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href="/nlab/show/local+Lagrangian">local Lagrangian</a>, <a class="existingWikiWord" href="/nlab/show/local+action+functional">local action functional</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/source+form">source form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lepage+form">Lepage form</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principle+of+extremal+action">principle of extremal action</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a>, <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noether%27s+theorem">Noether's theorem</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/conserved+current">conserved current</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetry">symmetry</a></p> </li> </ul> </li> </ul> <h2 id="derived_differential_geometric_version">Derived differential geometric version</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+complex">BV-BRST complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-module">D-module</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/variational+calculus+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="physics">Physics</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a></p> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </p> </div></div></div> <h4 id="lie_theory"><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>-Lie theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>)</p> <p><strong>Background</strong></p> <p><em>Smooth structure</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/concrete+smooth+%E2%88%9E-groupoid">concrete smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></p> </li> </ul> <p><em>Higher groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> <p><em>Lie theory</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie groupoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie algebroids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> <p><strong>Formal Lie groupoids</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a>, <a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></li> </ul> <p><strong>Cohomology</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a></p> </li> </ul> <p><strong>Homotopy</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> <p><strong>Related topics</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> <p><strong>Examples</strong></p> <p><em><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>-Lie groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a></p> </li> </ul> <p><em><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>-Lie groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <p><em><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>-Lie algebroids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <p><em><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>-Lie algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></p> </li> </ul> </div></div> <h4 id="symplectic_geometry">Symplectic geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/almost+symplectic+structure">almost symplectic structure</a>, <a class="existingWikiWord" href="/nlab/show/metaplectic+structure">metaplectic structure</a>, <a class="existingWikiWord" href="/nlab/show/metalinear+structure">metalinear structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+form">symplectic form</a>, <a class="existingWikiWord" href="/nlab/show/n-plectic+form">n-plectic form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>, <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> <p><a class="existingWikiWord" href="/nlab/show/Poisson+n-algebra">Poisson n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+infinity-groupoid">symplectic infinity-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectomorphism">symplectomorphism</a>, <a class="existingWikiWord" href="/nlab/show/symplectomorphism+group">symplectomorphism group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+vector+field">symplectic vector field</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+field">Hamiltonian vector field</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a>, <a class="existingWikiWord" href="/nlab/show/Hamiltonian+form">Hamiltonian form</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+gradient">symplectic gradient</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+action">Hamiltonian action</a>, <a class="existingWikiWord" href="/nlab/show/moment+map">moment map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+reduction">symplectic reduction</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+formalism">BRST-BV formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isotropic+submanifold">isotropic submanifold</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a>, <a class="existingWikiWord" href="/nlab/show/polarization">polarization</a></p> </li> </ul> <h2 id="classical_mechanics_and_quantization">Classical mechanics and quantization</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization">quantization</a></p> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a>,</p> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></strong>, <a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization+of+symplectic+groupoids">geometric quantization of symplectic groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/prequantum+line+bundle">prequantum line bundle</a>, <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+manifold">contact manifold</a>, <a class="existingWikiWord" href="/nlab/show/contactomorphism">contactomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contact+form">contact form</a>, <a class="existingWikiWord" href="/nlab/show/Reeb+vector+field">Reeb vector field</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a>, <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a>, <a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+bracket+Lie+n-algebra">Poisson bracket Lie n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+algebra">Heisenberg Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+n-algebra">Heisenberg Lie n-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a></p> </li> </ul> </li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/symplectic+geometry+-+contents">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> <li><a href='#LagrangianBV'>Lagrangian BV</a></li> <ul> <li><a href='#ClassicalBVAsHomologicalResolutionOfReducedPhaseSpace'>Classical BV as homological resolution of reduced phase space</a></li> <li><a href='#HomologicalIntegration'>Quantum BV as homological (path-)integration</a></li> <ul> <li><a href='#IdeaOfPathIntegralQuantization'>The idea of path integal quantization</a></li> <li><a href='#MultivectorFieldsDualToDifferentialForms'>Multivector fields dual to differential forms</a></li> <li><a href='#TheQuantumMasterEquationAsClosureOfIntegralMeasure'>The quantum master equation: the path integral measure is a closed form</a></li> <li><a href='#IntegrationOverManifoldsByBVCohomology'>Integration over manifolds by BV-cohomology</a></li> <li><a href='#BVQuantization'>BV quantization</a></li> <li><a href='#PathIntegrationAndQuantumObservablesByBVCohomology'>Quantum observables by BV-cohomology</a></li> </ul> <li><a href='#PoincareDualityOnHochschild'>Poincaré duality on Hochschild (co)homology and framed little disk algebra</a></li> <li><a href='#nonperturbative'>Non-perturbative</a></li> </ul> <li><a href='#HamiltonianBV'>Hamiltonian BFV – Homotopical Poisson reduction</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#ReferencesGeneral'>General</a></li> <li><a href='#ReferencesForLagrangianBV'>Lagrangian BV</a></li> <ul> <li><a href='#ReferencesForLagrangianBVForLagrangianTheories'>For Lagrangian theories</a></li> <li><a href='#ReferencesForNonLagrangianEquations'>For non-Lagrangian theories</a></li> <li><a href='#for_cftvertex_algebras'>For CFT/vertex algebras</a></li> </ul> <li><a href='#hamiltonian_bfv'>Hamiltonian BFV</a></li> <li><a href='#ReferencesMultisymplectic'>Multisymplectic BRST</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/physics">physics</a> and specifically in <a class="existingWikiWord" href="/nlab/show/field+theory">field theory</a>, the <em>BV-BRST formalism</em> is a tool in <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a> and <a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a> to handle the <a class="existingWikiWord" href="/nlab/show/intersection">intersection</a>- and <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a>-constructions that appear</p> <ol> <li> <p>in the construction of <a class="existingWikiWord" href="/nlab/show/reduced+phase+space">reduced</a> <a class="existingWikiWord" href="/nlab/show/phase+spaces">phase spaces</a> of <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theories">Lagrangian field theories</a>, in particular including <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a>; (“Lagrangian BV”)</p> </li> <li> <p>in <a class="existingWikiWord" href="/nlab/show/symplectic+reduction">symplectic reduction</a> of <a class="existingWikiWord" href="/nlab/show/phase+spaces">phase spaces</a> (“Hamiltonian BV”)</p> </li> </ol> <p>In either case the <em>BRST-BV</em> complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>P</mi> <mi>BV</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(P^{BV})</annotation></semantics></math> is a model in <a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a> of a joint <a class="existingWikiWord" href="/nlab/show/homotopy+intersection">homotopy intersection</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a>, hence of an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a> and <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a>, of a <a class="existingWikiWord" href="/nlab/show/space">space</a> in <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>/<a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a>.</p> <p>Accordingly, the BRST-BV complex is built from two main pieces:</p> <ol> <li>it contains in positive degree a <a class="existingWikiWord" href="/nlab/show/BRST-complex">BRST-complex</a>: the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> of the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a> which is the homotopy <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> (<a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a>) of the <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a> (in Lagrangian BV) or of the group of flows generated by the constraints (in Hamiltonian BFV) – which is in general an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> in either case – acting on configuration space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>;</li> </ol> <ul> <li>it contains in negative degree a <a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a> of the <a class="existingWikiWord" href="/nlab/show/critical+locus">critical locus</a> of the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> (for Lagrangian BV) or of the constraint surface (in Hamiltonian BFV).</li> </ul> <h2 id="LagrangianBV">Lagrangian BV</h2> <h3 id="ClassicalBVAsHomologicalResolutionOfReducedPhaseSpace">Classical BV as homological resolution of reduced phase space</h3> <p>The <em>classical Lagrangian BV-BRST complex</em> of a <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a> is, under suitable conditions, a <a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a> of the <a class="existingWikiWord" href="/nlab/show/homotopy+intersection">homotopy intersection</a> with the <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> (this is the BV part) of the <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> by the <a class="existingWikiWord" href="/nlab/show/infinitesimal+symmetries+of+the+Lagrangian">infinitesimal symmetries of the Lagrangian</a> (this is the BRST part), and hence a homological model of the <a class="existingWikiWord" href="/nlab/show/reduced+phase+space">reduced phase space</a> of the Lagrangian field theory.</p> <p>A detailed introduction to the classical Lagrangian BV-BRST formalism is at</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/A+first+idea+of+quantum+field+theory">A first idea of quantum field theory</a></em>, chapter <em><a href="geometry+of+physics+--+A+first+idea+of+quantum+field+theory#ReducedPhaseSpace">11. Reduced phase space</a></em></li> </ul> <h3 id="HomologicalIntegration">Quantum BV as homological (path-)integration</h3> <p>We discuss here the interpretation of the <em>quantum BV-complex</em> as a homological implementation of <a class="existingWikiWord" href="/nlab/show/integration">integration</a> thought of as <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a>-<a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> (in <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>).</p> <p>We indicate how on a finite dimensional smooth manifold the <a class="existingWikiWord" href="/nlab/show/BV-algebra">BV-algebra</a> appearing in Lagrangian BV-formalism is the dual of the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> of <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a> in the presence of a <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a> and how, by extention, this allows to interpret the BV-complex as a means for defining (<a class="existingWikiWord" href="/nlab/show/path+integral">path</a>-)<a class="existingWikiWord" href="/nlab/show/integration">integration</a> over general <a class="existingWikiWord" href="/nlab/show/configuration+spaces">configuration spaces</a> of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> by passing to BV-<a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a>.</p> <p>(The interpretation of the BV-differential as the dual de Rham differential necessary for this is due to (<a href="#Witten90">Witten 90</a>) (<a href="#Schwarz92">Schwarz 92</a>). A particularly clear-sighted account of the general relation is in <a href="#Gwilliam">Gwilliam 2013</a> ).</p> <p>Further <a href="#PoincareDualityOnHochschild">below</a> we discuss the generalization of these relation in terms of <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a> on <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild (co)homology</a>.</p> <ol> <li> <p><a href="#IdeaOfPathIntegralQuantization">The idea of path integral quantization</a></p> </li> <li> <p><a href="#MultivectorFieldsDualToDifferentialForms">Multivector fields dual to differential forms</a></p> </li> <li> <p><a href="#TheQuantumMasterEquationAsClosureOfIntegralMeasure">The quantum master equation: the path integral measure is a closed form</a></p> </li> <li> <p><a href="#IntegrationOverManifoldsByBVCohomology">Integration over manifolds by BV cohomology</a></p> </li> <li> <p><a href="#BVQuantization">BV-quantization</a></p> </li> <li> <p><a href="#PathIntegrationAndQuantumObservablesByBVCohomology">Path integration and quantum observables by BV-cohomology</a></p> </li> </ol> <h4 id="IdeaOfPathIntegralQuantization">The idea of path integal quantization</h4> <p>The <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> is supposed to be the <a class="existingWikiWord" href="/nlab/show/integral">integral</a> over a <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> using a <a class="existingWikiWord" href="/nlab/show/measure">measure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\mu_S</annotation></semantics></math> which is thought of in 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>μ</mi> <mi>S</mi></msub><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>exp</mi><mrow><mo>(</mo><mfrac><mi>i</mi><mi>ℏ</mi></mfrac><mi>S</mi><mrow><mo>(</mo><mi>ϕ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>⋅</mo><mi>μ</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>ϕ</mi><mo>∈</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mu_S(\phi) \coloneqq \exp\left(\frac{i}{\hbar} S\left(\phi\right)\right) \cdot \mu(\phi) \;\;\;\; \phi \in X \,, </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> some other measure and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">S : X \to \mathbb{R}</annotation></semantics></math> the <em><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a></em> of the <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> on the space of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> its value as an <a class="existingWikiWord" href="/nlab/show/observable">observable</a> of the system is supposed to be what would be the <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>f</mi><msub><mo stretchy="false">⟩</mo> <mi>S</mi></msub><mo>=</mo><mfrac><mrow><msub><mo>∫</mo> <mrow><mi>ϕ</mi><mo>∈</mo><mi>Fields</mi></mrow></msub><mi>f</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>μ</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mo>∫</mo> <mrow><mi>ϕ</mi><mo>∈</mo><mi>Fields</mi></mrow></msub><mi>μ</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex"> \langle f \rangle_S = \frac{\int_{\phi \in Fields} f(\phi) \cdot \mu(\phi)}{\int_{\phi \in Fields} \mu(\phi) } </annotation></semantics></math></div> <p>if the measure existed. Of course this does not make sense in terms of the usual notion of <a class="existingWikiWord" href="/nlab/show/integration">integration</a> against <a class="existingWikiWord" href="/nlab/show/measures">measures</a> since such measures do not exists except in the simplest situation. But there is a <a class="existingWikiWord" href="/nlab/show/cohomology">cohomological</a> notion of integration where instead of actually performing an integral, we identify its value, if it exists, with a cohomology class and generally interpret that cohomology class as the expectation value, even if an actual integral against a measure does not exist. This is what BV formalism achieves, which we discuss after some preliminaries below in <em><a href="#IntegrationOverManifoldsByBVCohomology">Integration over manifolds by BV cohomology</a></em>.</p> <h4 id="MultivectorFieldsDualToDifferentialForms">Multivector fields dual to differential forms</h4> <p>If one thinks of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as an ordinary <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo><</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d \lt \infty)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\mu_S</annotation></semantics></math> will be given by a <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>S</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mi>d</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu_S \in \Omega^d(X)</annotation></semantics></math>. By contraction of <a class="existingWikiWord" href="/nlab/show/multivector+fields">multivector fields</a> with <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a>, every choice of volume form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> induces an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> between <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> and <a class="existingWikiWord" href="/nlab/show/polyvector+fields">polyvector fields</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo lspace="verythinmathspace">:</mo><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mo>∧</mo> <mrow><mi>d</mi><mo>−</mo><mo>•</mo></mrow></msup><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mu \colon \Omega^\bullet(X) \stackrel{\simeq}{\longrightarrow} \wedge^{d-\bullet} \Gamma(T X) \,, </annotation></semantics></math></div> <p>which is usefully thought of as reversing degrees. Under this isomorphism the <a class="existingWikiWord" href="/nlab/show/deRham+differential">deRham differential</a> maps to a <a class="existingWikiWord" href="/nlab/show/divergence">divergence</a> operator, the <em><a class="existingWikiWord" href="/nlab/show/BV-operator">BV-operator</a></em>, conventionally denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>μ</mi><mo>∘</mo><msub><mi>d</mi> <mi>dR</mi></msub><mo>∘</mo><msup><mi>μ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> \Delta \;\coloneqq\; \mu \circ d_{dR} \circ \mu^{-1} </annotation></semantics></math></div> <p>which interacts naturally with the canonical bracket on multivector fields: the <a class="existingWikiWord" href="/nlab/show/Schouten+bracket">Schouten bracket</a>. (See at <em><a class="existingWikiWord" href="/nlab/show/polyvector+field">polyvector field</a></em> for more details.)</p> <div class="num_defn" id="TheDualBVComplexOfTheDeRhamComplexOnAManifolds"> <h6 id="definition">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu \in \Omega^n(X)</annotation></semantics></math>a <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>BV</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>Δ</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> BV(X, \mu) \coloneqq (\wedge^\bullet \Gamma(T X), \Delta_\mu) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complex</a> induced on <a class="existingWikiWord" href="/nlab/show/multivector+fields">multivector fields</a> by dualizing the <a class="existingWikiWord" href="/nlab/show/de+Rham+differential">de Rham differential</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>.</p> </div> <div class="num_remark" id="BVComplexOfManifoldIsPoisson0"> <h6 id="remark">Remark</h6> <p>The <a class="existingWikiWord" href="/nlab/show/Schouten+bracket">Schouten bracket</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>BV</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">BV(X,\mu)</annotation></semantics></math> makes this cochain complex a <a class="existingWikiWord" href="/nlab/show/Poisson+0-algebra">Poisson 0-algebra</a>.</p> </div> <p>For more see at <em><a class="existingWikiWord" href="/nlab/show/relation+between+BV+and+BD">relation between BV and BD</a></em>.</p> <h4 id="TheQuantumMasterEquationAsClosureOfIntegralMeasure">The quantum master equation: the path integral measure is a closed form</h4> <p>Observe that</p> <ul> <li> <p>if we think of</p> <ul> <li> <p>the measure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> as some closed reference differential form on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>the exponentiated action functional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mrow><mo>(</mo><mfrac><mi>i</mi><mi>ℏ</mi></mfrac><mi>S</mi><mrow><mo>(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>)</mo></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">exp\left(\frac{i}{\hbar}S\left(-\right)\right)</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/multivector+field">multivector field</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>;</p> </li> <li> <p>the expression <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mfrac><mi>i</mi><mi>ℏ</mi></mfrac><mi>S</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">exp(\frac{i}{\hbar}S(-)) \mu</annotation></semantics></math> as the contraction of this multivector field with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></p> </li> </ul> </li> <li> <p>then the <strong>BV quantum master equaton</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mi>exp</mi><mo stretchy="false">(</mo><mfrac><mi>i</mi><mi>ℏ</mi></mfrac><mi>S</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Delta \exp(\frac{i}{\hbar}S) = 0</annotation></semantics></math> says nothing but that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mfrac><mi>i</mi><mi>ℏ</mi></mfrac><mi>S</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">exp(\frac{i}{\hbar}S(-)) \mu</annotation></semantics></math> is a <em>closed differential form</em>.</p> </li> <li> <p>If we furthermore take into account that in the presence of gauge symmetries the space <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 not a plain manifold but the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Lie+infinity-algebroid">algebroid</a> of the gauge symmetries acting on the space of fields, hence an <a class="existingWikiWord" href="/nlab/show/NQ-supermanifold">NQ-supermanifold</a> (whose Chevalley-Eilenberg algebra is the <strong>BRST complex</strong>), then this just says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mfrac><mi>i</mi><mi>ℏ</mi></mfrac><mi>S</mi><mo stretchy="false">)</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">\exp(\frac{i}{\hbar}S) \mu</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integrable form</a> in the sense of <a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration theory of supermanifolds</a>.</p> </li> </ul> <p>This means that Lagrangian BV formalism is nothing but a way of describing closed differential forms on <a class="existingWikiWord" href="/nlab/show/Lie+infinity-algebroid">Lie infinity-algebroid</a> in terms of multivectors contracted into a reference differention form. The multivectors dual to degree 0 elements in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Lie+infinity-algebroid">algebroid</a> are the so-called “<strong>anti-fields</strong>”, while those dual to the higher degree elements are the so-called “<strong>anti-ghosts</strong>”.</p> <h4 id="IntegrationOverManifoldsByBVCohomology">Integration over manifolds by BV-cohomology</h4> <p>The following proposition about integration of differential <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>-forms is the archetype for interpreting cohomology in BV-complexes in terms of <a class="existingWikiWord" href="/nlab/show/integration">integration</a>.</p> <div class="num_example"> <h6 id="example">Example</h6> <p>On the <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a> of compact support <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∫</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Ω</mi> <mi>cp</mi> <mi>n</mi></msubsup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\int \;\colon\; \Omega^n_{cp} \to \mathbb{R}</annotation></semantics></math> is equivalently given by the projection onto the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> by the exact forms, hence by passing to <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> in the truncated <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>B</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi><mo>→</mo><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msup><mi>B</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msup><mi>B</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(B^n) \to \cdots \to \Omega^{n-1}(B^n) \to \Omega^n(B^n)</annotation></semantics></math>.</p> </div> <p>See at <em><a class="existingWikiWord" href="/nlab/show/kernel+of+integration+is+the+exact+differential+forms">kernel of integration is the exact differential forms</a></em> for details.</p> <p>This “integration without integration” is discussed in more detail at <em><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a></em>.</p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed</a> <a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/finite+number">finite number</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>S</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu_S \in \Omega^n(X)</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a>. Let again</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>BV</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>μ</mi> <mi>S</mi></msub><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>Δ</mi> <mrow><msub><mi>μ</mi> <mi>S</mi></msub></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> BV(X,\mu_S) \coloneqq( \wedge^\bullet \Gamma(T X), \Delta_{\mu_S} ) </annotation></semantics></math></div> <p>be the corresponding dual <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complex</a> of the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> by def. <a class="maruku-ref" href="#TheDualBVComplexOfTheDeRhamComplexOnAManifolds"></a> above.</p> <div class="num_defn" id="ExpectationValueOfFunctionOnManifold"> <h6 id="definition_2">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in C^\infty(X)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>, its <strong><a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a></strong> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\mu_S</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>f</mi><msub><mo stretchy="false">⟩</mo> <mrow><msub><mi>μ</mi> <mi>S</mi></msub></mrow></msub><mo>≔</mo><mfrac><mrow><msub><mo>∫</mo> <mi>X</mi></msub><mi>f</mi><mo>⋅</mo><msub><mi>μ</mi> <mi>S</mi></msub></mrow><mrow><msub><mo>∫</mo> <mi>X</mi></msub><msub><mi>μ</mi> <mi>S</mi></msub></mrow></mfrac><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \langle f\rangle_{\mu_S} \coloneqq \frac{ \int_X f \cdot \mu_S }{\int_X \mu_S } \,. </annotation></semantics></math></div></div> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mi>BV</mi></msub></mrow><annotation encoding="application/x-tex">[-]_{BV}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> classes in the BV complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>BV</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>μ</mi> <mi>S</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">BV(X, \mu_S)</annotation></semantics></math>.</p> <div class="num_prop" id="ExpectationValueOfFunctionOnManifoldByBVCohomology"> <h6 id="proposition">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>BV</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>μ</mi> <mi>S</mi></msub><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>≃</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in BV(X,\mu_S)_0 \simeq C^\infty(X)</annotation></semantics></math> the cohomology class of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> in the BV complex is the <a class="existingWikiWord" href="/nlab/show/expectation+value">expectation value</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, def. <a class="maruku-ref" href="#ExpectationValueOfFunctionOnManifold"></a> times the cohomology class of the unit function 1:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>f</mi><msub><mo stretchy="false">]</mo> <mi>BV</mi></msub><mo>=</mo><mo stretchy="false">⟨</mo><mi>f</mi><msub><mo stretchy="false">⟩</mo> <mrow><msub><mi>μ</mi> <mi>S</mi></msub></mrow></msub><mo stretchy="false">[</mo><mn>1</mn><msub><mo stretchy="false">]</mo> <mi>BV</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [f]_{BV} = \langle f\rangle_{\mu_S} [1]_{BV} \,. </annotation></semantics></math></div></div> <p>See (<a href="#Gwilliam">Gwilliam 13, lemma 2.2.2</a>).</p> <h4 id="BVQuantization">BV quantization</h4> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed manifold</a> as above and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>BV</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">BV(X, \mu)</annotation></semantics></math> for the BV-complex def. <a class="maruku-ref" href="#TheDualBVComplexOfTheDeRhamComplexOnAManifolds"></a>, induced by a given <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu \in \Omega^n(X)</annotation></semantics></math>.</p> <div class="num_prop" id="ShiftInBVDifferentialOnManifoldDueToFunctional"> <h6 id="proposition_2">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S \in C^\infty(X)</annotation></semantics></math> then the BV-complex induced via def. <a class="maruku-ref" href="#TheDualBVComplexOfTheDeRhamComplexOnAManifolds"></a> by the volume form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>S</mi></msub><mo>≔</mo><mi>exp</mi><mrow><mo>(</mo><mfrac><mn>1</mn><mi>ℏ</mi></mfrac><mi>S</mi><mo>)</mo></mrow><mo>⋅</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex"> \mu_S \coloneqq \exp\left(\frac{1}{\hbar} S\right) \cdot \mu </annotation></semantics></math></div> <p>(for any constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math> to be read as <a class="existingWikiWord" href="/nlab/show/Planck%27s+constant">Planck's constant</a>) has BV-differential related to that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> itself by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Δ</mi> <mrow><msub><mi>μ</mi> <mi>S</mi></msub></mrow></msub><mo>=</mo><msub><mi>Δ</mi> <mi>μ</mi></msub><mo>+</mo><mfrac><mn>1</mn><mi>ℏ</mi></mfrac><msub><mi>ι</mi> <mrow><mi>d</mi><mi>S</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Delta_{\mu_S} = \Delta_\mu + \frac{1}{\hbar}\iota_{d S} \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ι</mi> <mrow><mi>d</mi><mi>S</mi></mrow></msub><mo>:</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mo>∧</mo> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\iota_{d S} : \wedge^\bullet \Gamma(T X) \to \wedge^{\bullet-1} \Gamma(T X)</annotation></semantics></math> is the operation of acting with a <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a> on <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> by <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, extended as a graded <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> to <a class="existingWikiWord" href="/nlab/show/multivector+fields">multivector fields</a>.</p> </div> <div class="num_prop" id="ClassicalBVComplexOnManifoldAsDerivedCriticalLocus"> <h6 id="proposition_3">Proposition</h6> <p>The complex</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>BV</mi> <mi>cl</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>S</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>ι</mi> <mrow><mi>d</mi><mi>S</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> BV_{cl}(X, S) \coloneqq (\wedge^\bullet \Gamma(T X), \iota_{d S}) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/derived+critical+locus">derived critical locus</a> of the function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>.</p> </div> <p>By the discussion at <em><a class="existingWikiWord" href="/nlab/show/derived+critical+locus">derived critical locus</a></em>.</p> <div class="num_remark" id="ClassicalAndQuantumBVComplexOverManifold"> <h6 id="remark_2">Remark</h6> <p>Prop. <a class="maruku-ref" href="#ShiftInBVDifferentialOnManifoldDueToFunctional"></a> and prop. <a class="maruku-ref" href="#ClassicalBVComplexOnManifoldAsDerivedCriticalLocus"></a> together say that the BV-complex of a manifold <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> for a volume form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>S</mi></msub></mrow><annotation encoding="application/x-tex">\mu_S</annotation></semantics></math> shifted from a background volume form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> by a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mrow><mo>(</mo><mfrac><mn>1</mn><mi>ℏ</mi></mfrac><mi>S</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex">\exp\left(\frac{1}{\hbar} S\right)</annotation></semantics></math> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math>-deformation of the <a class="existingWikiWord" href="/nlab/show/derived+critical+locus">derived critical locus</a> of <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> by a contribution of the background volume form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>.</p> <p>We call <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>ι</mi> <mrow><mi>d</mi><mi>S</mi></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\wedge^\bullet \Gamma(T X), \iota_{d S})</annotation></semantics></math> the <strong>classical BV complex</strong> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>Γ</mi><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>ι</mi> <mrow><mi>d</mi><mi>S</mi></mrow></msub><mo>+</mo><mi>ℏ</mi><msub><mi>Δ</mi> <mi>μ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\wedge^\bullet \Gamma(T X), \iota_{d S} + \hbar \Delta_{\mu} )</annotation></semantics></math> the <strong>quantum BV complex</strong> of the manifold <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> equipped with the function <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 the voume form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>.</p> </div> <p>The crucial idea now is the following.</p> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p><strong>(central idea of BV quantization)</strong></p> <p>In the above discussion of BV complexes over finite-dimensional manifolds, the construction of the <strong>classical BV complex</strong> in remark <a class="maruku-ref" href="#ClassicalAndQuantumBVComplexOverManifold"></a> as a <a class="existingWikiWord" href="/nlab/show/derived+critical+locus">derived critical locus</a> directly makes sense in great generality for <a class="existingWikiWord" href="/nlab/show/action+functionals">action functionals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> defined on spaces of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> more general than finite-dimensional <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>. (It makes sense in a general context of <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a>, see at <em><a href="differential+cohesive+infinity-topos#CriticalLocus">differential cohesive infinity-topos – critical locus</a></em>). On the other hand, the construction of the quantum BV complex as the dual to the de Rham complex by a <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a> by def. <a class="maruku-ref" href="#TheDualBVComplexOfTheDeRhamComplexOnAManifolds"></a> breaks down as soon as the space of fields is no longer a finite dimensional manifold, hence breaks down for all but the most degenerate <a class="existingWikiWord" href="/nlab/show/quantum+field+theories">quantum field theories</a>. But by remark <a class="maruku-ref" href="#ClassicalAndQuantumBVComplexOverManifold"></a> we may instead think of the quantum BV complex as a certain <strong>deformation</strong> of the classical BV complex, and <em>that</em> notion continues to make sense in full generality.</p> <p>And once such a deformation of a critical locus has been obtained, we may read prop. <a class="maruku-ref" href="#ExpectationValueOfFunctionOnManifoldByBVCohomology"></a> the other way round and regard the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of the deformed complex as the <em>definition</em> of quantum expectation values of <a class="existingWikiWord" href="/nlab/show/observables">observables</a>.</p> </div> <p>See for instance (<a href="#Park">Park, 2.1</a>)</p> <p>In order to implement this idea, we need to axiomatize those properties of classical BV complexes and their quantum deformation as above which we demand to be preserved by the generalization away from finite dimensional manifolds. This is what the following definitions do.</p> <div class="num_defn" id="ClassicalBVComplexAsPoisson0Algebra"> <h6 id="definition_3">Definition</h6> <p>A <strong>classical BV complex</strong> is a <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complex</a> equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/Poisson+0-algebra">Poisson 0-algebra</a>.</p> </div> <div class="num_defn" id="BeilinsonDrinfeldAlgebra"> <h6 id="definition_4">Definition</h6> <p>A <strong>quantum BV complex</strong> or <strong>Beilinson-Drinfeld algebra</strong> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+algebra">graded algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> over the ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{R} [ [ \hbar ] ]</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/formal+power+series">formal power series</a> in a formal constant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math>, equipped with a <a class="existingWikiWord" href="/nlab/show/Poisson+0-algebra">Poisson bracket</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{-,-\}</annotation></semantics></math> of degree 1 and with an operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\Delta \colon A \to A</annotation></semantics></math> of degree 1 which satisfies:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Delta^2 = 0</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>Δ</mi><mi>a</mi><mo stretchy="false">)</mo><mi>b</mi><mo>+</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mi>a</mi><mo stretchy="false">|</mo></mrow></msup><mi>a</mi><mo stretchy="false">(</mo><mi>Δ</mi><mi>b</mi><mo stretchy="false">)</mo><mo>+</mo><mi>ℏ</mi><mo stretchy="false">{</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\Delta( a b) = (\Delta a) b + (-1)^{\vert a\vert} a (\Delta b) + \hbar \{a,b\}</annotation></semantics></math> for all homogenous elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a, b \in A</annotation></semantics></math></p> </li> </ol> </div> <p>In (<a href="#Gwilliam">Gwilliam 2013</a>) this is def. 2.2.5.</p> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>A <a class="existingWikiWord" href="/nlab/show/Beilinson-Drinfeld+algebra">Beilinson-Drinfeld algebra</a> is <em>not</em> a <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a> with <a class="existingWikiWord" href="/nlab/show/differential">differential</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math>: the Poisson bracket <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\hbar \{-,-\}</annotation></semantics></math> measures the failure for the differential to satisfy the <a class="existingWikiWord" href="/nlab/show/Leibniz+rule">Leibniz rule</a>. In particular the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> is <em>not</em> an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a>.</p> <p>In this respect the notion of BV-quantization via BD-algebras differs from other traditional notions of BV-quantization, where one demands the quantum BV-complex to be a noncommutative dg-algebra deformation of the classical BV complex. But instead the BD-algebras induced by a <a class="existingWikiWord" href="/nlab/show/local+action+functional">local action functional</a> and varying over open subsets of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> form a <a class="existingWikiWord" href="/nlab/show/factorization+algebra">factorization algebra</a> and <em>that</em> encodes the <a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a>: the <em><a class="existingWikiWord" href="/nlab/show/factorization+algebra+of+observables">factorization algebra of observables</a></em> (see there for more).</p> </div> <p>But:</p> <div class="num_defn" id="ClassicalLimitOfBeilinsonDrinfeldAlgebra"> <h6 id="definition_5">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>ℏ</mi></msub></mrow><annotation encoding="application/x-tex">A_\hbar</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Beilinson-Drinfeld+algebra">Beilinson-Drinfeld algebra</a>, its <a class="existingWikiWord" href="/nlab/show/classical+limit">classical limit</a> is the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+algebras">tensor product of algebras</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mrow><mi>ℏ</mi><mo>=</mo><mn>0</mn></mrow></msub><mo>≔</mo><msub><mi>A</mi> <mi>ℏ</mi></msub><msub><mo>⊗</mo> <mrow><mi>ℝ</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>ℏ</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow></msub><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> A_{\hbar = 0} \coloneqq A_\hbar \otimes_{\mathbb{R}[ [ \hbar ] ]} \mathbb{R} </annotation></semantics></math></div> <p>hence the result of setting the formal parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/Planck%27s+constant">Planck's constant</a>”) to 0.</p> </div> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>The classical limit of a <a class="existingWikiWord" href="/nlab/show/Beilinson-Drinfeld+algebra">Beilinson-Drinfeld algebra</a> is canonically a classical BV-complex, def. <a class="maruku-ref" href="#ClassicalBVComplexAsPoisson0Algebra"></a>.</p> </div> <div class="num_defn" id="BVQuantizationByBDAlgebra"> <h6 id="definition_6">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mrow><mi>ℏ</mi><mo>=</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">A_{\hbar = 0}</annotation></semantics></math> a classical BV complex, def. <a class="maruku-ref" href="#ClassicalBVComplexAsPoisson0Algebra"></a>, a <strong>BV quantization</strong> of it is a <a class="existingWikiWord" href="/nlab/show/Beilinson-Drinfeld+algebra">Beilinson-Drinfeld algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>ℏ</mi></msub></mrow><annotation encoding="application/x-tex">A_{\hbar}</annotation></semantics></math>, def. <a class="maruku-ref" href="#BeilinsonDrinfeldAlgebra"></a> whose classical limit, def. <a class="maruku-ref" href="#ClassicalLimitOfBeilinsonDrinfeldAlgebra"></a>, is the given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mrow><mi>ℏ</mi><mo>=</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">A_{\hbar = 0}</annotation></semantics></math>.</p> </div> <p>In (<a href="#Gwilliam">Gwilliam 2013</a>) this is def. 2.2.6.</p> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a></th><th><a class="existingWikiWord" href="/nlab/show/kinetic+action">kinetic action</a></th><th><a class="existingWikiWord" href="/nlab/show/interaction">interaction</a></th><th><a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> <a class="existingWikiWord" href="/nlab/show/measure">measure</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_66547e90e66a0bf33e020921b1eaa3943837feb2_1"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>S</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi>μ</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\exp(-S(\phi)) \cdot \mu = </annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_66547e90e66a0bf33e020921b1eaa3943837feb2_2"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>ϕ</mi><mo>,</mo><mi>Q</mi><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\exp(-(\phi, Q \phi)) \cdot</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_66547e90e66a0bf33e020921b1eaa3943837feb2_3"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\exp(I(\phi)) \cdot</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_66547e90e66a0bf33e020921b1eaa3943837feb2_4"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/BV-complex">BV</a> <a class="existingWikiWord" href="/nlab/show/differential">differential</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/elliptic+complex">elliptic complex</a> +</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/antibracket">antibracket</a> with interaction +</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/BV-Laplacian">BV-Laplacian</a></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_66547e90e66a0bf33e020921b1eaa3943837feb2_5"><semantics><mrow><msub><mi>d</mi> <mi>q</mi></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">d_q =</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_66547e90e66a0bf33e020921b1eaa3943837feb2_6"><semantics><mrow><mi>Q</mi></mrow><annotation encoding="application/x-tex">Q</annotation></semantics></math> +</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_66547e90e66a0bf33e020921b1eaa3943837feb2_7"><semantics><mrow><mo stretchy="false">{</mo><mi>I</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{I,-\}</annotation></semantics></math> +</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_66547e90e66a0bf33e020921b1eaa3943837feb2_8"><semantics><mrow><mi>ℏ</mi><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\hbar \Delta</annotation></semantics></math></td></tr> </tbody></table> </div> <h4 id="PathIntegrationAndQuantumObservablesByBVCohomology">Quantum observables by BV-cohomology</h4> <div class="num_defn"> <h6 id="definition_7">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/quantum+BV-complex">quantum BV-complex</a>, its <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> are the <strong><a class="existingWikiWord" href="/nlab/show/expectation+values">expectation values</a> of <a class="existingWikiWord" href="/nlab/show/observables">observables</a></strong> of the <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory</a>.</p> <p>Specifically, an observable is a closed element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> in the quantum BV-complex and its <em>expectation value</em> is its image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>f</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[f]</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a>.</p> </div> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/quantum+BV-complex">quantum BV-complex</a> by def. <a class="maruku-ref" href="#BVQuantizationByBDAlgebra"></a> its <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> is, by definition, a perturbation of that of its <a class="existingWikiWord" href="/nlab/show/classical+limit">classical limit</a> BV complex, def. <a class="maruku-ref" href="#ClassicalLimitOfBeilinsonDrinfeldAlgebra"></a>. Accordingly, the quantum observables may be computed from the classical observables by the <a class="existingWikiWord" href="/nlab/show/homological+perturbation+lemma">homological perturbation lemma</a>. For <a class="existingWikiWord" href="/nlab/show/free+field+theories">free field theories</a> this yields <a class="existingWikiWord" href="/nlab/show/Wick%27s+lemma">Wick's lemma</a> and <a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a> for computing observables. (<a href="#Gwilliam">Gwilliam 2013, section 2.3</a>).</p> </div> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>For local theories (…) <a class="existingWikiWord" href="/nlab/show/gauge+fixing+operator">gauge fixing operator</a> (…) <a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a> (…)</p> </div> <p>(…)</p> <h3 id="PoincareDualityOnHochschild">Poincaré duality on Hochschild (co)homology and framed little disk algebra</h3> <p>The <a href="#HomologicalIntegration">above</a> duality between <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> and <a class="existingWikiWord" href="/nlab/show/multivector+field">multivector field</a> may be understood in a more general context.</p> <p>Multivector fields may be understood in terms of <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. Under the identification of <a class="existingWikiWord" href="/nlab/show/Hochschild+homology">Hochschild homology</a>/<a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a> with the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> the product of the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(i S(-))</annotation></semantics></math> with a formal <a class="existingWikiWord" href="/nlab/show/measure">measure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>vol</mi></mrow><annotation encoding="application/x-tex">vol</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is regarded as a <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a> in <a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a>. Or rather, an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> with <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> is picked, and interpreted as a choice of <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>vol</mi></mrow><annotation encoding="application/x-tex">vol</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\exp(i S(-))</annotation></semantics></math> is regarded as a cocycle in cyclic cohomology, hence as a <a class="existingWikiWord" href="/nlab/show/multivector+field">multivector field</a> whose closure condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Delta \exp(i S(-)) = 0 </annotation></semantics></math> is the quantum master equation of BV-formalism.</p> <p>By the identification of <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a><br />with functions on <a class="existingWikiWord" href="/nlab/show/derived+loop+space">derived loop space</a>s we know that the operator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> encodes the rotation of loops. Accordingly, the resuling <a class="existingWikiWord" href="/nlab/show/BV-algebra">BV-algebra</a> has an interpretation as an algebra over (the homology of) the <a class="existingWikiWord" href="/nlab/show/framed+little+disk+operad">framed little disk operad</a>.</p> <p>For certain algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> there exists <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a> between <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> and <a class="existingWikiWord" href="/nlab/show/Hochschild+homology">Hochschild homology</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo>:</mo><msub><mi>HH</mi> <mi>i</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>HH</mi> <mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tau : HH_i(A) \to HH^{n-i}(A) </annotation></semantics></math></div> <p>(<a href="#VanDenBergh">VanDenBergh</a>) and this takes the <a class="existingWikiWord" href="/nlab/show/Connes+coboundary+operator">Connes coboundary operator</a> to the <a class="existingWikiWord" href="/nlab/show/BV+operator">BV operator</a> (<a href="#Ginzburg">Ginzburg</a>).</p> <h3 id="nonperturbative">Non-perturbative</h3> <p>On <a class="existingWikiWord" href="/nlab/show/non-perturbative+quantum+field+theory">non-perturbative</a> enhancements of BV-BRST formalism cast in (<a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential</a>) <a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos">cohesive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>∞</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(\infty,1)</annotation> </semantics> </math>-topos</a> theory:</p> <ul> <li id="AlfonsiYoung23"><a class="existingWikiWord" href="/nlab/show/Luigi+Alfonsi">Luigi Alfonsi</a>, <a class="existingWikiWord" href="/nlab/show/Charles+A.+S.+Young">Charles A. S. Young</a>, <em>Towards non-perturbative BV-theory via derived differential cohesive geometry</em> [<a href="https://arxiv.org/abs/2307.15106">arXiv:2307.15106</a>]</li> </ul> <h2 id="HamiltonianBV">Hamiltonian BFV – Homotopical Poisson reduction</h2> <p>The following is a rough survey of homotopical Poisson reduction, following (<a href="#Stasheff96">Stasheff 96</a>).</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \{-,-\})</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth</a> <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>≔</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \coloneqq C^\infty(X)</annotation></semantics></math> be its <a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a> of <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>s.</p> <p>Consider</p> <ul> <li> <p>an <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊂</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">I \subset A</annotation></semantics></math></p> </li> <li> <p>that is closed under the Poisson bracket</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>I</mi><mo>,</mo><mi>I</mi><mo stretchy="false">}</mo><mo>⊂</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\{I,I\} \subset I</annotation></semantics></math></p> <p>(one says that we have <em><a class="existingWikiWord" href="/nlab/show/first+class+constraint">first class constraint</a></em> or that the 0-locus of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is <em>coisotropic</em>)</p> </li> </ul> <p>By the Poisson bracket <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> acts on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. The <strong>Poisson reduction</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is the combined</p> <ol> <li> <p>passage to the 0-locus of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>, which algebraically (dually) is passage to the quotient algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">/</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">A/I</annotation></semantics></math>;</p> </li> <li> <p>passage to the quotient of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/action">action</a>, which dually is the passage to the invariant subalgebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mi>I</mi></msup></mrow><annotation encoding="application/x-tex">A^I</annotation></semantics></math>.</p> </li> </ol> <p>This may be achieved in different orders:</p> <div class="num_defn"> <h6 id="definition_8">Definition</h6> <p>The <strong>Sniatycky-Weinstein reduction is the object</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>SW</mi></msub><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mi>I</mi><msup><mo stretchy="false">)</mo> <mi>I</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A_{SW} := (A/I)^I \,. </annotation></semantics></math></div> <p>The <strong>Dirac reduction</strong> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>Dirac</mi></msub><mo>:</mo><mo>=</mo><mi>N</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>I</mi></mrow><annotation encoding="application/x-tex"> A_{Dirac} := N(I)/I </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">{</mo><mi>f</mi><mo>∈</mo><mi>A</mi><mo stretchy="false">|</mo><mo stretchy="false">{</mo><mi>f</mi><mo>,</mo><mi>I</mi><mo stretchy="false">}</mo><mo>⊂</mo><mi>I</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">N(I) = \{f \in A | \{f, I\} \subset I\}</annotation></semantics></math> is the “subalgebra of <a class="existingWikiWord" href="/nlab/show/observable">observable</a>s”.</p> </div> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>These two algebras are <a class="existingWikiWord" href="/nlab/show/isomorphic">isomorphic</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>red</mi></msub><mo>:</mo><mo>=</mo><msub><mi>A</mi> <mi>SW</mi></msub><mo>≃</mo><msub><mi>A</mi> <mi>Dirac</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A_{red} := A_{SW} \simeq A_{Dirac} \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>Suppose a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathfrak{g}</annotation></semantics></math> acts on the <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson 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>, by <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a>. This is equivalently encoded in a <a class="existingWikiWord" href="/nlab/show/moment+map">moment map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mu : X \to \mathfrak{g}^*</annotation></semantics></math>.</p> <p>Let then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> of functions that vanish on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>μ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu^{-1}(0)</annotation></semantics></math>. This is always coisotropic.</p> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>red</mi></msub></mrow><annotation encoding="application/x-tex">A_{red}</annotation></semantics></math> is the algebraic dual to the preimage <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>μ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu^{-1}(0)</annotation></semantics></math> quotiented by the Lie algebra action: the “constraint surface” quotiented by the symmetries.</p> <p>In fact, if 0 is a <a class="existingWikiWord" href="/nlab/show/regular+value">regular value</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>red</mi></msub><mo>:</mo><mo>=</mo><msup><mi>μ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X_{red} := \mu^{-1}(0)/G</annotation></semantics></math> is a submanifold and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>red</mi></msub><mo>≃</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>red</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A_{red} \simeq C^\infty(X_{red}) \,. </annotation></semantics></math></div></div> <p>We now discuss the BRST-BV complex for the set of constraints <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \{-,-\})</annotation></semantics></math>, which will be a resolution of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>red</mi></msub></mrow><annotation encoding="application/x-tex">A_{red}</annotation></semantics></math> in the following sense:</p> <ul> <li> <p>instead of forming the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X/G</annotation></semantics></math> we form the <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> or <a class="existingWikiWord" href="/nlab/show/quotient+stack">quotient stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X//G</annotation></semantics></math>. More precisely we do this for the infinitesimal action and consider a quotient <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>;</p> </li> <li> <p>instead of forming the intersecton <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo stretchy="false">|</mo> <mrow><mi>I</mi><mo>=</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X|_{I = 0}</annotation></semantics></math> we consider its derived locus.</p> </li> </ul> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>T</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>T</mi> <mi>N</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{T_1, \cdots, T_N\}</annotation></semantics></math> be any finite set of generators of the <a class="existingWikiWord" href="/nlab/show/ideal">ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>. Then there exists a non-positively graded <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complex</a> on the graded algebra</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>Sym</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> A \otimes Sym(V) \,, </annotation></semantics></math></div> <p>where <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> is a <a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a> in non-positive degree and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sym</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sym(V)</annotation></semantics></math> is its symmetric <a class="existingWikiWord" href="/nlab/show/tensor+algebra">tensor algebra</a>: the <a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">C^\infty(X)/I</annotation></semantics></math>.</p> <p>Then on</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>Sym</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>Sym</mi><mo stretchy="false">(</mo><msup><mi>V</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A \otimes Sym(V) \otimes Sym(V^*) </annotation></semantics></math></div> <p>(with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V^*</annotation></semantics></math> in non-negative degree)</p> <p>there is an evident graded generalization of the Poisson bracket on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, which is on <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V^*</annotation></semantics></math> just the canonical pairing.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>c</mi> <mi>α</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{c^\alpha\}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/basis">basis</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V^*</annotation></semantics></math>, called the <strong><a class="existingWikiWord" href="/nlab/show/ghost">ghost</a></strong>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>π</mi> <mi>α</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\pi_\alpha\}</annotation></semantics></math> for the dual basis on <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>, called the <strong>ghost momenta</strong>.</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p><strong>(Henneaux, Stasheff et al.)</strong></p> <p>(<a class="existingWikiWord" href="/nlab/show/homological+perturbation+theory">homological perturbation theory</a>)</p> <p>There exists an element</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mo>∈</mo><mi>A</mi><mo>⊗</mo><mi>S</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>S</mi><mo stretchy="false">(</mo><msup><mi>V</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega \in A \otimes S(V) \otimes S(V^*) </annotation></semantics></math></div> <p>the <strong>BRST-BV charge</strong> such that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>Ω</mi><mo>,</mo><mi>Ω</mi><mo stretchy="false">}</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\{\Omega, \Omega\} = 0</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>S</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>S</mi><mo stretchy="false">(</mo><msup><mi>V</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mi>Ω</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A\otimes S(V) \otimes S(V^*), d := \{\Omega, -\})</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complex</a>, in fact a <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>S</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>S</mi><mo stretchy="false">(</mo><msup><mi>V</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo>=</mo><mo stretchy="false">{</mo><mi>Ω</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><mo stretchy="false">/</mo><mi>I</mi><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex"> H^0(A \otimes S(V) \otimes S(V^*), d = \{\Omega, -\}) = A/I \; </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mrow><mo><</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>S</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>S</mi><mo stretchy="false">(</mo><msup><mi>V</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo>=</mo><mo stretchy="false">{</mo><mi>Ω</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex"> H^{\lt 0}(A \otimes S(V) \otimes S(V^*), d = \{\Omega, -\}) = 0 \; </annotation></semantics></math></div> <p>(which says that this is in non-positive degree a resolution of the constraint locus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">/</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">A/I</annotation></semantics></math>)</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is a regular ideal (meaing that <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> can be chosen to be concentrated in degree 1) or the vanishing ideal of a coisotropic submanifold, then the cohomology in positive degree</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>S</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>S</mi><mo stretchy="false">(</mo><msup><mi>V</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo>=</mo><mo stretchy="false">{</mo><mi>Ω</mi><mo>,</mo><mn>0</mn><mo stretchy="false">}</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mi>I</mi><mo>,</mo><mi>I</mi><mo stretchy="false">/</mo><msup><mi>I</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^{\bullet \geq 0}(A \otimes S(V) \otimes S(V^*), d = \{\Omega, 0\}) \simeq H^\bullet(CE(A/I, I/I^2)) </annotation></semantics></math></div> <p>is isomorphic to the <a class="existingWikiWord" href="/nlab/show/infinity-Lie+algebra+cohomology">Lie algebroid cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a> whose <a class="existingWikiWord" href="/nlab/show/Lie-Rinehart+algebra">Lie-Rinehart algebra</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">/</mo><mi>I</mi><mo>,</mo><mi>I</mi><mo stretchy="false">/</mo><msup><mi>I</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A/I, I/I^2)</annotation></semantics></math></p> <p>(which says that in positive degree the BRST-BV complex is a resolution of the <a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>I</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{I,-\}</annotation></semantics></math> acting on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>).</p> </li> </ul> </div> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p><strong>(Oh-Park, Cattaneo-Felder)</strong> If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">C \subset X</annotation></semantics></math> is coisotropic, there is an <a class="existingWikiWord" href="/nlab/show/L-infinity+algebra">L-infinity algebra</a>-structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><mi>Γ</mi><mo stretchy="false">(</mo><mi>N</mi><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\wedge^\bullet \Gamma(N C)</annotation></semantics></math> such that the induced bracket on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>0</mn></msup><mo>=</mo><msub><mi>A</mi> <mi>red</mi></msub></mrow><annotation encoding="application/x-tex">H^0 = A_{red}</annotation></semantics></math> is the given one;</p> </div> <div class="num_theorem"> <h6 id="theorem_3">Theorem</h6> <p><strong>(Schätz)</strong> The BRST-BV complex with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{-,-\}</annotation></semantics></math> as its Lie bracket is <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphic</a> to the above.</p> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/BRST+complex">BRST complex</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/variational+BV-BRST+bicomplex">variational BV-BRST bicomplex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+BV+algebra">homotopy BV algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+L-infinity+algebra">quantum L-infinity algebra</a></p> </li> </ul> <h2 id="References">References</h2> <h3 id="ReferencesGeneral">General</h3> <p>A classical standard references for the Lagrangian formalism is</p> <ul> <li id="Henneaux90"><a class="existingWikiWord" href="/nlab/show/Marc+Henneaux">Marc Henneaux</a>, <em>Lectures on the Antifield-BRST formalism for gauge theories</em>, Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106 (<a href="http://www.math.uni-hamburg.de/home/schweigert/ws07/henneaux2.pdf">pdf</a>)</li> </ul> <p>Similarly the bulk of the textbook</p> <ul> <li id="HenneauxTeitelboim"><a class="existingWikiWord" href="/nlab/show/Marc+Henneaux">Marc Henneaux</a>, <a class="existingWikiWord" href="/nlab/show/Claudio+Teitelboim">Claudio Teitelboim</a>, <em><a class="existingWikiWord" href="/nlab/show/Quantization+of+Gauge+Systems">Quantization of Gauge Systems</a></em>, Princeton University Press 1992. xxviii+520 pp.</li> </ul> <p>considers the Hamiltonian formulation. Chapters 17 and 18 are about the Lagrangian (“antifield”) formulation, with section 18.4 devoted to the relation between the two.</p> <p>The <a class="existingWikiWord" href="/nlab/show/L-infinity+algebroid">L-infinity algebroid</a>-structure of the <a class="existingWikiWord" href="/nlab/show/local+BV-BRST+complex">local BV-BRST complex</a> on the <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> is made manifest in</p> <ul> <li id="Barnich10"><a class="existingWikiWord" href="/nlab/show/Glenn+Barnich">Glenn Barnich</a>, equation(3) of <em>A note on gauge systems from the point of view of Lie algebroids</em>, in P. Kielanowski, V. Buchstaber, A. Odzijewicz, <p>M. Schlichenmaier, T Voronov, (eds.) XXIX Workshop on Geometric Methods in Physics, vol. 1307 of AIP Conference Proceedings, 1307, 7 (2010) (<a href="https://arxiv.org/abs/1010.0899">arXiv:1010.0899</a>, <a href="https://doi.org/10.1063/1.3527427">doi:/10.1063/1.3527427</a>)</p> </li> </ul> <p>Formulation as <a class="existingWikiWord" href="/nlab/show/homotopy+AQFT">homotopy AQFT</a>:</p> <ul> <li id="BeniniBruinsmaSchenkel19"><a class="existingWikiWord" href="/nlab/show/Marco+Benini">Marco Benini</a>, Simen Bruinsma, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, <em>Linear Yang-Mills theory as a homotopy AQFT</em> (<a href="https://arxiv.org/abs/1906.00999">arXiv:1906.00999</a>)</li> </ul> <p>Review in the context of <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Luigi+Alfonsi">Luigi Alfonsi</a>, §5 in: <em>Higher geometry in physics</em>, in: <em><a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a></em>, Elsevier (2024) [<a href="https://arxiv.org/abs/2312.07308">arXiv:2312.07308</a>]</li> </ul> <h3 id="ReferencesForLagrangianBV">Lagrangian BV</h3> <h4 id="ReferencesForLagrangianBVForLagrangianTheories">For Lagrangian theories</h4> <p>The original articles are</p> <ul> <li id="BV81"> <p><a class="existingWikiWord" href="/nlab/show/Igor+Batalin">Igor Batalin</a>, <a class="existingWikiWord" href="/nlab/show/Grigori+Vilkovisky">Grigori Vilkovisky</a>, <em>Gauge Algebra and Quantization</em>, Phys. Lett. B 102 (1): 27–31, 1981 (<a href="https://doi.org/10.1016/0370-2693(81)90205-7">doi:10.1016/0370-2693(81)90205-7</a>)</p> </li> <li id="BV83"> <p><a class="existingWikiWord" href="/nlab/show/Igor+Batalin">Igor Batalin</a>, <a class="existingWikiWord" href="/nlab/show/Grigori+Vilkovisky">Grigori Vilkovisky</a>, (1983). <em>Quantization of Gauge Theories with Linearly Dependent Generators</em>, Phys. Rev. D 28 (10): 2567–2582. doi:10.1103/PhysRevD.28.2567. Erratum-ibid. 30 (1984) 508 doi:10.1103/PhysRevD.30.508</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Batalin">Igor Batalin</a>, <a class="existingWikiWord" href="/nlab/show/Grigori+Vilkovisky">Grigori Vilkovisky</a>, <em>Existence Theorem For Gauge Algebra</em>, J. Math. Phys. 26 (1985) 172-184.</p> </li> </ul> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Joaquim+Gomis">Joaquim Gomis</a>, J. Paris, S. Samuel, <em>Antibrackets, Antifields and Gauge Theory Quantization</em> (<a href="http://arxiv.org/abs/hep-th/9412228">arXiv:hep-th/9412228</a>)</p> </li> <li id="Park"> <p>J. Park, <em>Pursuing the quantum world</em> (<a href="http://cds.cern.ch/record/638963/files/0308130.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pavel+Mnev">Pavel Mnev</a>. <em>Lectures on Batalin-Vilkovisky formalism and its applications in topological quantum field theory</em> (2017). (<a href="https://www.arxiv.org/abs/1707.08096">arXiv:1707.08096</a>).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alberto+S.+Cattaneo">Alberto S. Cattaneo</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Mnev">Pavel Mnev</a>, <a class="existingWikiWord" href="/nlab/show/Michele+Schiavina">Michele Schiavina</a>, <em>BV Quantization</em>, <a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a> [<a href="https://arxiv.org/abs/2307.07761">arXiv:2307.07761</a>]</p> </li> </ul> <p>Geometrical aspects were pioneered in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Albert+Schwarz">Albert Schwarz</a>, <em><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">Semiclassical approximation</a> in Batalin-Vilkovisky formalism</em>, Comm. Math. Phys. <strong>158</strong> (1993), no. 2, 373–396, <a href="http://projecteuclid.org/euclid.cmp/1104254246">euclid</a></p> </li> <li> <p>M. Alexandrov, <a class="existingWikiWord" href="/nlab/show/M.+Kontsevich">M. Kontsevich</a>, <a class="existingWikiWord" href="/nlab/show/Albert+Schwarz">Albert Schwarz</a>, O. Zaboronsky, <em>The geometry of the master equation and topological quantum field theory</em>, Int. J. Modern Phys. A 12(7):1405–1429, 1997, <a href="http://arxiv.org/abs/hep-th/9502010">hep-th/9502010</a></p> </li> </ul> <p>A systematic account of the classical master equation is also in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/David+Kazhdan">David Kazhdan</a>, <em>The classical master equation in the finite-dimensional case</em> (<a href="http://math.berkeley.edu/~theojf/KazhdanNotes.pdf">pdf</a>)</p> </li> <li id="FelderKazhdan"> <p><a class="existingWikiWord" href="/nlab/show/Giovanni+Felder">Giovanni Felder</a>, <a class="existingWikiWord" href="/nlab/show/David+Kazhdan">David Kazhdan</a>, <em>The classical master equation</em> (<a href="http://arxiv.org/abs/1212.1631">arXiv:1212.1631</a>)</p> </li> </ul> <p>Generalization from <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>-actions to actual <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>-actions:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Marco+Benini">Marco Benini</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Safronov">Pavel Safronov</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, <em>Classical BV formalism for group actions</em> (<a href="https://arxiv.org/abs/2104.14886">arXiv:2104.14886</a>)</li> </ul> <p>Other discussions include</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <em>An introduction to the Batalin-Vilkovisky formalism</em>, Lecture given at the Recontres Mathématiques de Glanon, July 2003, <a href="http://arxiv.org/abs/math/0402057">arXiv:math/0402057</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alberto+Cattaneo">Alberto Cattaneo</a>, <em>From topological field theory to deformation quantization and reduction</em>, ICM 2006. (<a href="http://www.math.uzh.ch/fileadmin/math/preprints/icm.pdf">pdf</a>)</p> </li> <li> <p>M. Bächtold, <em>On the finite dimensional BV formalism</em>, 2005. (<a href="http://www.math.uzh.ch/reports/04_05.pdf">pdf</a>)</p> </li> <li> <p>Carlo Albert, Bea Bleile, <a class="existingWikiWord" href="/nlab/show/J%C3%BCrg+Fr%C3%B6hlich">Jürg Fröhlich</a>, <em>Batalin-Vilkovisky integrals in finite dimensions</em>, <a href="http://eprintweb.org/S/article/math-ph/0812.0464">arXiv/0812.0464</a></p> </li> <li> <p>Qiu and Zabzine, <em>Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and their Applications</em>, <a href="http://arxiv.org/pdf/1105.2680v2">arXiv/1105.2680</a>.</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantization</a> of <a class="existingWikiWord" href="/nlab/show/gauge+theories">gauge theories</a> (<a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a>) in <a class="existingWikiWord" href="/nlab/show/causal+perturbation+theory">causal perturbation theory</a>/<a class="existingWikiWord" href="/nlab/show/perturbative+AQFT">perturbative AQFT</a> is discussed (for trivial <a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a> and restricted to <a class="existingWikiWord" href="/nlab/show/gauge+invariant+observables">gauge invariant observables</a>) via <a class="existingWikiWord" href="/nlab/show/BRST-complex">BRST-complex</a>/<a class="existingWikiWord" href="/nlab/show/BV-formalism">BV-formalism</a> in</p> <ul> <li id="Hollands07"> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Hollands">Stefan Hollands</a>, <em>Renormalized Quantum Yang-Mills Fields in Curved Spacetime</em>, Rev. Math. Phys.20:1033-1172, 2008 (<a href="https://arxiv.org/abs/0705.3340">arXiv:0705.3340</a>)</p> </li> <li id="FredenhagenRejzner11a"> <p><a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, <em>Batalin-Vilkovisky formalism in the functional approach to classical field theory</em>, Commun. Math. Phys. 314(1), 93–127 (2012) (<a href="https://arxiv.org/abs/1101.5112">arXiv:1101.5112</a>)</p> </li> <li id="FredenhagenRejzner11b"> <p><a class="existingWikiWord" href="/nlab/show/Klaus+Fredenhagen">Klaus Fredenhagen</a>, <a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, <em>Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory</em>, Commun. Math. Phys. 317(3), 697–725 (2012) (<a href="https://arxiv.org/abs/1110.5232">arXiv:1110.5232</a>)</p> </li> <li id="Rejzner11"> <p><a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, <em>Batalin-Vilkovisky formalism in locally covariant field theory</em> (<a href="https://arxiv.org/abs/1111.5130">arXiv:1111.5130</a>)</p> </li> <li id="Rejzner13"> <p><a class="existingWikiWord" href="/nlab/show/Katarzyna+Rejzner">Katarzyna Rejzner</a>, <em>Remarks on local symmetry invariance in perturbative algebraic quantum field theory</em> (<a href="https://arxiv.org/abs/1301.7037">arXiv:1301.7037</a>)</p> </li> </ul> <p>and surveyed in</p> <ul> <li id="Rejzner16"><a class="existingWikiWord" href="/nlab/show/Kasia+Rejzner">Kasia Rejzner</a>, section 7 of <em>Perturbative algebraic quantum field theory</em> Springer 2016 (<a href="https://link.springer.com/book/10.1007%2F978-3-319-25901-7">web</a>)</li> </ul> <p>Discussion for field theories with <a class="existingWikiWord" href="/nlab/show/boundary+conditions">boundary conditions</a> and going in the direction of <a class="existingWikiWord" href="/nlab/show/extended+TQFT">extended field theory</a>/<a class="existingWikiWord" href="/nlab/show/local+quantum+field+theory">local quantum field theory</a> is in</p> <ul> <li id="CattaneoMnevReshetikhin12"><a class="existingWikiWord" href="/nlab/show/Alberto+Cattaneo">Alberto Cattaneo</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Mnev">Pavel Mnev</a>, <a class="existingWikiWord" href="/nlab/show/Nicolai+Reshetikhin">Nicolai Reshetikhin</a>, <em>Classical BV theories on manifolds with boundary</em>, <a href="http://arxiv.org/abs/1201.0290">arXiv:1201.0290</a>; <em>Classical and quantum Lagrangian field theories with boundary</em>, <a href="http://arxiv.org/abs/1207.0239">arXiv:1207.0239</a>; <em>Perturbative quantum gauge theories on manifolds with boundary</em>, <a href="http://arxiv.org/abs/1507.01221">arxiv/1507.01221</a></li> </ul> <p>A discussion of BV-BRST formalism in the general context of <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a> is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a>, <em><a class="existingWikiWord" href="/nlab/show/Renormalization+and+Effective+Field+Theory">Renormalization and Effective Field Theory</a></em></li> </ul> <p>Relation to <a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a> is made explicit in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Owen+Gwilliam">Owen Gwilliam</a>, <a class="existingWikiWord" href="/nlab/show/Theo+Johnson-Freyd">Theo Johnson-Freyd</a>, <em>How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism</em> (2011) (<a href="http://arxiv.org/abs/1202.1554">arXiv:1202.1554</a>)</li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Theo+Johnson-Freyd">Theo Johnson-Freyd</a>, <em>Star-quantization via lattice topological field theory</em>, talk at <a href="http://scgp.stonybrook.edu/events/event-pages/string-math-2013">String-Math 2013</a> (<a href="http://media.scgp.stonybrook.edu/presentations/20130618_johnson-freyd.pdf">pdf</a>)</li> </ul> <p>The interpretation of the BV quantum master equation as a description of closed differential forms acting as measures on infinite-dimensional spaces of fields is described in</p> <ul id="Witten90"> <li><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>A note on the antibracket formalism</em>, Modern Physics Letters A, <strong>5</strong>, n. 7, 487–494, <a href="http://www.ams.org/mathscinet-getitem?mr=91h:81178">MR91h:81178</a>, <a href="http://dx.doi.org/10.1142/S0217732390000561">doi</a>, <a href="http://ccdb4fs.kek.jp/cgi-bin/img_index?9004090">scan</a></li> </ul> <ul> <li id="Schwarz92"><a class="existingWikiWord" href="/nlab/show/Albert+Schwarz">Albert Schwarz</a>, <em>Geometry of Batalin-Vilkovisky quantization</em> (<a href="http://arxiv.org/abs/hep-th/9205088">arXiv:hep-th/9205088</a>)</li> </ul> <p>This isomorphisms between the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> and the complex of <a class="existingWikiWord" href="/nlab/show/polyvector+field">polyvector field</a>s is reviewed for instance on p. 3 of</p> <ul id="WillwacherCalaque"> <li>Thomas Willwacher, Damien Calaque <em>Formality of cyclic cochains</em> (<a href="http://arxiv.org/abs/0806.4095">arXiv:0806.4095</a>)</li> </ul> <p>and in section 2 of</p> <ul id="CattaneoFiorenzaLongoni"> <li><a class="existingWikiWord" href="/nlab/show/Alberto+Cattaneo">Alberto Cattaneo</a>, <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, Riccardo Longoni, <em>On the Hochschild-Kostant-Rosenberg map for graded manifolds</em> (<a href="http://www.math.uzh.ch/fileadmin/math/preprints/05-06.pdf">pdf</a>)</li> </ul> <p>A discussion in the general context of <a class="existingWikiWord" href="/nlab/show/BV-algebras">BV-algebras</a> is in</p> <ul> <li id="Roger"><a class="existingWikiWord" href="/nlab/show/Claude+Roger">Claude Roger</a>, <em>Gerstenhaber and Batalin-Vilkovisky algebras</em>, Archivum mathematicum, Volume 45 (2009), No. 4 (<a href="http://www.emis.de/journals/AM/09-4/roger.pdf">pdf</a>)</li> </ul> <p>The generalization of this to <a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+duality">Poincaré duality</a> on Hochschild (co)homollogy is in</p> <ul id="VanDenBergh"> <li> <p>M. Van den Bergh, <em>A relation between Hochschild homology and cohomology for Gorenstein rings</em> . Proc. Amer. Math. Soc. 126 (1998), 1345–1348; (<a href="http://www.jstor.org/stable/118786">JSTOR</a>)</p> <p>Correction: Proc. Amer. Math. Soc. 130 (2002), 2809–2810.</p> </li> </ul> <p>with more on that in</p> <ul id="Ginzburg"> <li> <p>U. Krähmer, <em>Poincaré duality in Hochschild cohomology</em> (<a href="http://www.maths.gla.ac.uk/~ukraehmer/brussels.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Victor+Ginzburg">Victor Ginzburg</a>, <em>Calabi-Yau Algebras</em> (<a href="http://arxiv.org/abs/math.AG/0612139">arXiv</a>)</p> </li> </ul> <p>The application in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a>/<a class="existingWikiWord" href="/nlab/show/string+field+theory">string field theory</a> is discussed in</p> <ul> <li>B. Zwiebach, <em>Closed string field theory: Quantum action and the B-V master equation</em>, Nucl. Phys. B 390, 33-152 (1993)</li> </ul> <p>A mathematically oriented reformulation of some of this (in the context of <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> ) is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a>, <em>The Gromov-Witten potential associated to a TCFT</em> (<a href="http://www.math.northwestern.edu/~costello/0509264.pdf">pdf</a>)</li> </ul> <p>Here the analog of the <a class="existingWikiWord" href="/nlab/show/virtual+fundamental+class">virtual fundamental class</a> on the <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of surfaces is realized as a solution to the BV-master equation.</p> <p>The perspective on the BV-complex as a <a class="existingWikiWord" href="/schreiber/show/derived+critical+locus">derived critical locus</a> is indicated in</p> <ul> <li id="CostelloGwilliam"><a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a>, <a class="existingWikiWord" href="/nlab/show/Owen+Gwilliam">Owen Gwilliam</a>, <em>Factorization algebras in perturbative quantum field theory – Derived critical locus</em> (weblass=‘newWikiWord’>Derived%20critical%20locus<a href="/nlab/new/Derived%2520critical%2520locus">?</a></span>))</li> </ul> <p>A clear discussion of the BV-complex as a means for homological <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> is in</p> <ul> <li id="Gwilliam"><a class="existingWikiWord" href="/nlab/show/Owen+Gwilliam">Owen Gwilliam</a>, <em>Factorization algebras and free field theories</em> PhD thesis (2013) (<a class="existingWikiWord" href="/nlab/files/GwilliamThesis.pdf" title="pdf">pdf</a>)</li> </ul> <p>Related Chern-Simons type graded action functionals are discussed also in</p> <ul> <li>M.V. Movshev, <a class="existingWikiWord" href="/nlab/show/Albert+Schwarz">Albert Schwarz</a>, <em>Generalized Chern-Simons action and maximally supersymmetric gauge theories</em> (<a href="http://arxiv.org/abs/1304.7500">arXiv:1304.7500</a>)</li> </ul> <p>Lectures, discussing also the relation to the <a class="existingWikiWord" href="/nlab/show/graph+complex">graph complex</a> are</p> <ul> <li>Jian Qiu, <a class="existingWikiWord" href="/nlab/show/Maxim+Zabzine">Maxim Zabzine</a>, <em>Introduction to graded geometry, Batalin-Vilkovisky formalism and their applications</em>, <a href="http://arxiv.org/abs/1105.2680">arxiv/1105.2680</a>; <em>Knot weight systems from graded symplectic geometry</em>, <a href="http://arxiv.org/abs/1110.5234">arxiv/1110.5234</a>; <em>Odd Chern-Simons theory, Lie algebra cohomology and characteristic classes</em>, <a href="http://arxiv.org/abs/0912.1243">arxiv/0912.1243</a></li> <li>Klaus Fredenhagen, Katarzyna Rejzner, <em>Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory</em>, <a href="http://arxiv.org/abs/1110.5232">arxiv/1110.5232</a></li> </ul> <p>Gluing aspects are in focus of the program explained in</p> <ul> <li>Alberto S. Cattaneo, Pavel Mnev, Nicolai Reshetikhin, <em>Perturbative BV theories with Segal-like gluing</em>, <a href="http://arxiv.org/abs/1602.00741">arxiv/1602.00741</a></li> </ul> <p>On extracting <a class="existingWikiWord" href="/nlab/show/L-infinity+algebras"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>L</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">L_\infty</annotation> </semantics> </math>-algebras</a> from <a class="existingWikiWord" href="/nlab/show/BV-formalism">BV-formalism</a> around a solutio, encoding <a class="existingWikiWord" href="/nlab/show/tree+level">tree level</a> <a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Maxim+Grigoriev">Maxim Grigoriev</a>, <a class="existingWikiWord" href="/nlab/show/Dmitry+Rudinsky">Dmitry Rudinsky</a>, <em>Notes on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-approach to local gauge field theories</em>, Journal of Geometry and Physics <strong>190</strong> (2023) 104863 [<a href="https://arxiv.org/abs/2303.08990">arXiv2303.08990</a>, <a href="https://doi.org/10.1016/j.geomphys.2023.104863">doi:10.1016/j.geomphys.2023.104863</a>]</li> </ul> <h4 id="ReferencesForNonLagrangianEquations">For non-Lagrangian theories</h4> <p>The whole formalism also applies to the locus of solutions of <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a>s that are not necessarily the <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>s of an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>. Discussion of this more general case is in</p> <ul> <li> <p>D.S. Kaparulin, S.L. Lyakhovich, <a class="existingWikiWord" href="/nlab/show/A.A.+Sharapov">A.A. Sharapov</a>, <em>Local BRST cohomology in (non-)Lagrangian field theory</em> (<a href="http://arxiv.org/abs/1106.4252">arXiv:1106.4252</a>)</p> </li> <li> <p>D.S. Kaparulin, S.L. Lyakhovich, A.A. Sharapov, <em>Rigid Symmetries and Conservation Laws in Non-Lagrangian Field Theory</em> (<a href="http://arxiv.org/abs/1001.0091">arXiv:1001.0091</a>)</p> </li> <li> <p>S.L. Lyakhovich, A.A. Sharapov, <em>Quantizing non-Lagrangian gauge theories: an augmentation method</em> (<a href="http://arxiv.org/abs/hep-th/0612086">arXiv:hep-th/0612086</a>)</p> </li> <li> <p>S.L. Lyakhovich, A.A. Sharapov, <em>BRST theory without Hamiltonian and Lagrangian</em> (<a href="http://arxiv.org/abs/hep-th/0411247">arXiv:hep-th/0411247</a>)</p> </li> </ul> <p>Section 4.5 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Gennadi+Sardanashvily">Gennadi Sardanashvily</a>, <em>Advanced Classical Field Theory</em> (2009) (<a href="http://www.g-sardanashvily.ru/book09.pdf">pdf</a>)</li> </ul> <p>This also makes the connection to</p> <ul> <li>P. Olver, <em>Applications of Lie Groups to Differential Equations</em> (Springer-Verlag, Berlin) (1986)</li> </ul> <p>See also via <a class="existingWikiWord" href="/nlab/show/quantum+L-infinity+algebras">quantum L-infinity algebras</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Martin+Doubek">Martin Doubek</a>, <a class="existingWikiWord" href="/nlab/show/Branislav+Jur%C4%8Do">Branislav Jurčo</a>, <a class="existingWikiWord" href="/nlab/show/J%C3%A1n+Pulmann">Ján Pulmann</a>, <em>Quantum L∞ Algebras and the Homological Perturbation Lemma</em> Comm. Math. Phys. <strong>367</strong> (2019) 215–240 [<a href="https://arxiv.org/abs/1712.02696">arXiv:1712.02696</a>, <a href="https://doi.org/10.1007/s00220-019-03375-x">doi:10.1007/s00220-019-03375-x</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Branislav+Jur%C4%8Do">Branislav Jurčo</a>, <a class="existingWikiWord" href="/nlab/show/Tommaso+Macrelli">Tommaso Macrelli</a>, <a class="existingWikiWord" href="/nlab/show/Christian+S%C3%A4mann">Christian Sämann</a>, <a class="existingWikiWord" href="/nlab/show/Martin+Wolf">Martin Wolf</a>, <em>Loop Amplitudes and Quantum Homotopy Algebras</em>, Journal of High Energy Physics <strong>2020</strong> 3 (2020) [<a href="https://doi.org/10.1007/JHEP07(2020)003">doi:10.1007/JHEP07(2020)003</a>, <a href="https://arxiv.org/abs/1912.06695">arXiv:1912.06695</a>]</p> </li> </ul> <p>Review:</p> <ul> <li>Sebastian Albrecht: <em>Formulation of Batalin-Vilkovisky Field Theories as Homotopy Lie Algebras</em>, Munich (2022) [<a href="https://www.theorie.physik.uni-muenchen.de/TMP/theses/sebastian_albrecht.pdf">pdf</a>]</li> </ul> <h4 id="for_cftvertex_algebras">For CFT/vertex algebras</h4> <p>A class of “free” vertex algebras are also quantized using Batalin-Vilkovisky formalism, with results on quantization of <a class="existingWikiWord" href="/nlab/show/BCOV+theory">BCOV theory</a> important for understanding <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a>, in</p> <ul> <li>Si Li, <em>Vertex algebras and quantum master equation</em>, <a href="https://arxiv.org/abs/1612.01292">arxiv/1612.01292</a></li> </ul> <h3 id="hamiltonian_bfv">Hamiltonian BFV</h3> <p>BRST formalism is discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Glenn+Barnich">Glenn Barnich</a>, Friedemann Brandt, <a class="existingWikiWord" href="/nlab/show/Marc+Henneaux">Marc Henneaux</a>, <em>Local BRST cohomology in gauge theories</em>, Phys. Rep. <strong>338</strong> (2000), no. 5, 439–569, <a href="http://xxx.lanl.gov/abs/hep-th/0002245">hep-th/0002245</a>, <a href="http://dx.doi.org/10.1016/S0370-1573(00)00049-1">doi</a></li> </ul> <p>The original references on Hamiltonian BFV formalism are</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Batalin">Igor Batalin</a>, <a class="existingWikiWord" href="/nlab/show/Grigori+Vilkovisky">Grigori Vilkovisky</a>, <em>Relativistic S-matrix of dynamical systems with boson and fermion constraints</em> , Phys. Lett. <strong>B69</strong> (1977) 309-312;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Batalin">Igor Batalin</a>, <a class="existingWikiWord" href="/nlab/show/Efim+Fradkin">Efim Fradkin</a>, <em>A generalized canonical formalism and quantization of reducible gauge theories</em> , Phys. Lett. B122 (1983) 157-164.</p> </li> </ul> <p>Homological Poisson reduction is discussed in</p> <ul> <li id="Stasheff96"><a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>Homological Reduction of Constrained Poisson Algebras</em>, J. Differential Geom. Volume 45, Number 1 (1997), 221-240 (<a href="http://arxiv.org/abs/q-alg/9603021">arXiv:q-alg/9603021</a>, <a href="https://projecteuclid.org/euclid.jdg/1214459757">Euclid</a>)</li> </ul> <p>Remarks on the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> interpretation of BRST-BV are in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em>The (secret?) homological algebra of the Batalin-Vilkovisky approach</em> (<a href="http://arxiv.org/abs/hep-th/9712157">arXiv</a>)</li> </ul> <p>A standard textbook on the application of BRST-BV to <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> is</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Marc+Henneaux">Marc Henneaux</a>, <a class="existingWikiWord" href="/nlab/show/Claudio+Teitelboim">Claudio Teitelboim</a>, <em>Quantization of gauge systems</em>, Princeton University Press 1992. xxviii+520 pp.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Glenn+Barnich">Glenn Barnich</a>, Friedemann Brandt, <a class="existingWikiWord" href="/nlab/show/Marc+Henneaux">Marc Henneaux</a>, <em>Local BRST cohomology in the antifield formalism. I. General theorems</em>, <a href="http://projecteuclid.org/euclid.cmp/1104275094">euclid</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=97c:81186">MR97c:81186</a></p> </li> <li> <p><em>Basics of Poisson reduction</em> (<a href="http://golem.ph.utexas.edu/category/2008/07/poisson_reduction.html">blog</a>)</p> </li> <li> <p>Alejandro Cabrera, <em>Homological BV-BRST methods: from QFT to Poisson reduction</em> (<a href="http://www.math.uni-hamburg.de/home/schreiber/Charla_IMPA_BRST.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jeremy+Butterfield">Jeremy Butterfield</a>, <em>On symplectic reduction in classical mechanis</em> (<a href="http://philsci-archive.pitt.edu/archive/00002373/01/ButterfieldNHSympRed.pdf">pdf</a>)</p> </li> <li> <p>S. Lyakhovich, A. Sharapov, <em>BRST theory without Hamiltonian and Lagrangian</em> (<a href="http://arxiv.org/PS_cache/hep-th/pdf/0411/0411247v2.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Florian+Sch%C3%A4tz">Florian Schätz</a>, <em>BFV-complex and higher homotopy structures</em> (<a href="http://www.math.ist.utl.pt/~fschaetz/BFV-complex.pdf">pdf</a>)</p> </li> <li> <p>MO: <a href="http://mathoverflow.net/questions/30352/what-is-the-batalin-vilkovisky-formalism-and-what-are-its-uses-in-mathematics/32443#32443">what is the BV formalism and its uses</a></p> </li> </ul> <h3 id="ReferencesMultisymplectic">Multisymplectic BRST</h3> <p>In the context of <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li> <p>Sean Hrabak, <em>Ambient Diffeomorphism Symmetries of Embedded Submanifolds, Multisymplectic BRST and Pseudoholomorphic Embeddings</em> (<a href="http://arxiv.org/abs/math-ph/9904026">arXiv:math-ph/9904026</a>)</p> </li> <li> <p>Sean Hrabak, <em>On a Multisymplectic Formulation of the Classical BRST symmetry for First Order Field Theories Part I: Algebraic Structures</em> (<a href="http://arxiv.org/abs/math-ph/9901012">arXiv:math-ph/9901012</a>)</p> </li> <li> <p>Sean Hrabak, <em>On a Multisymplectic Formulation of the Classical BRST Symmetry for First Order Field Theories Part II: Geometric Structures</em> (<a href="http://arxiv.org/abs/math-ph/9901013">arXiv:math-ph/9901013</a>)</p> </li> </ul> <p>based on</p> <ul> <li>I. Kanatchikov, <em>On field theoretic generalizations of a Poisson algebra</em>, Rept.Math.Phys. 40 (1997) 225 (<a href="http://arxiv.org/abs/hep-th/9710069">arXiv:hep-th/9710069</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 6, 2024 at 00:18:20. 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