group">locally compact topological group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+group">string group</a> </li> </ul> Lie groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kac-Moody+group">Kac-Moody group</a> </li> </ul> Super-Lie groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a> </li> </ul> Higher groups <ul> <li> <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>, <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/n-group">n-group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/k-tuply+groupal+n-groupoid">k-tuply groupal n-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>, <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-group">fivebrane Lie 6-group</a> </li> </ul> Cohomology and Extensions <ul> <li> <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a>, <a class="existingWikiWord" href="/nlab/show/Ext-group">Ext-group</a> </li> </ul> Related concepts <ul> <li><a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> </div></div> <h4 id="motivic_cohomology">Motivic cohomology</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/motive">motive</a>, <a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a> <h2 id="ingredients">Ingredients</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> </li> </ul> <h2 id="definitions">Definitions</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/pure+motive">pure motive</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Chow+motive">Chow motive</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/numerical+motive">numerical motive</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/periods">periods</a>, <a class="existingWikiWord" href="/nlab/show/motivic+Galois+group">motivic Galois group</a>, <a class="existingWikiWord" href="/nlab/show/cosmic+Galois+group">cosmic Galois group</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/mixed+motive">mixed motive</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Nori+motive">Nori motive</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Voevodsky+motive">Voevodsky motive</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/A1-homotopy+theory">A1-homotopy theory</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/noncommutative+motive">noncommutative motive</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/bivariant+algebraic+K-theory">bivariant algebraic K-theory</a>, <a class="existingWikiWord" href="/nlab/show/KK-theory">KK-theory</a></li> </ul> </li> </ul> <h2 id="related">Related</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/motivic+function">motivic function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/motivic+integration">motivic integration</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/motivic+Donaldson-Thomas+invariant">motivic Donaldson-Thomas invariant</a> </li> </ul> <div> <a href="/nlab/edit/motivic+cohomology+-+contents">Edit this sidebar</a> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> What is called the cosmic Galois group is a <a class="existingWikiWord" href="/nlab/show/motivic+Galois+group">motivic Galois group</a> that naturally <a class="existingWikiWord" href="/nlab/show/action">acts</a> on structures in <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>. The actual <a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a> is a 1-parameter <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> of the cosmic Galois group. In more detail, in (<a href="#ConnesMarcolli04">Connes-Marcolli 04</a>) the authors consider a <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a> satisfied by divergences appearing in the <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> formulation of <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> (see there). This leads to a <a class="existingWikiWord" href="/nlab/show/category">category</a> of “equisingular flat connections” that turns out to be a <a class="existingWikiWord" href="/nlab/show/Tannakian+category">Tannakian category</a>, meaning that it is equivalent to the <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> over some pro-algebraic 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 authors observe 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/action">acts</a> on any renormalizable <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a> in a nice way. Due to this property <a class="existingWikiWord" href="/nlab/show/Pierre+Cartier">Pierre Cartier</a> referred to this as the cosmic Galois group: <blockquote> La parenté de plus en plus manifeste entre le <a class="existingWikiWord" href="/nlab/show/groupe+de+Grothendieck-Teichm%C3%BCller">groupe de Grothendieck-Teichmüller</a> d’une part, et le groupe de renormalisation de la Théorie Quantique des Champs n’est sans doute que la première manifestation d’un groupe de symétrie des constantes fondamentales de la physique, une espèce de groupe de Galois cosmique!“ (<a class="existingWikiWord" href="/nlab/show/Pierre+Cartier">Pierre Cartier</a> according to <a href="#Connes12">Connes 12</a>, see also the end of <a href="#Cartier01">Cartier 01</a>). </blockquote> This cosmic Galois 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> is (non-canonically) isomorphic to some <a class="existingWikiWord" href="/nlab/show/motivic+Galois+group">motivic Galois group</a>. In (<a href="#KitchlooMorava12">Kitchloo-Morava 12</a>) the cosmic Galois group is related to the motivic/<a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannakian group</a> of a motivic stabilization of the <a class="existingWikiWord" href="/nlab/show/symplectic+category">symplectic category</a> of <a class="existingWikiWord" href="/nlab/show/symplectic+manifolds">symplectic manifolds</a> and <a class="existingWikiWord" href="/nlab/show/Lagrangian+correspondences">Lagrangian correspondences</a> between them, the <a class="existingWikiWord" href="/nlab/show/stable+symplectic+category">stable symplectic category</a> (<a href="#Kitchloo12">Kitchloo 12</a>). <h2 id="related_concepts">Related concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/motivic+Galois+group">motivic Galois group</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Grothendieck-Teichm%C3%BCller+group">Grothendieck-Teichmüller group</a> </li> </ul> <h2 id="references">References</h2> The “cosmic Galois group” in <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> theory was introduced in <ul> <li id="ConnesMarcolli04"><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <a class="existingWikiWord" href="/nlab/show/Matilde+Marcolli">Matilde Marcolli</a>, Renormalization and motivic Galois theory (<a href="http://arxiv.org/abs/math/0409306">arXiv:math/0409306</a>)</li> </ul> The name originates in <ul> <li id="Cartier01"><a class="existingWikiWord" href="/nlab/show/Pierre+Cartier">Pierre Cartier</a>, A mad day’s work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry, Bull. Amer. Math. Soc. 38 (2001), 389-408, <a href="http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf">pdf</a></li> </ul> A similar statement about a <a class="existingWikiWord" href="/nlab/show/motivic+Galois+group">motivic Galois group</a> acting on the space of <a class="existingWikiWord" href="/nlab/show/deformation+quantizations">deformation quantizations</a> of a <a class="existingWikiWord" href="/nlab/show/free+field+theory">free field theory</a> appeared earlier in <ul> <li><a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, Operads and Motives in Deformation Quantization, Lett. Math. Phys.48:35-72,1999 (<a href="http://arxiv.org/abs/math/9904055">arXiv:math/9904055</a>)</li> </ul> Review and exposition includes <ul> <li id="Connes12"> <a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, talk at <a href="http://www.galois.ihp.fr/manifestations/colloque/">Galois 200th birthday conference</a> January (2012) <a href="http://www.alainconnes.org/docs/slidesgaloisihp1.pdf">pdf slides</a> </li> <li id="Brown15"> <a class="existingWikiWord" href="/nlab/show/Francis+Brown">Francis Brown</a>, Feynman Amplitudes and Cosmic Galois group (<a href="https://arxiv.org/abs/1512.06409">arXiv:1512.06409</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Francis+Brown">Francis Brown</a>, Periods and Feynman amplitudes (<a href="https://arxiv.org/abs/1512.09265">arXiv:1512.09265</a>) </li> </ul> A technical review of aspects of this is in <ul> <li><a class="existingWikiWord" href="/nlab/show/Jack+Morava">Jack Morava</a>, The motivic Thom isomorphism (<a href="http://www.math.uiuc.edu/K-theory/0654/">web</a>, <a href="http://www.math.uiuc.edu/K-theory/0654/gamma2.pdf">pdf</a>)</li> </ul> See also <ul> <li> <a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, Toward a Kozmik Galois group, talk at Northwester, April 2010 (<a href="http://www.math.northwestern.edu/~pgoerss/midwest10/MidW010.pdf">pdf</a>) </li> <li id="Morava11"> <a class="existingWikiWord" href="/nlab/show/Jack+Morava">Jack Morava</a>, The Cosmic Galois group as Koszul dual to Waldhausen’s A(pt) (<a href="http://arxiv.org/abs/1108.4627">arXiv:1108.4627</a>) </li> </ul> A review of cosmic Galois/Grothendieck-Teichmüller group and a discussion from the perspective of the <a class="existingWikiWord" href="/nlab/show/motivic+symplectic+category">motivic symplectic category</a> is section 5 of <ul> <li id="KitchlooMorava12"><a class="existingWikiWord" href="/nlab/show/Nitu+Kitchloo">Nitu Kitchloo</a>, <a class="existingWikiWord" href="/nlab/show/Jack+Morava">Jack Morava</a>, The Grothendieck–Teichmüller group and the stable symplectic category, 2012. (<a href="http://arxiv.org/abs/1212.6905">arxiv:1212.6905</a>)</li> </ul> based on <ul id="Kitchloo12"> <li><a class="existingWikiWord" href="/nlab/show/Nitu+Kitchloo">Nitu Kitchloo</a>, The Stable Symplectic Category and Geometric Quantization (<a href="http://arxiv.org/abs/1204.5720v1">arXiv:1204.5720</a>)</li> </ul> <ul> <li><a href="http://mathoverflow.net/questions/94954/connes-kreimer-hopf-algebra-and-cosmic-galois-group">MO discussion</a></li> </ul> </body></html> </div> <div class="revisedby"> Last revised on October 10, 2020 at 17:51:30. 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