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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="algebraic_quantum_field_theory">Algebraic Quantum Field Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative</a>, <a class="existingWikiWord" href="/nlab/show/AQFT+on+curved+spacetime">on curved spacetimes</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+algebraic+quantum+field+theory">homotopical</a>)</p> <p><a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">Introduction</a></p> <h2 id="concepts">Concepts</h2> <p><strong><a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a></strong>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">pre-quantum</a>, <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relativistic+field+theory">relativistic</a>, <a class="existingWikiWord" href="/nlab/show/Euclidean+field+theory">Euclidean</a>, <a class="existingWikiWord" href="/nlab/show/thermal+quantum+field+theory">thermal</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+history">field history</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+density">Lagrangian density</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/presymplectic+current">presymplectic current</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange</a><a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+variational+field+theory">locally variational field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Peierls-Poisson+bracket">Peierls-Poisson bracket</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/advanced+and+retarded+propagator">advanced and retarded propagator</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+propagator">causal propagator</a></p> </li> </ul> </li> </ul> </li> </ul> <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><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+deformation+quantization">algebraic deformation quantization</a>, <a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/quantum+probability">quantum probability</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subsystem">subsystem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/observables">observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+observables">field observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/polynomial+observables">polynomial observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microcausal+observables">microcausal observables</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a>, <a class="existingWikiWord" href="/nlab/show/C%2A-algebra">C*-algebra</a>, <a class="existingWikiWord" href="/nlab/show/von+Neumann+algebra">von Neumann algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+locality">causal locality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+net">field net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/state+on+a+star-algebra">state on a star-algebra</a>, <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+state">pure state</a></p> <p><a class="existingWikiWord" href="/nlab/show/wave+function">wave function</a></p> <p><a class="existingWikiWord" href="/nlab/show/collapse+of+the+wave+function">collapse of the wave function</a>/<a class="existingWikiWord" href="/nlab/show/conditional+expectation+value">conditional expectation value</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mixed+state">mixed state</a>, <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+state">vacuum state</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-free+state">quasi-free state</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+state">Hadamard state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+propagator">Wightman propagator</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/picture+of+quantum+mechanics">picture of quantum mechanics</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/free+field">free field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/star+algebra">star algebra</a>, <a class="existingWikiWord" href="/nlab/show/Moyal+deformation+quantization">Moyal deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick+algebra">Wick algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+commutation+relations">canonical commutation relations</a>, <a class="existingWikiWord" href="/nlab/show/Weyl+relations">Weyl relations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+ordered+product">normal ordered product</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fock+space">Fock space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</a>, <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+master+equation">quantum master equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/master+Ward+identity">master Ward identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+anomaly">gauge anomaly</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/interacting+field+theory">interacting field</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>, <a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction">interaction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a>, <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/causal+additivity">causal additivity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/time-ordered+product">time-ordered product</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+propagator">Feynman propagator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>, <a class="existingWikiWord" href="/nlab/show/Feynman+perturbation+series">Feynman perturbation series</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+action">effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+stability">vacuum stability</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+field+algebra">interacting field algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bogoliubov%27s+formula">Bogoliubov's formula</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+M%C3%B8ller+operator">quantum Møller operator</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adiabatic+limit">adiabatic limit</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/infrared+divergence">infrared divergence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interacting+vacuum">interacting vacuum</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+scheme">("re-")normalization scheme</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/extension+of+distributions">extension of distributions</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+condition">("re"-)normalization condition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group">renormalization group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/interaction+vertex+redefinition">interaction vertex redefinition</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/St%C3%BCckelberg-Petermann+renormalization+group">Stückelberg-Petermann renormalization group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization+group+flow">renormalization group flow</a>/<a class="existingWikiWord" href="/nlab/show/running+coupling+constants">running coupling constants</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/UV+cutoff">UV cutoff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/counterterms">counterterms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+effective+action">relative effective action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilsonian+RG">Wilsonian RG</a>, <a class="existingWikiWord" href="/nlab/show/Polchinski+flow+equation">Polchinski flow equation</a></p> </li> </ul> </li> </ul> <h2 id="Theorems">Theorems</h2> <h3 id="states_and_observables">States and observables</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/order-theoretic+structure+in+quantum+mechanics">order-theoretic structure in quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Alfsen-Shultz+theorem">Alfsen-Shultz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Harding-D%C3%B6ring-Hamhalter+theorem">Harding-Döring-Hamhalter theorem</a></p> </li> </ul> </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/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wigner+theorem">Wigner theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bub-Clifton+theorem">Bub-Clifton theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kadison-Singer+problem">Kadison-Singer problem</a></p> </li> </ul> <h3 id="operator_algebra">Operator algebra</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wick%27s+theorem">Wick's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GNS+construction">GNS construction</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cyclic+vector">cyclic vector</a>, <a class="existingWikiWord" href="/nlab/show/separating+vector">separating vector</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fell%27s+theorem">Fell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stone-von+Neumann+theorem">Stone-von Neumann theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag%27s+theorem">Haag's theorem</a></p> </li> </ul> <h3 id="local_qft">Local QFT</h3> <ul> <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/Bisognano-Wichmann+theorem">Bisognano-Wichmann 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/spin-statistics+theorem">spin-statistics theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/DHR+superselection+theory">DHR superselection theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> (<a class="existingWikiWord" href="/nlab/show/Wick+rotation">Wick rotation</a>)</p> </li> </ul> <h3 id="perturbative_qft">Perturbative QFT</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Schwinger-Dyson+equation">Schwinger-Dyson equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/main+theorem+of+perturbative+renormalization">main theorem of perturbative renormalization</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> <li><a href='#example'>Example</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> <p>Given a <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E,\mathbf{L})</annotation></semantics></math> with <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><mover><mo>→</mo><mi>fb</mi></mover><mi>Σ</mi></mrow><annotation encoding="application/x-tex">E \overset{fb}{\to} \Sigma</annotation></semantics></math> over 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>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> and <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></mrow><annotation encoding="application/x-tex">\mathbf{L}</annotation></semantics></math>, then its <em>local BV-BRST complex</em> (or <em>local BRST complex</em>, for short) is the realization of the <a class="existingWikiWord" href="/nlab/show/BV-BRST+complex">BV-BRST complex</a> not on <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><msub><mi>τ</mi> <mi>Σ</mi></msub><mi>α</mi></mrow><annotation encoding="application/x-tex">A = \tau_{\Sigma} \alpha</annotation></semantics></math> given by <a class="existingWikiWord" href="/nlab/show/functionals">functionals</a> on the <a class="existingWikiWord" href="/nlab/show/space+of+field+histories">space of field histories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" 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><mo>=</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\Gamma_{\Sigma}(E)_{\delta_{EL} = 0}</annotation></semantics></math> which are <a class="existingWikiWord" href="/nlab/show/transgression+of+variational+differential+forms">transgressions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex">\tau_{\Sigma}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/variational+differential+forms">variational differential forms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∈</mo><msubsup><mi>Ω</mi> <mi>Σ</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha \in \Omega^{\bullet, \bullet}_\Sigma(E)</annotation></semantics></math> on the jet bundle, but on these variational differential forms themselves (whence “local”, i.e. before <a class="existingWikiWord" href="/nlab/show/transgression+of+variational+differential+forms">transgression</a>).</p> <p>If <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> denotes the <a class="existingWikiWord" href="/nlab/show/BV-BRST+differential">BV-BRST differential</a> in a BV-resolution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>Σ</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><msub><mi>ℰ</mi> <mi>BV</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\Omega^{\bullet,\bullet}_\Sigma(E)\vert_{\mathcal{E}_{BV}}</annotation></semantics></math> of the restriction to the <a class="existingWikiWord" href="/nlab/show/shell">shell</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</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">\mathcal{E} \hookrightarrow J^\infty_\Sigma(E)</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>Σ</mi> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^{\bullet,\bullet}_\Sigma(E)</annotation></semantics></math> with its <a class="existingWikiWord" href="/nlab/show/total+spacetime+derivative">total spacetime derivative</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> (horizontal derivative), then the <em>local BV-BRST cohomology</em> is the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>+</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">s + d</annotation></semantics></math>, hence of the <a class="existingWikiWord" href="/nlab/show/total+complex">total complex</a> of the <a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a> given by <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> and <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>.</p> <p>Generally, considering <a class="existingWikiWord" href="/nlab/show/variational+differential+forms">variational differential forms</a> up to <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>-exact terms is the “local” incarnation of what under the <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration</a> involved in the <a class="existingWikiWord" href="/nlab/show/transgression+of+variational+differential+forms">transgression</a> is <a class="existingWikiWord" href="/nlab/show/integration+by+parts">integration by parts</a> and it is in this way that “local BV-BRST cohomology” knows about the actual BV-BRST cohomology on <a class="existingWikiWord" href="/nlab/show/local+observables">local observables</a>.</p> <h2 id="example">Example</h2> <p>Consider local coordinates <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><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\phi^a)</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> of the <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a>. The corresponding <a class="existingWikiWord" href="/nlab/show/antifield">antifield</a> coordinates are to be denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>ϕ</mi><mo>¯</mo></mover> <mi>a</mi></msub></mrow><annotation encoding="application/x-tex">\overline{\phi}_a</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/BV-BRST+differential">BV-BRST differential</a> takes them to the corresponding component</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><msub><mover><mi>ϕ</mi><mo>¯</mo></mover> <mi>a</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mi>δ</mi> <mi>El</mi></msub><mi>L</mi></mrow><mrow><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex"> s(\overline{\phi}_a) = \frac{\delta_{El} L}{\delta \phi^a} </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>.</p> <p>In 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>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(p+1,0)</annotation></semantics></math> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>+</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">s+d</annotation></semantics></math>-closed elements in vanishing <a class="existingWikiWord" href="/nlab/show/ghost">ghost</a> degree are <a class="existingWikiWord" href="/nlab/show/pairs">pairs</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><msub><mi>J</mi> <mi>v</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v,J_v)</annotation></semantics></math> consisting of an <a class="existingWikiWord" href="/nlab/show/infinitesimal+symmetry+of+the+Lagrangian">infinitesimal symmetry of the Lagrangian</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>, regarded as an <a class="existingWikiWord" href="/nlab/show/antifield">antifield</a> <a class="existingWikiWord" href="/nlab/show/density">density</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>v</mi> <mi>a</mi></msup><msub><mover><mi>ϕ</mi><mo>¯</mo></mover> <mi>a</mi></msub><msub><mi>dvol</mi> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex">v^a \overline{\phi}_a dvol_\Sigma</annotation></semantics></math>, together with a corresponding <a class="existingWikiWord" href="/nlab/show/conserved+current">conserved</a> <a class="existingWikiWord" href="/nlab/show/Noether%27s+theorem">Noether current</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>J</mi> <mi>v</mi></msub></mrow><annotation encoding="application/x-tex">J_v</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><mo stretchy="false">{</mo><msub><mi>J</mi> <mi>v</mi></msub><mo stretchy="false">}</mo></mtd> <mtd><mover><mo>⟶</mo><mi>d</mi></mover></mtd> <mtd><mo stretchy="false">{</mo><mover><mover><mrow><mi>d</mi><msub><mi>J</mi> <mi>v</mi></msub><mo>−</mo><msub><mi>ι</mi> <mi>v</mi></msub><msub><mi>δ</mi> <mi>EL</mi></msub><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow><mo>⏞</mo></mover><mrow><mo>=</mo><mn>0</mn></mrow></mover><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo><mpadded width="0"><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>s</mi></mrow></mpadded></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">{</mo><msup><mi>v</mi> <mi>a</mi></msup><msub><mover><mi>ϕ</mi><mo>¯</mo></mover> <mi>a</mi></msub><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo stretchy="false">}</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \{J_v\} &\overset{d}{\longrightarrow}& \{ \overset{= 0}{\overbrace{ d J_v - \iota_v \delta_{EL}\mathbf{L} }} \} \\ && \uparrow\mathrlap{-s} \\ && \{ v^a \overline{\phi}_a dvol_\Sigma\} } </annotation></semantics></math></div> <p>Such pairs are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(s+d)</annotation></semantics></math>-exact if <a class="existingWikiWord" href="/nlab/show/on-shell">on-shell</a> the infintiesimal symmetry coincides with an <a class="existingWikiWord" href="/nlab/show/infinitesimal+gauge+symmetry">infinitesimal gauge symmetry</a>. To see this, recall:</p> <p>An <a class="existingWikiWord" href="/nlab/show/infinitesimal+gauge+symmetry">infinitesimal gauge symmetry</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>ϵ</mi></msub></mrow><annotation encoding="application/x-tex">v_\epsilon</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/gauge+parameter">gauge parameter</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>ϵ</mi> <mi>α</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\epsilon^\alpha)</annotation></semantics></math> is a vector field on the jet bundle with components 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>ℒ</mi> <mrow><msub><mi>v</mi> <mi>ϵ</mi></msub></mrow></msub><msup><mi>ϕ</mi> <mi>a</mi></msup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msubsup><mi>R</mi> <mi>α</mi> <mi>a</mi></msubsup><msup><mi>ϵ</mi> <mi>α</mi></msup><mo>+</mo><msubsup><mi>R</mi> <mi>α</mi> <mrow><mi>a</mi><mi>μ</mi></mrow></msubsup><mfrac><mrow><mi>d</mi><msup><mi>ϵ</mi> <mi>α</mi></msup></mrow><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex"> \mathcal{L}_{v_\epsilon} \phi^a \;\coloneqq\; R^a_\alpha \epsilon^\alpha + R^{a \mu}_\alpha \frac{d \epsilon^\alpha}{d x^\mu} </annotation></semantics></math></div> <p>such that this is an <a class="existingWikiWord" href="/nlab/show/infinitesimal+symmetry+of+the+Lagrangian">infinitesimal symmetry of the Lagrangian</a> in that</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>ι</mi> <mrow><msub><mi>v</mi> <mi>ϵ</mi></msub></mrow></msub><msub><mi>δ</mi> <mi>EL</mi></msub><mstyle mathvariant="bold"><mi>L</mi></mstyle></mtd> <mtd><mo>=</mo><msup><mi>v</mi> <mi>a</mi></msup><mfrac><mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mi>L</mi></mrow><mrow><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow></mfrac><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msup><mi>ϵ</mi> <mi>α</mi></msup><mrow><mo>(</mo><msubsup><mi>R</mi> <mi>α</mi> <mi>a</mi></msubsup><mfrac><mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mi>L</mi></mrow><mrow><mi>δ</mi><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><mrow><mo>(</mo><msubsup><mi>R</mi> <mi>α</mi> <mrow><mi>a</mi><mi>μ</mi></mrow></msubsup><mfrac><mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mi>L</mi></mrow><mrow><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow></mfrac><mo>)</mo></mrow><mo>)</mo></mrow><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo>+</mo><mi>d</mi><mrow><mo>(</mo><msup><mi>ϵ</mi> <mi>α</mi></msup><msubsup><mi>R</mi> <mi>α</mi> <mrow><mi>a</mi><mi>μ</mi></mrow></msubsup><mfrac><mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mi>L</mi></mrow><mrow><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow></mfrac><mo>)</mo></mrow><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><mn>0</mn><mo>+</mo><mi>d</mi><mo stretchy="false">(</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \iota_{v_\epsilon} \delta_{EL} \mathbf{L} & = v^a \frac{\delta_{EL} L}{\delta \phi^a} dvol_\Sigma \\ & = \epsilon^\alpha \left( R^a_\alpha \frac{\delta_{EL} L}{ \delta \phi^a} - \frac{d}{d x^\mu} \left( R^{a \mu}_\alpha \frac{\delta_{EL} L}{\delta \phi^a} \right) \right) dvol_\Sigma + d\left( \epsilon^\alpha R^{a \mu}_\alpha \frac{\delta_{EL} L}{\delta \phi^a} \right) \iota_{\partial_\mu} dvol_\Sigma \\ & = 0 + d(...) \end{aligned} </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>ϵ</mi> <mi>α</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\epsilon^\alpha)</annotation></semantics></math>.</p> <p>The corresponding <a class="existingWikiWord" href="/nlab/show/antighosts">antighosts</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>c</mi><mo>¯</mo></mover> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">\overline{c}_\alpha</annotation></semantics></math> are taken by the BV-BRST differential to the antifield-preimage of the term on the left:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>s</mi><mrow><mo>(</mo><msub><mover><mi>c</mi><mo>¯</mo></mover> <mi>α</mi></msub><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msubsup><mi>R</mi> <mi>α</mi> <mi>a</mi></msubsup><msub><mover><mi>ϕ</mi><mo>¯</mo></mover> <mi>a</mi></msub><mo>−</mo><mfrac><mi>d</mi><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mrow><mo>(</mo><msubsup><mi>R</mi> <mi>α</mi> <mrow><mi>a</mi><mi>μ</mi></mrow></msubsup><msub><mover><mi>ϕ</mi><mo>¯</mo></mover> <mi>a</mi></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> s\left(\overline{c}_\alpha\right) \;=\; R^a_\alpha \overline{\phi}_a - \frac{d}{d x^\mu} \left( R^{a \mu}_\alpha \overline{\phi}_a \right) \,. </annotation></semantics></math></div> <p>Moreover, an <a class="existingWikiWord" href="/nlab/show/on-shell">on-shell</a> vanishing <a class="existingWikiWord" href="/nlab/show/infinitesimal+symmetry+of+the+Lagrangian">infinitesimal symmetry of the Lagrangian</a> is a vector field with components of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>κ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mfrac><mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mi>L</mi></mrow><mrow><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex"> \kappa^{a b} \frac{\delta_{EL} L}{\delta \phi^a} </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>κ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mi>κ</mi> <mrow><mi>b</mi><mi>a</mi></mrow></msup></mrow><annotation encoding="application/x-tex">\kappa^{a b} = - \kappa^{b a}</annotation></semantics></math> a skew-symmetric system of smooth functions on the jet bundle.</p> <p>The linear combination of such an infinitesimal gauge symmetry and an on-shell vanishing infinitesimal symmetry is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>+</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(s+d)</annotation></semantics></math>-exact:</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><msup><mi>v</mi> <mi>a</mi></msup><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><msubsup><mi>R</mi> <mi>α</mi> <mi>a</mi></msubsup><msup><mi>ϵ</mi> <mi>α</mi></msup><mo>+</mo><msubsup><mi>R</mi> <mi>α</mi> <mrow><mi>a</mi><mi>μ</mi></mrow></msubsup><mfrac><mrow><mi>d</mi><msup><mi>ϵ</mi> <mi>α</mi></msup></mrow><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mo>+</mo><msup><mi>κ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mfrac><mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mi>L</mi></mrow><mrow><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow></mfrac><mo>)</mo></mrow><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>s</mi><mrow><mo>(</mo><msup><mi>ϵ</mi> <mi>α</mi></msup><msub><mover><mi>c</mi><mo>¯</mo></mover> <mi>α</mi></msub><mo>−</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msup><mi>κ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><msub><mover><mi>ϕ</mi><mo>¯</mo></mover> <mi>a</mi></msub><msub><mover><mi>ϕ</mi><mo>¯</mo></mover> <mi>b</mi></msub><mo>)</mo></mrow><msub><mi>dvol</mi> <mi>σ</mi></msub><mo>+</mo><mi>d</mi><mrow><mo>(</mo><msup><mi>ϵ</mi> <mi>α</mi></msup><msubsup><mi>R</mi> <mi>α</mi> <mrow><mi>a</mi><mi>μ</mi></mrow></msubsup><mo>)</mo></mrow><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>μ</mi></msub></mrow></msub><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} v^a dvol_\Sigma & = \left( R^a_\alpha \epsilon^\alpha + R^{a \mu}_\alpha \frac{d \epsilon^\alpha}{d x^\mu} + \kappa^{a b} \frac{\delta_{EL} L }{ \delta \phi^a } \right) dvol_\Sigma \\ & = s \left( \epsilon^\alpha \overline{c}_\alpha - \tfrac{1}{2}\kappa^{a b} \overline{\phi}_a \overline{\phi}_b \right) dvol_\sigma + d\left( \epsilon^\alpha R^{a \mu}_\alpha \right) \iota_{\partial_\mu} dvol_\Sigma \end{aligned} </annotation></semantics></math></div> <p>(<a href="#BarnichBrandtHenneaux94">Barnich-Brandt-Henneaux 94, p. 20</a>)</p> <p>It may be useful to organize this expression into the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>+</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">s+d</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/bicomplex">bicomplex</a> like so:</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><mi>K</mi><mo stretchy="false">}</mo></mtd> <mtd><mover><mo>⟶</mo><mi>d</mi></mover></mtd> <mtd><mo stretchy="false">{</mo><mi>d</mi><mi>K</mi><mo>+</mo><msup><mi>ϵ</mi> <mi>α</mi></msup><msubsup><mi>R</mi> <mi>α</mi> <mrow><mi>a</mi><mi>μ</mi></mrow></msubsup><mfrac><mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow><mrow><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow></mfrac><mo stretchy="false">}</mo></mtd> <mtd><mover><mo>⟶</mo><mi>d</mi></mover></mtd> <mtd><mo stretchy="false">{</mo><mover><mover><mrow><mi>d</mi><msub><mi>J</mi> <mi>v</mi></msub><mo>−</mo><msub><mi>ι</mi> <mi>v</mi></msub><msub><mi>δ</mi> <mi>EL</mi></msub><mstyle mathvariant="bold"><mi>L</mi></mstyle></mrow><mo>⏞</mo></mover><mrow><mo>=</mo><mn>0</mn></mrow></mover><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mpadded width="0" lspace="-100%width"><mi>s</mi></mpadded><mo stretchy="false">↑</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo><mpadded width="0"><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>s</mi></mrow></mpadded></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>ϵ</mi> <mi>α</mi></msup><msubsup><mi>R</mi> <mi>α</mi> <mrow><mi>a</mi><mi>μ</mi></mrow></msubsup><msub><mover><mi>ϕ</mi><mo>¯</mo></mover> <mi>a</mi></msub><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>μ</mi></msub></mrow></msub><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd> <mtd><munder><mo>⟶</mo><mi>d</mi></munder></mtd> <mtd><mrow><mo>{</mo><mi>d</mi><mrow><mo>(</mo><msup><mi>ϵ</mi> <mi>α</mi></msup><msubsup><mi>R</mi> <mi>α</mi> <mrow><mi>a</mi><mi>μ</mi></mrow></msubsup><msub><mover><mi>ϕ</mi><mo>¯</mo></mover> <mi>a</mi></msub><mo>)</mo></mrow><msub><mi>ι</mi> <mrow><msub><mo>∂</mo> <mi>μ</mi></msub></mrow></msub><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo>+</mo><mrow><mo>(</mo><msubsup><mi>R</mi> <mi>α</mi> <mi>a</mi></msubsup><msup><mi>ϵ</mi> <mi>α</mi></msup><mo>+</mo><msubsup><mi>R</mi> <mi>α</mi> <mrow><mi>a</mi><mi>μ</mi></mrow></msubsup><mfrac><mrow><mi>d</mi><msup><mi>ϵ</mi> <mi>α</mi></msup></mrow><mrow><mi>d</mi><msup><mi>x</mi> <mi>μ</mi></msup></mrow></mfrac><mo>+</mo><msup><mi>κ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><mfrac><mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mi>L</mi></mrow><mrow><mi>δ</mi><msup><mi>ϕ</mi> <mi>a</mi></msup></mrow></mfrac><mo>)</mo></mrow><msub><mover><mi>ϕ</mi><mo>¯</mo></mover> <mi>a</mi></msub><mspace width="thinmathspace"></mspace><msub><mi>dvol</mi> <mi>Σ</mi></msub><mo>}</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo><mpadded width="0"><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>s</mi></mrow></mpadded></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mi>ϵ</mi> <mi>α</mi></msup><msub><mover><mi>c</mi><mo>¯</mo></mover> <mi>α</mi></msub><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msup><mi>κ</mi> <mrow><mi>a</mi><mi>b</mi></mrow></msup><msub><mover><mi>ϕ</mi><mo>¯</mo></mover> <mi>a</mi></msub><msub><mover><mi>ϕ</mi><mo>¯</mo></mover> <mi>b</mi></msub><mo>)</mo></mrow><msub><mi>dvol</mi> <mi>Σ</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \{K\} &\overset{d}{\longrightarrow}& \{ d K + \epsilon^\alpha R^{a \mu}_\alpha \frac{\delta_{EL}\mathbf{L}}{ \delta \phi^a} \} &\overset{d}{\longrightarrow}& \{ \overset{= 0}{\overbrace{ d J_v - \iota_v \delta_{EL}\mathbf{L} }} \} \\ && \mathllap{s}\uparrow && \uparrow\mathrlap{-s} \\ && \epsilon^\alpha R^{a \mu}_\alpha \overline{\phi}_a \iota_{\partial_\mu} dvol_\Sigma &\underset{d}{\longrightarrow}& \left\{ d\left( \epsilon^\alpha R^{a \mu}_\alpha \overline{\phi}_a \right) \iota_{\partial_\mu} dvol_\Sigma + \left( R^a_\alpha \epsilon^\alpha + R^{a \mu}_\alpha \frac{d \epsilon^\alpha}{d x^\mu} + \kappa^{a b} \frac{\delta_{EL} L }{ \delta \phi^a } \right) \overline{\phi}_a \, dvol_\Sigma \right\} \\ && && \uparrow\mathrlap{-s} \\ && && \left( - \epsilon^\alpha \overline{c}_\alpha + \tfrac{1}{2}\kappa^{a b } \overline{\phi}_a \overline{\phi}_b \right) dvol_\Sigma } </annotation></semantics></math></div> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/variational+BV-BRST+bicomplex">variational BV-BRST bicomplex</a></li> </ul> <h2 id="References">References</h2> <p>The general theory:</p> <ul> <li id="BarnichBrandtHenneaux94"><a class="existingWikiWord" href="/nlab/show/Glenn+Barnich">Glenn Barnich</a>, <a class="existingWikiWord" href="/nlab/show/Friedemann+Brandt">Friedemann Brandt</a>, <a class="existingWikiWord" href="/nlab/show/Marc+Henneaux">Marc Henneaux</a>, <em>Local BRST cohomology in the antifield formalism: I. General theorems</em>, Commun. Math. Phys. <strong>174</strong> (1995) 57-92 [<a href="https://arxiv.org/abs/hep-th/9405109">arXiv:hep-th/9405109</a>, <a href="https://doi.org/10.1007/BF02099464">doi:10.1007/BF02099464</a>]</li> </ul> <p>Review:</p> <ul> <li id="BarnichBrandtHenneaux00"><a class="existingWikiWord" href="/nlab/show/Glenn+Barnich">Glenn Barnich</a>, <a class="existingWikiWord" href="/nlab/show/Friedemann+Brandt">Friedemann Brandt</a>, <a class="existingWikiWord" href="/nlab/show/Marc+Henneaux">Marc Henneaux</a>, <em>Local BRST cohomology in gauge theories</em>, Phys. Rept. <strong>338</strong> (2000) 439-569 [<a href="https://doi.org/10.1016/S0370-1573(00)00049-1">doi:10.1016/S0370-1573(00)00049-1</a><a href="https://arxiv.org/abs/hep-th/0002245">arXiv:hep-th/0002245</a>]</li> </ul> <p>Details on the <a class="existingWikiWord" href="/nlab/show/local+antibracket">local antibracket</a>:</p> <ul> <li id="BarnichHenneaux96"><a class="existingWikiWord" href="/nlab/show/Glenn+Barnich">Glenn Barnich</a>, <a class="existingWikiWord" href="/nlab/show/Marc+Henneaux">Marc Henneaux</a>, section 2 and appendix B of: <em>Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket</em>, J. Math. Phys. <strong>37</strong> (1996) 5273-5296 [<a href="https://arxiv.org/abs/hep-th/9601124">arXiv:hep-th/9601124</a>, <a href="https://doi.org/10.1063/1.531726">doi:10.1063/1.531726</a>]</li> </ul> <p>Application to <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> and/or <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> (<a class="existingWikiWord" href="/nlab/show/Einstein-Yang-Mills+theory">Einstein-Yang-Mills theory</a>) is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Glenn+Barnich">Glenn Barnich</a>, <a class="existingWikiWord" href="/nlab/show/Friedemann+Brandt">Friedemann Brandt</a>, <a class="existingWikiWord" href="/nlab/show/Marc+Henneaux">Marc Henneaux</a>, <em>Local BRST cohomology in Einstein-Yang-Mills theory</em>, Nucl. Phys. B <strong>455</strong> (1995) 357-408 [<a href="https://doi.org/10.1016/0550-3213(95)00471-4">doi:10.1016/0550-3213(95)00471-4</a><a href="https://arxiv.org/abs/hep-th/9505173">arXiv:hep-th/9505173</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 29, 2024 at 07:37:39. 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