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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{A first idea of quantum field theory -- Reduced phase space} \hypertarget{ReducedPhaseSpace}{}\subsection*{{Reduced phase space}}\label{ReducedPhaseSpace} In this chapter we discuss these topics: \begin{itemize}% \item Global gauge reduction for strictly [[invariant]] functions ([[action functionals]]): \begin{itemize}% \item \emph{\hyperlink{DerivedCriticalLocusInsideLieAlgebroids}{Derived critical loci inside Lie algebroids}} \item \emph{\hyperlink{SchoutenBracketAntibracket}{Schouten bracket on Lie algebroids}} \end{itemize} \item Local gauge reduction for weakly invariant local functions ([[Lagrangian densities]]): \begin{itemize}% \item \emph{\hyperlink{LocalJetBundleAntibracket}{Local antibracket}} \item \emph{\hyperlink{DerivedCriticalLocusOnJetBundle}{Local BV-BRST complex}} \item \emph{\hyperlink{BVBRSTComplexGlobal}{Global BV-BRST complex}} \end{itemize} \end{itemize} For a [[Lagrangian field theory]] with [[infinitesimal gauge symmetries]], the \emph{[[reduced phase space]]} is the [[quotient]] of the [[shell]] (the [[solution]]-locus of the [[equations of motion]]) by the [[action]] of the [[gauge symmetries]]; or rather it is the combined \emph{[[homotopy quotient]]} by the [[gauge symmetries]] and its \emph{[[homotopy intersection]]} with the [[shell]]. Passing to the [[reduced phase space]] may lift the [[obstruction]] for a [[gauge theory]] to have a [[covariant phase space]] and hence a [[quantization]]. The [[higher differential geometry]] of [[homotopy quotients]] and [[homotopy intersections]] is usefully modeled by tools from [[homological algebra]], here known as the \emph{[[BV-BRST complex]]}. In order to exhibit the key structure without getting distracted by the local [[jet bundle]] geometry, we first discuss the simple form in which the reduced phase space would appear after [[transgression of variational differential forms|transgression]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}) if [[spacetime]] were [[compact space|compact]], so that, by the [[principle of extremal action]] (prop. \ref{PrincipleOfExtremalAction}), it would be the [[derived critical locus]] ($d S \simeq 0$) of a globally defined [[action functional]] $S$. This ``global'' version of the [[BV-BRST complex]] is example \ref{ArchetypeOfBVBRSTComplex} below. The genuine \emph{[[local field theory|local]]} construction of the derived [[shell]] is in the [[jet bundle]] of the [[field bundle]], where the [[action functional]] appears ``de-transgressed'' in the form of the [[Lagrangian density]], which however is invariant under gauge transformations generally only up to horizontally exact terms. This \emph{local} incarnation of the redcuced phase space is modeled by the genuine \emph{[[local BV-BRST complex]]}, example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm} below. Finally, under [[transgression of variational differential forms]] this yields a [[differential]] on the graded [[local observables]] of the field theory. This is the \emph{global [[BV-BRST complex]]} of the [[Lagrangian field theory]] (def. \ref{ComplexBVBRSTGlobal} below). $\,$ \textbf{[[derived critical loci]] inside [[Lie algebroids]]} By analogy with the algebraic formulation of [[smooth functions]] between [[Cartesian spaces]] (the [[embedding of smooth manifolds into formal duals of R-algebras|embedding of Cartesian spaces into formal duals of R-algebras]], prop. \ref{AlgebraicFactsOfDifferentialGeometry}) it is clear how to define a map ([[homomorphism]]) between [[Lie algebroids]]: \begin{defn} \label{HomomorphismBetweenLieAlgebroids}\hypertarget{HomomorphismBetweenLieAlgebroids}{} \textbf{([[homomorphism]] between [[Lie algebroids]])} Given two [[derived Lie algebroids]] $\mathfrak{a}$, $\mathfrak{a}'$ (def. \ref{LInfinityAlgebroid}), then a [[homomorphism]] between them \begin{displaymath} f \;\colon\; \mathfrak{a} \longrightarrow \mathfrak{a}' \end{displaymath} is a [[dg-algebra]]-[[homomorphism]] between their [[Chevalley-Eilenberg algebras]] going the other way around \begin{displaymath} CE(\mathfrak{a}) \longleftarrow CE(\mathfrak{a}') \;\colon\; f^\ast \end{displaymath} such that this covers an algebra homomorphism on the function algebras: \begin{displaymath} \itexarray{ CE(\mathfrak{a}) &\overset{f^\ast}{\longleftarrow}& CE(\mathfrak{a}') \\ \downarrow && \downarrow \\ C^\infty(X) &\underset{(f\vert_X)^\ast}{\longleftarrow}& C^\infty(Y) } \,. \end{displaymath} (This is also called a ``[[curved sh-map|non-curved sh-map]]''.) \end{defn} \begin{example} \label{GaugeInvariantFunctionsIntermsOfLieAlgebroids}\hypertarget{GaugeInvariantFunctionsIntermsOfLieAlgebroids}{} \textbf{([[invariant]] [[functions]] in terms of [[Lie algebroids]])} Let $\mathfrak{g}$ be a [[super Lie algebra]] equipped with a [[Lie algebra action]] (def. \ref{InfinitesimalActionByLieAlgebra}) \begin{displaymath} \itexarray{ \mathfrak{g} \times X && \overset{R}{\longrightarrow} && T X \\ & {}_{\mathllap{pr_2}}\searrow && \swarrow_{\mathrlap{rb}} \\ && X } \end{displaymath} on a [[supermanifold]] $X$. Then there is a canonical homomorphism of [[Lie algebroids]] (def. \ref{HomomorphismBetweenLieAlgebroids}) \begin{equation} \itexarray{ X &&& CE(X) &=& C^\infty(X) &\oplus& 0 \\ \downarrow^{\mathrlap{p}} &\phantom{AAA}&& \uparrow^{\mathrlap{p^\ast}} && \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{0}} \\ X/\mathfrak{g} &&& CE(X/\mathfrak{g}) &=& C^\infty(X) &\oplus& C^\infty(X) \otimes \wedge^\bullet \mathfrak{g}^\ast } \label{ProjectionMapForActionLieAlgebroid}\end{equation} from the manifold $X$ regarded as a Lie algebroid by example \ref{BasicExamplesOfLieAlgebroids} to the [[action Lie algebroid]] $X/\mathfrak{g}$ (example \ref{ActionLieAlgebroid}), which may be called the \emph{[[homotopy quotient]] [[coprojection]] map}. The dual homomorphism of [[differential graded-commutative superalgebras]] is given simply by the identity on $C^\infty(X)$ and the [[zero map]] on $\mathfrak{g}^\ast$. Next regard the [[real line]] [[manifold]] $\mathbb{R}^1$ as a Lie algebroid by example \ref{BasicExamplesOfLieAlgebroids}. Then homomorphisms of Lie algebroids (def. \ref{HomomorphismBetweenLieAlgebroids}) of the form \begin{displaymath} S \;\colon\; X/\mathfrak{g} \longrightarrow \mathbb{R}^1 \,, \end{displaymath} hence \emph{smooth functions on the Lie algebroid}, are equivalently \begin{itemize}% \item ordinary [[smooth functions]] $S \;\colon\; X \longrightarrow \mathbb{R}^1$ on the underlying [[smooth manifold]], \item which are [[invariant]] under the Lie algebra action in that $R(-)(S) = 0$. \end{itemize} In terms of the canonical [[homotopy quotient]] [[coprojection]] map $p$ \eqref{ProjectionMapForActionLieAlgebroid} this says that a smooth function on $X$ [[extension]] extends to the [[action Lie algebroid]] precisely if it is [[invariant]]: \begin{displaymath} \itexarray{ X &\overset{S}{\longrightarrow}& \mathbb{R}^1 \\ {}^{\mathllap{p}}\downarrow & \nearrow_{ \mathrlap{ \text{exists precisely if} \; S \; \text{is invariant} } } \\ X/\mathfrak{g} } \end{displaymath} \end{example} \begin{proof} An $\mathbb{R}$-algebra homomorphism \begin{displaymath} CE( X/\mathfrak{g} ) \overset{S^\ast}{\longleftarrow} C^\infty(\mathbb{R}^1) \end{displaymath} is fixed by what it does to the canonical [[coordinate function]] $x$ on $\mathbb{R}^1$, which is taken by $S^\ast$ to $S \in C^\infty(X) \hookrightarrow CE(X/\mathfrak{g})$. For this to be a dg-algebra homomorphism it needs to respect the differentials on both sides. Since the differential on the right is trivial, the condition is that $0 = d_{CE} S = R(-)(f)$: \begin{displaymath} \itexarray{ \left\{ S \right\} &\overset{S^\ast}{\longleftarrow}& \left\{ x \right\} \\ {}^{\mathllap{d_{CE(X/\mathfrak{g})}}}\downarrow && \downarrow^{\mathrlap{d_{CE(\mathbb{R}^1)} = 0 } } \\ \left\{ R(-)(S) = 0 \right\} &\underset{S^\ast}{\longleftarrow}& \left\{ 0 \right\} } \end{displaymath} \end{proof} Given a gauge invariant function, hence a function $S \colon X/\mathfrak{g} \to \mathbb{R}$ on a Lie algebroid (example \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids}), its [[exterior derivative]] $d S$ should be a [[section]] of the [[cotangent bundle]] of the Lie algebroid. Moreover, if all field variations are infinitesimal (as in def. \ref{LocalObservablesOnInfinitesimalNeighbourhood}) then it should in fact be a section of the [[infinitesimal neighbourhood]] (example \ref{InfinitesimalNeighbourhood}) of the [[zero section]] inside the [[cotangent bundle]], the \emph{infinitesimal cotangent bundle} $T^\ast_{inf}(X/\mathfrak{g})$ of the Lie algebroid (def. \ref{LieAlgebroidInfinitesimalCotangentBundle} ebelow). To motivate the definition \ref{LieAlgebroidInfinitesimalCotangentBundle} below of \emph{infinitesimal cotangent bundle of a Lie algebroid} recall from example \ref{InfinitesimalNeighbourhood} that the [[algebra of functions]] on the infinitesimal cotangent bundle should be fiberwise the [[formal power series algebra]] in the [[linear functions]]. But a fiberwise linear function on a [[cotangent bundle]] is by definition a [[vector field]]. Finally observe that [[derivations of smooth functions are vector fields|vector fields are equivalently derivations of smooth functions]] (prop. \ref{AlgebraicFactsOfDifferentialGeometry}). This leads to the following definition: \begin{defn} \label{LieAlgebroidInfinitesimalCotangentBundle}\hypertarget{LieAlgebroidInfinitesimalCotangentBundle}{} \textbf{([[automorphism ∞-Lie algebra|infinitesimal cotangent Lie algebroid]])} Let $\mathfrak{a}$ be a [[Lie ∞-algebroid]] (def. \ref{LInfinityAlgebroid}) over some manifold $X$. Then its \emph{infinitesimal cotangent bundle} $T^\ast_{inf} \mathfrak{a}$ is the [[Lie ∞-algebroid]] over $X$ whose underlying [[graded module]] over $C^\infty(X)$ is the [[direct sum]] of the original module with the [[derivations]] of the graded algebra underlying $CE(\mathfrak{a})$: \begin{displaymath} (T^\ast_{inf} \mathfrak{a})^\ast_\bullet \;\coloneqq\; \mathfrak{a}^\ast_\bullet \oplus Der(CE(\mathfrak{a}))_\bullet \end{displaymath} with [[differential]] on the summand $\mathfrak{a}$ being the original differential and on $Der(CE(\mathfrak{a}))$ being the graded [[commutator]] with the differential $d_{CE(\mathfrak{a})}$ on $CE(\mathfrak{a})$ (which is itself a graded derivation of degree +1): \begin{displaymath} \itexarray{ \mathllap{ d_{CE(T^\ast_{inf} \mathfrak{a})} } &\mathrlap{ \vert_{\mathfrak{a}^\ast} }& & \coloneqq & d_{CE(\mathfrak{a})} \\ \mathllap{ d_{CE(T^\ast_{inf} \mathfrak{a})} } & \mathrlap{ \vert_{Der(\mathfrak{a})} } & \phantom{ \vert_{Der(\mathfrak{a})} } & \coloneqq & [d_{CE(\mathfrak{a})},-] } \end{displaymath} Just as for ordinary [[cotangent bundles]] (def. \ref{Differential1FormsOnCartesianSpaces}) there is a canonical homomorphism of Lie algebroids (def. \ref{HomomorphismBetweenLieAlgebroids}) from the infinitesimal cotangent Lie algebroid down to the base Lie algebroid: \begin{equation} \itexarray{ T^\ast_{inf} \mathfrak{a} &\phantom{AAA}&& CE(T^\ast_{inf} \mathfrak{g}) &=& CE(\mathfrak{a}) &\oplus& \wedge^{\bullet \geq 1}_{CE(\mathfrak{a})} Der(\mathfrak{a}) \\ \downarrow^{\mathrlap{cb}} &&& \uparrow^{\mathrlap{cb^\ast}} && \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{0}} \\ \mathfrak{a} &&& CE(\mathfrak{a}) &=& CE(\mathfrak{a}) &\oplus& 0 } \label{CotangentLieAlgebrpoidProjection}\end{equation} given dually by the identity on the original generators. \end{defn} \begin{example} \label{CotangentBundleOfActionLieAlgebroid}\hypertarget{CotangentBundleOfActionLieAlgebroid}{} \textbf{([[automorphism ∞-Lie algebra|infinitesimal cotangent bundle]] of [[action Lie algebroid]])} Let $X/\mathfrak{g}$ be an [[action Lie algebroid]] (def. \ref{ActionLieAlgebroid}) whose [[Chevalley-Eilenberg differential]] is given in local coordinates by \eqref{DifferentialOnActionLieAlgebroid} \begin{displaymath} d_{CE(X/\mathfrak{g})} \;=\; \tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma \frac{\partial}{\partial c^\alpha} + c^\alpha R_a^\alpha \frac{\partial}{\partial \phi^a} \,. \end{displaymath} Then its infinitesimal cotangent Lie algebroid $T^\ast_{inf} (X/\mathfrak{g})$ (def. \ref{LieAlgebroidInfinitesimalCotangentBundle}) has the generators \begin{displaymath} \itexarray{ & \left( \frac{\partial}{\partial c^\alpha} \right) & \left( \phi^a \right) , \left( \frac{\partial}{\partial \phi^a} \right) & \left( c^\alpha \right) \\ deg = & -1 & 0 & +1 } \end{displaymath} and we find that CE-differential on the new derivation generators is given by \begin{equation} \begin{aligned} d_{CE(T^\ast_{inf}(X/\mathfrak{g}))} \left( \frac{\partial}{\partial c^\alpha} \right) & \coloneqq \left[d_{CE(X/\mathfrak{g})}, \frac{\partial}{\partial c^\alpha} \right] \\ & = R_\alpha^a \frac{\partial}{\partial \phi^a} + \gamma^\beta{}_{\alpha \gamma} c^\gamma \frac{\partial}{\partial c^\beta} \end{aligned} \label{CotangentLieAlgebroidDifferentialForActionLieAlgebroidOnGhostFieldCoordinates}\end{equation} and \begin{equation} \begin{aligned} d_{CE(T^\ast_{inf}(X/\mathfrak{g}))} \left( \frac{\partial}{\partial \phi^a} \right) & \coloneqq \left[ d_{CE(X/\mathfrak{g})}, \frac{\partial}{\partial \phi^a} \right] \\ & = - c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \frac{\partial}{\partial \phi^b} \end{aligned} \,. \label{CotangentLieAlgebroidDifferentialForActionLieAlgebroidOnFieldCoordinates}\end{equation} To amplify that the [[derivations]] \emph{on} $CE(X/\mathfrak{g})$, such as $\frac{\partial}{\partial \phi^a}$ and $\frac{\partial}{\partial c^\alpha}$, are now [[coordinate functions]] \emph{in} $CE(T^\ast_{inf}(X/\mathfrak{g}))$ one writes them as \begin{equation} \phi^\ddagger_a \;\coloneqq\; \frac{\partial}{\partial \phi^a} \phantom{AAAAA} c\ddagger_\alpha \;\coloneqq\; \frac{\partial}{\partial c^\alpha} \,. \label{AntiNotationForDerivations}\end{equation} so that the generator content then reads as follows: \begin{equation} \itexarray{ & \left( c^\ddagger_\alpha \right) & \left( \phi^a \right) , \left( \phi^\ddagger_a \right) & \left( c^\alpha \right) \\ deg = & -1 & 0 & +1 } \,. \label{GeneratorsOfDerivedCriticalLocusInActionLieAlgebroid}\end{equation} In this notation the full action of the CE-differential for $T^\ast_{inf}(X/\mathfrak{g})$ is therefore the following: \begin{equation} \itexarray{ & d_{CE(T^\ast_{inf}(X/\mathfrak{g}))} \\ \phi^a &\mapsto& c^\alpha R^a_\alpha \\ c^\alpha & \mapsto& \tfrac{1}{2} \gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma \\ \phi^\ddagger_a &\mapsto& - c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \phi^\ddagger_b \\ c^\ddagger_\alpha &\mapsto& R_\alpha^a \phi^\ddagger_a + \gamma^\beta{}_{\alpha \gamma} c^\gamma c^\ddagger_\beta } \label{CEDifferentialOnGeneratorsForInfinitesimalCotangentBundleOfActionLieAlgebroid}\end{equation} \end{example} With a concept of [[cotangent bundles]] for [[Lie algebroids]] in hand, we want to see next that their [[sections]] are [[differential 1-forms]] on a [[Lie algebroid]] in an appropriate sense: \begin{prop} \label{ExteriorDifferentialOfGaugeInvariantFunctionIsSectionOfInfinitesimalCotangentLieAlgebroid}\hypertarget{ExteriorDifferentialOfGaugeInvariantFunctionIsSectionOfInfinitesimalCotangentLieAlgebroid}{} \textbf{([[exterior differential]] of [[invariant]] function is [[section]] of [[automorphism ∞-Lie algebra|infinitesimal cotangent bundle]])} For $\mathfrak{a}$ a [[Lie ∞-algebroid]] (def. \ref{LInfinityAlgebroid}) over some $X$; and $S \;\colon\;\mathfrak{a} \longrightarrow \mathbb{R}$ a [[invariant]] smooth function on it (example \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids}) there is an induced [[section]] $d S$ of the infinitesimal cotangent Lie algebroid (def. \ref{LieAlgebroidInfinitesimalCotangentBundle}) bundle projection \eqref{CotangentLieAlgebrpoidProjection}: \begin{displaymath} \itexarray{ && T^\ast_{inf} \mathfrak{a} \\ & {}^{\mathllap{d S}}\nearrow & \downarrow^{\mathrlap{cb}} \\ \mathfrak{a} &=& \mathfrak{a} } \,, \end{displaymath} given dually by the [[homomorphism]] of [[differential graded-commutative superalgebras]] \begin{displaymath} (d S)^\ast \;\colon\; CE(T^\ast_{inf} \mathfrak{a}) \longrightarrow CE(\mathfrak{a}) \end{displaymath} which sends \begin{enumerate}% \item the generators in $\mathfrak{a}^\ast$ to themselves; \item a [[vector field]] $v$ on $X$, regarded as a degree-0 [[derivation]] to $d S(v) = v(S) \in C^\infty(X)$; \item all other derivations to zero. \end{enumerate} \end{prop} \begin{proof} We discuss the proof in the special case that $\mathfrak{a} = X/\mathfrak{g}$ is an [[action Lie algebroid]] (def. \ref{ActionLieAlgebroid}) hence where $T^\ast_{inf}(\mathfrak{a}) = T^\ast_{inf}(X/\mathfrak{g})$ is as in example \ref{CotangentBundleOfActionLieAlgebroid}. The general case is directly analogous. Since $(d S)^\ast$ has been defined on generators, it is uniquely a homomorphism of graded algebras. It is clear that if $(d S)^\ast$ is indeed a [[homomorphism]] of [[differential graded-commutative superalgebras]] in that it also respects the CE-differentials, then it yields a section as claimed, because by definition it is the identity on $\mathfrak{a}^\ast$. Hence all we need to check is that $(d S)^\ast$ indeed respects the CE-differentials. On the original generators in $\mathfrak{a}^\ast$ this is immediate, since on these the CE-differential on both sides are by definition the same. On the derivation $\phi^\ddagger_a \coloneqq \frac{\partial}{ \partial \phi^a}$ we find from \eqref{CotangentLieAlgebroidDifferentialForActionLieAlgebroidOnFieldCoordinates} \begin{displaymath} \itexarray{ \left\{ \frac{\partial S}{\partial \phi^a} \right\} &\overset{(d S)^\ast}{\longleftarrow}& \left\{ \phi^\ddagger_a \right\} \\ {}^{\mathllap{d_{CE(X/\mathfrak{g})}}}\downarrow && \downarrow^{\mathrlap{d_{CE(T^\ast_{inf} (X/\mathfrak{g}))}}} \\ \left\{ -c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \frac{\partial S}{\partial \phi^b} \right\} &\underset{(d S)^\ast}{\longleftarrow}& \left\{ -c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \phi^\ddagger_b \right\} } \end{displaymath} Notice that the left vertical map is indeed as shown, due to the invariance of $S$ (example \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids}), which allows an ``[[integration by parts]]'': \begin{displaymath} \begin{aligned} d_{CE(X/\mathfrak{g})}\left( \frac{\partial S}{\partial \phi_a} \right) & = c^\alpha R_\alpha^{b} \frac{\partial}{\partial \phi^b} \frac{\partial}{\partial \phi^a} S \\ & = \frac{\partial}{\partial \phi^a} \left( c^\alpha \underset{ = 0 }{ \underbrace{ R_\alpha^b \frac{\partial S}{\partial \phi^b} } } \right) \;-\; c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \frac{\partial S}{\partial \phi^b} \end{aligned} \end{displaymath} Similarly, on the derivation $c^\ddagger_\alpha \coloneqq \frac{\partial}{\partial c^\alpha}$ we find from \eqref{CotangentLieAlgebroidDifferentialForActionLieAlgebroidOnGhostFieldCoordinates} and using the invariance of $S$ (example \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids}) \begin{displaymath} \itexarray{ \left\{ 0 \right\} &\overset{(d S)^\ast}{\longleftarrow}& \left\{ c^\ddagger_\alpha \right\} \\ {}^{\mathllap{d_{CE(X/\mathfrak{g})}}}\downarrow && \downarrow^{\mathrlap{d_{CE(T^\ast_{inf}(X/\mathfrak{g}))}}} \\ \left\{ 0 = R_\alpha^a \frac{\partial S}{\partial \phi^a} \right\} &\underset{(d S)^\ast}{\longleftarrow}& \left\{ R_\alpha^a \phi^\ddagger_a + \gamma^\beta{}_{\alpha \gamma} c^\gamma c^\ddagger_\alpha \right\} } \,. \end{displaymath} This shows that the differentials are being respected. \end{proof} Next we describe the [[vanishing locus]] of $d S$, hence the [[critical locus]] of $S$. Notice that if $d S$ is regarded as an ordinary [[differential 1-form]] on an ordinary [[smooth manifold]] $X$, then its ordinary [[vanishing locus]] \begin{displaymath} X_{d S = 0} \;=\; \left\{ x \in X \;\vert\; d S(x) = 0 \right\} \end{displaymath} is simply the [[fiber product]] of $d S$ with the [[zero section]] of the [[cotangent bundle]], hence the [[universal property|universal]] space that makes the following [[commuting diagram|diagram commute]]: \begin{displaymath} \itexarray{ X_{d S = 0} &\overset{\phantom{AAA}}{\hookrightarrow}& X \\ \downarrow && \downarrow^{\mathrlap{0}} \\ X &\underset{d S}{\longrightarrow}& T^\ast_{inf} X } \,. \end{displaymath} This is just the [[category theory|general abstract]] way to express the [[equation]] $d S = 0$. In this [[category theory|general abstract]] form the concept of [[critical locus]] generalizes to [[invariant]] functions on [[super L-infinity algebra|super]] [[Lie algebroids]], where the vanishing of $d S$ is regarded only \emph{up to [[homotopy]]}, namely up to [[infinitesimal symmetry]] transformations by the [[Lie algebra]] $\mathfrak{g}$. In this [[homotopy theory|homotopy-theoretic]] refinement we speak of the \emph{[[derived critical locus]]}. The following definition simply states what this comes down to in components. For a detailed derivation see at \emph{[[derived critical locus]]} and for general introduction to [[higher differential geometry]] and [[higher Lie theory]] see at \emph{[[schreiber:Higher Structures|Higher structures in Physics]]}. \begin{defn} \label{DerivedCriticalLocusOfGaugeInvariantFunctionOnLieAlgebroid}\hypertarget{DerivedCriticalLocusOfGaugeInvariantFunctionOnLieAlgebroid}{} \textbf{([[derived critical locus]] of [[invariant]] function on [[Lie ∞-algebroid]])} Let $\mathfrak{a}$ be a [[Lie ∞-algebroid]] (def. \ref{LInfinityAlgebroid}) over some $X$, let \begin{displaymath} S \;\colon\; \mathfrak{a} \longrightarrow \mathbb{R} \end{displaymath} be an [[invariant]] function (example \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids}) and consider the [[section]] of its infinitesimal [[cotangent bundle]] $T^\ast_{inf} \mathfrak{a}$ (def. \ref{CotangentBundleOfActionLieAlgebroid}) corresponding to its exterior derivative via prop. \ref{ExteriorDifferentialOfGaugeInvariantFunctionIsSectionOfInfinitesimalCotangentLieAlgebroid}: \begin{displaymath} \itexarray{ \mathfrak{a} && \overset{d S}{\longrightarrow} && T^\ast_{inf} \mathfrak{a} \\ & {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{cb}} \\ && \mathfrak{a} } \end{displaymath} Then the \emph{[[derived critical locus]]} of $S$ is the [[derived Lie algebroid]] (def. \ref{LInfinityAlgebroid}) to be denoted $\mathfrak{a}_{d S \simeq 0}$ which is the [[homotopy pullback]] of the section $d S$ along the [[zero section]]: \begin{displaymath} \itexarray{ \mathfrak{a}_{d S \simeq 0} &\longrightarrow& \mathfrak{a} \\ \downarrow &(pb)& \downarrow^{\mathrlap{0}} \\ \mathfrak{a} &\underset{d S}{\longrightarrow}& T^\ast_{inf} \mathfrak{a} } \,. \end{displaymath} This means equivalently (details are at \emph{[[derived critical locus]]}) that the Chevalley-Eilenberg algebra of $\mathfrak{a}_{d S \simeq 0}$ is like that of the infinitesimal cotangent Lie algebroid $T^\ast_{inf} \mathfrak{a}$ (def. \ref{LieAlgebroidInfinitesimalCotangentBundle}) except for two changes: \begin{enumerate}% \item all [[derivations]] are shifted down in degree by one; rephrased in terms of [[graded manifold]] (remark \ref{dgManifolds}) this means that the [[graded manifold]] underlying $\mathfrak{a}_{d S \simeq 0}$ is $T^\ast_{inf}[-1]\mathfrak{a}$; \item the [[Chevalley-Eilenberg differential]] on the derivations coming from [[tangent vector fields]] $v$ on $X$ is that of the infinitesimal cotangent Lie algebroid $T^\ast_{inf} \mathfrak{a}$ plus $d S(v) = v(S)$. \end{enumerate} \end{defn} We now make the general concept of [[derived critical locus]] inside an [[L-∞ algebroid]] (def. \ref{DerivedCriticalLocusOfGaugeInvariantFunctionOnLieAlgebroid}) explicit in our running example of an [[action Lie algebroid]]; the reader not concerned with the general idea of [[homotopy pullbacks]] may consider the following example as the definition of derived critical locus for the purposes of our running examples: \begin{example} \label{ArchetypeOfBVBRSTComplex}\hypertarget{ArchetypeOfBVBRSTComplex}{} \textbf{([[derived critical locus]] inside [[action Lie algebroid]])} Consider an [[invariant]] function (def. \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids}) on an [[action Lie algebroid]] (def. \ref{ActionLieAlgebroid}) \begin{displaymath} S \;\colon\; X/\mathfrak{g} \overset{\phantom{AAA}}{\longrightarrow} \mathbb{R} \end{displaymath} for the case that the underlying [[supermanifold]] $X$ is a [[super Cartesian space]] (def. \ref{SuperCartesianSpace}) with global [[coordinates]] $(\phi^a)$ as in example \ref{CotangentBundleOfActionLieAlgebroid}. Then the [[derived critical locus]] (def. \ref{DerivedCriticalLocusOfGaugeInvariantFunctionOnLieAlgebroid}) \begin{displaymath} (X/\mathfrak{g})_{d S \simeq 0} \end{displaymath} is, in terms of its [[Chevalley-Eilenberg algebra]] $CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)$ (def. \ref{LInfinityAlgebroid}) given as follows: Its generators are those of $CE\left( T^\ast_{inf}(X/\mathfrak{g}) \right)$ as in \eqref{GeneratorsOfDerivedCriticalLocusInActionLieAlgebroid}, except for a shift of degree of the [[derivation]]-generators down by one: \begin{displaymath} \itexarray{ & \left( c^\ddagger_{\alpha} \right) & \left( \phi^\ddagger_a \right) & \left( \phi^a \right) & \left( c^\alpha \right) \\ deg = & -2 & -1 & 0 & +1 } \end{displaymath} Rephrased in terms of [[graded manifold]] (remark \ref{dgManifolds}) this means that the [[graded manifold]] underlying the derived critical locus is the \emph{shifted infinitesimal cotangent bundle} of the graded manifold $\mathfrak{g}[1] \times X$ \eqref{ActionLieAlgebroidGradedManifold} which underlies the [[action Lie algebroid]] (def. \ref{ActionLieAlgebroid}): \begin{equation} (X/\mathfrak{g})_{d S \simeq 0} \;=_{grmfd}\; T^\ast_{inf}[-1]\left( \mathfrak{g}[1] \times X \right) \label{ShiftedCotangentBundleForCriticalLocusInsideLieAlgebroid}\end{equation} and if $X = \mathbb{R}^{b\vert s}$ is a [[super Cartesian space]] this becomes more specifically \begin{displaymath} \begin{aligned} (\mathbb{R}^{p \vert q}/\mathfrak{g})_{d S \simeq 0} & =_{grmfd} T^\ast_{inf}[-1]\left( \mathfrak{g}[1] \times \mathbb{R}^{p \vert q} \right) \\ & =_{\phantom{grmfd}} \underset{ (c^\alpha) }{ \underbrace{ \mathfrak{g}[1] }} \times \underset{ (\phi^a) }{ \underbrace{ \mathbb{R}^{p\vert q} }} \times \underset{ (\phi^\ddagger_a) }{ \underbrace{ (\mathbb{R}^{p \vert q})^\ast_{inf}[-1] }} \times \underset{ (c^\ddagger_\alpha) }{ \underbrace{ \mathfrak{g}^\ast[-2] }} \end{aligned} \end{displaymath} Moreover, on these generators the CE-differential is given by \begin{equation} \itexarray{ & d_{CE\left((X/\mathfrak{g})_{d S \simeq 0}\right)} \\ \phi^a &\mapsto& c^\alpha R^a_\alpha \\ c^\alpha & \mapsto& \tfrac{1}{2} \gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma \\ \phi^\ddagger_a &\mapsto& \underset{ new }{ \underbrace{ \frac{\partial S}{\partial \phi^a} }} - c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \phi^\ddagger_b \\ c^\ddagger_\alpha &\mapsto& R_\alpha^a \phi^\ddagger_a + \gamma^\beta{}_{\alpha \gamma} c^\gamma c^\ddagger_b } \label{ExplicitCEDifferentialInCotangentBundleOfActionLieAlgebroid}\end{equation} which is just the expression for the differential \eqref{CEDifferentialOnGeneratorsForInfinitesimalCotangentBundleOfActionLieAlgebroid} in $CE\left( T^\ast_{inf}(X/\mathfrak{g}) \right)$ from example \ref{CotangentBundleOfActionLieAlgebroid}, except for the fact that (the derivations are shifted down in degree and) the new term $\frac{\partial S}{\partial \phi^a}$ over the brace. \end{example} The following example illustrates how the concept of [[derived critical locus]] $X_{d S \simeq 0}$ of $S$ is a [[homotopy theory|homotopy theoretic]] version of the ordinary concept of [[critical locus]] $X_{d S = 0}$: \begin{example} \label{OrdinaryCriticalLocusAsCohomologyOfDerivedCriticalLocus}\hypertarget{OrdinaryCriticalLocusAsCohomologyOfDerivedCriticalLocus}{} \textbf{(ordinary [[critical locus]] is [[cochain cohomology]] of [[derived critical locus]] in degree 0)} Let $X$ be an [[superpoint]] (def. \ref{SuperCartesianSpace}) or more generally the [[infinitesimal neighbourhood]] (example \ref{InfinitesimalNeighbourhood}) of a point in a [[super Cartesian space]] (def. \ref{SuperCartesianSpace}) with [[coordinate functions]] $(\phi^a)$, so that its [[algebra of functions]] $C^\infty(X)$ is a truncated [[polynomial algebra]] or [[formal power series algebra]] in the [[variables]] $\phi^a$. Consider for simplicity the special case that $\mathfrak{g} = 0$ so that there is no [[Lie algebra action]] on $X$. Then the [[Chevalley-Eilenberg algebra]] of the [[derived critical locus]] $X_{d S \simeq 0}$ of $S$ (example \ref{ArchetypeOfBVBRSTComplex}) has generators \begin{displaymath} \begin{aligned} & & \left( \phi^\ddagger_a \right) & \left( \phi^a \right) & \\ deg = & & -1 & 0 & \end{aligned} \end{displaymath} and [[differential]] given by \begin{displaymath} \itexarray{ & d_{CE\left( X_{d S \simeq 0} \right)} \\ \phi^a &\mapsto& 0 \\ \phi^\ddagger_a &\mapsto& \frac{\partial S}{\partial \phi^a} } \,. \end{displaymath} Hence the [[cochain cohomology]] of the [[Chevalley-Eilenberg algebra]] of the derived critical locus indegree 0 is the [[quotient]] of $C^\infty(X)$ by the ideal which is generated by $\left( \frac{\partial S}{\partial \phi^a} \right)$ \begin{displaymath} H^0\left( CE\left( X_{d S \simeq 0} \right) \right) \;=\; C^\infty(X)/\left( \frac{\partial S}{\partial \phi^a} \right) \,. \end{displaymath} But under the assumption that $X$ is a [[superpoint]] or [[infinitesimal neighbourhood]] of a point, this quotient algebra is just the [[algebra of functions]] on the ordinary [[critical locus]] $X_{d S = 0}$. (The quotient says that every function on $X$ which vanishes where $\frac{\partial S}{\partial \phi^a}$ vanishes is [[zero]] in the quotient. This means that the quotient algebra consists of the functions on $X$ modulo the [[equivalence relation]] that identifies two if they agree on the critical locus $X_{d S = 0}$, which is the functions on $X_{d S = 0}$.) Hence the [[derived critical locus]] yields the ordinary [[critical locus]] in [[cochain cohomology]]: \begin{displaymath} H^0\left( CE\left( X_{d S \simeq 0} \right) \right) \;\simeq\; C^\infty\left( X_{d S = 0} \right) \,. \end{displaymath} However, it is not in general the case that the [[derived critical locus]] is a [[resolution]] of the ordinary [[critical locus]], in that all its cohomology in [[negative number|negative]] degree vanishes. Instead, the cohomology of the [[Chevalley-Eilenberg algebra]] of a [[derived critical locus]] in [[negative number|negative]] degree detects [[Lie algebra action]] and more generally [[L-∞ algebra action]] on $X$ under which $S$ is invariant. If this action is incorporated into $X$ by passing to the [[action Lie algebroid]] $X/\mathfrak{g}$ and then forming the [[derived critical locus]] $(X/\mathfrak{g})_{d S \simeq 0}$ in there, as in example \ref{ArchetypeOfBVBRSTComplex}. This issue we discuss in detail in the chapter \emph{\hyperlink{GaugeFixing}{Gauge fixing}}, see prop. \ref{BVComplexIsHomologicalResolutionPreciselyIfNoNonTrivialImplicitGaugeSymmetres} below. \end{example} In order to generalize the statement of example \ref{OrdinaryCriticalLocusAsCohomologyOfDerivedCriticalLocus} to the case that a [[Lie algebra action]] is taken into account, we need to realize the [[Chevalley-Eilenberg algebra]] of a [[derived critical locus]] in a [[Lie algebroid]] is the [[total complex]] of a [[double complex]]: \begin{prop} \label{DerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure}\hypertarget{DerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure}{} \textbf{([[Chevalley-Eilenberg algebra]] of [[derived critical locus]] is [[total complex]] of [[BV-BRST formalism|BV-BRST]] [[bicomplex]])} Let $(X/\mathfrak{g})_{d S \simeq 0}$ be a [[derived critical locus]] inside an [[action Lie algebroid]] as in example \ref{ArchetypeOfBVBRSTComplex}. Then its [[Chevalley-Eilenberg differential]] \eqref{ExplicitCEDifferentialInCotangentBundleOfActionLieAlgebroid} may be decomposed as the sum of two anti-commuting differential \begin{displaymath} d_{CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)} \;=\; s_{BRST} + s_{BS} \end{displaymath} which are defined on the generators of the [[Chevalley-Eilenberg algebra]] as follows: \begin{equation} \itexarray{ & s_{BV} \\ \phi^a &\mapsto& 0 \\ c^\alpha & \mapsto& 0 \\ \phi^\ddagger_a &\mapsto& \frac{\partial S}{\partial \phi^a} \\ c^\ddagger_\alpha &\mapsto& R_\alpha^a \phi^\ddagger_a \\ \phantom{A} \\ & s_{BRST} \\ \phi^a &\mapsto& c^\alpha R^a_\alpha \\ c^\alpha & \mapsto& \tfrac{1}{2} \gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma \\ \phi^\ddagger_a &\mapsto& - c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \phi^\ddagger_b \\ c^\ddagger_\alpha &\mapsto& \gamma^\beta{}_{\alpha \gamma} c^\gamma c^\ddagger_b } \label{ExplicitBVandBRSTDifferentialInCotangentBundleOfActionLieAlgebroid}\end{equation} If we moreover decompose the degree of the generators into two degrees \begin{displaymath} \itexarray{ & \left( c^\ddagger_{\alpha} \right) & \left( \phi^\ddagger_a \right) & \left( \phi^a \right) & \left( c^\alpha \right) \\ deg_{gh} = & 0 & 0 & 0 & +1 \\ deg_{af} = & -2 & -1 & 0 & 0 } \end{displaymath} then these two differentials constitute a [[bicomplex]] \begin{displaymath} \itexarray{ CE^{0,0}\left( (X/\mathfrak{g})_{d S \simeq 0}\right) &\overset{s_{BRST}}{\longrightarrow}& CE^{1,0}\left( (X/\mathfrak{g})_{d S \simeq 0}\right) &\overset{s_{BRST}}{\longrightarrow}& CE^{2,0}\left( (X/\mathfrak{g})_{d S \simeq 0}\right) &\overset{s_{BRST}}{\longrightarrow}& \cdots \\ \big\uparrow^{\mathrlap{s_{BV}}} && \big\uparrow^{\mathrlap{s_{BV}}} && \big\uparrow^{\mathrlap{s_{BV}}} && \\ CE^{0,-1}\left( (X/\mathfrak{g})_{d S \simeq 0}\right) &\overset{s_{BRST}}{\longrightarrow}& CE^{1,-1}\left( (X/\mathfrak{g})_{d S \simeq 0}\right) &\overset{s_{BRST}}{\longrightarrow}& CE^{2,-1}\left( (X/\mathfrak{g})_{d S \simeq 0}\right) &\overset{s_{BRST}}{\longrightarrow}& \cdots \\ \big\uparrow^{\mathrlap{s_{BV}}} && \big\uparrow^{\mathrlap{s_{BV}}} && \big\uparrow^{\mathrlap{s_{BV}}} && \\ CE^{0,-2}\left( (X/\mathfrak{g})_{d S \simeq 0}\right) &\overset{s_{BRST}}{\longrightarrow}& CE^{1,-2}\left( (X/\mathfrak{g})_{d S \simeq 0}\right) &\overset{s_{BRST}}{\longrightarrow}& CE^{2,-2}\left( (X/\mathfrak{g})_{d S \simeq 0}\right) &\overset{s_{BRST}}{\longrightarrow}& \cdots \\ \big\uparrow^{\mathrlap{s_{BV}}} && \big\uparrow^{\mathrlap{s_{BV}}} && \big\uparrow^{\mathrlap{s_{BV}}} && \\ \vdots && \vdots && \vdots } \end{displaymath} whose [[total complex]] is the [[Chevalley-Eilenberg dg-algebra]] of the derived critical locus \begin{displaymath} \begin{aligned} CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right) & = \underset{ gh, af }{\bigoplus} CE^{gh,af}\left( (X/\mathfrak{g})_{d S \simeq 0} \right) \\ d_CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right){} & = s_{BV} + s_{BRST} \end{aligned} \,. \end{displaymath} \end{prop} \begin{proof} It is clear from the definition that the graded [[derivations]] $s_{BV}$ and $s_{BRST}$ have (i.e. increase) bidegree as follows: \begin{displaymath} \itexarray{ & s_{BRST} & s_{BV} \\ deg_{gh} = & +1 & 0 \\ deg_{af} = & 0 & +1 } \,. \end{displaymath} This implies that in \begin{displaymath} \begin{aligned} 0 & = \left( d_{CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)} \right)^2 \\ & = \left( s_{BV} + s_{BRST}\right)^2 \\ & = \underset{ = 0 }{ \underbrace{ \left( s_{BV}\right)^2 }} + \underset{ = 0 }{ \underbrace{ \left( s_{BRST} \right)^2 }} + \underset{ = 0 }{ \underbrace{ \left[ s_{BV}, s_{BRST} \right] } } \end{aligned} \end{displaymath} all three terms have to vanish separately, as shown, since they each have different bidegree (the last term denotes the graded commutator, hence the [[anticommutator]]). This is the statement to be proven. Notice that the nilpotency of $s_{BV}$ is also immediately checked explicitly, due to the [[invariant|invariance]] of $S$ (example \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids}): \begin{displaymath} \begin{aligned} s_{BV} \left( s_{BV} \left( c^\ddagger_\alpha \right) \right) & = s_BV\left( R_\alpha^a \phi^\ddagger_a \right) \\ & = R_\alpha^a \frac{\partial S}{\partial \phi^a} \\ & = 0 \end{aligned} \end{displaymath} \end{proof} As a corollary of prop. $\backslash$refDerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure\{\} we obtain the generalization of example \ref{OrdinaryCriticalLocusAsCohomologyOfDerivedCriticalLocus} to non-trivial $\mathfrak{g}$-actions: \begin{prop} \label{CochainCohomologyOfBVBRSTComplexInDegreeZero}\hypertarget{CochainCohomologyOfBVBRSTComplexInDegreeZero}{} \textbf{([[cochain cohomology]] of [[BV-BRST complex]] in degree 0 is the [[invariant]] function on the [[critical locus]])} Let $(X/\mathfrak{g})_{d S \simeq 0}$ be a [[derived critical locus]] inside an [[action Lie algebroid]] as in example \ref{ArchetypeOfBVBRSTComplex}. Then if the vertical [[differential]] (prop. \ref{DerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure}) \begin{displaymath} \itexarray{ CE^{\bullet, \bullet+1}\left( (X/\mathfrak{g})_{d S \simeq 0} \right) \\ \uparrow^{\mathrlap{s_{BV}}} \\ CE^{\bullet, \bullet}\left( (X/\mathfrak{g})_{d S \simeq 0} \right) } \end{displaymath} has vanishing [[cochain cohomology]] in [[negative number|negative]] $af$-degree \begin{equation} H^{\bullet \leq 1}(s_{BV}) = 0 \label{VanishingOfNaiveLieAlgebroidBVCohomlogyInNegativeDegree}\end{equation} then the [[cochain cohomology]] of the full [[Chevalley-Eilenberg dg-algebra]] is given by the cochain cohomology of $s_{BRST}$ on $H^0(s_{BV})$: \begin{displaymath} H^k\left( CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right) \right) \;\simeq\; H^k\left( H^0(s_{BV}), s_{BRST} \right) \,. \end{displaymath} Moreover if $X$ is inside the [[infinitesimal neighbourhood]] of a point as in example \ref{OrdinaryCriticalLocusAsCohomologyOfDerivedCriticalLocus} then the full cochain cohomology in degree 0 is the space of those functions on the ordinary [[critical locus]] $X_{d S = 0}$ which are $\mathfrak{g}$-[[invariant]]: \begin{displaymath} H^0 \left( CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right) \right) \;=\; \left\{ X_{d S = 0} \overset{f}{\to} \mathbb{R} \;\vert\; \left(R_\alpha^a \frac{\partial f}{\partial \phi^a} = 0\right) \right\} \end{displaymath} \end{prop} \begin{proof} The first statement follows from the [[spectral sequence]] [[spectral sequence of a double complex|of the double complex]] \begin{displaymath} H^{gh} \left( H^{af} \left( CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right) \right) \right) \;\Rightarrow\; H^{gh + af}\left( CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right) \right) \,. \end{displaymath} Under the given assumption the second page of this [[spectral sequence]] is concentrated on the row $af = 0$. This implies that all differentials on this page vanish, so that the sequence collapses on this page. Moreover, since the spectral sequence consists of [[vector spaces]] ([[modules]] over the [[real numbers]]) the \href{spectral+sequence#ExtensionProblem}{extension problem} is trivial, and hence the claim follows. Now if $X$ is inside the [[infinitesimal neighbourhood]] of a point, then example \ref{OrdinaryCriticalLocusAsCohomologyOfDerivedCriticalLocus} says that $H^0(s_{BV})$ in $deg_{gh} = 0$ consists of the functions on the ordinary critical locus and hence the abvove result implies that \begin{displaymath} \begin{aligned} H^0\left( CE\left( (X/\mathfrak{g})_{d S \simeq 0}\right) \right) & = ker(s_{BRST})\vert_{C^\infty\left( X_{d S = 0} \right) } \,/\, \underset{= 0}{ \underbrace{ im(s_{BRST})\vert_{C^\infty\left( X_{d S = 0} \right)} } } \\ & = ker(s_{BRST})\vert_{C^\infty\left( X_{d S = 0} \right) } \\ & = \left\{ X_{d S = 0} \overset{f}{\longrightarrow} \mathbb{R} \,\vert\, \left( R_\alpha^a \frac{\partial S}{\partial \phi^a} = 0 \right) \right\} \end{aligned} \end{displaymath} \end{proof} This means that under condition \eqref{VanishingOfNaiveLieAlgebroidBVCohomlogyInNegativeDegree} the construction of a [[derived critical locus]] inside an [[action Lie algebroid]] provides a [[resolution]] of the space of those functions which are \begin{enumerate}% \item \emph{[[restriction|restricted]]} to the [[critical locus]] (a [[homotopy intersection]]); \item \emph{[[invariant]]} under the [[Lie algebra action]] (a [[homotopy quotient]]). \end{enumerate} We apply this general mechanism \hyperlink{DerivedCriticalLocusOnJetBundle}{below} to [[Lagrangian field theory]], where it serves to provide a [[resolution]] by the \emph{[[BV-BRST complex]]} of the space of [[observables]] which are \begin{enumerate}% \item [[on-shell]], \item \emph{[[gauge invariance|gauge invariant]]}. \end{enumerate} But in order to control this application, we first establish the tool of the \emph{[[Schouten bracket]]/[[antibracket]]}. $\,$ \textbf{[[Schouten bracket]]/[[antibracket]]} Since the infinitesimal cotangent Lie algebroid $T^\ast_{inf} \mathfrak{a}$ has function algebra given by tensor products of [[tangent vector fields]]/[[derivations]], we expect that a graded analogue of the [[Lie bracket]] of ordinary [[tangent vector fields]] exists on the [[Chevalley-Eilenberg algebra]] $CE\left( T^\ast_{inf} \mathfrak{a}\right)$. This is indeed the case, and crucial for the theory: \begin{defn} \label{SchoutenBracketAndAntibracket}\hypertarget{SchoutenBracketAndAntibracket}{} \textbf{([[Schouten bracket]] and [[antibracket]] for [[action Lie algebroid]])} Consider a [[derived critical locus]] $(X/\mathfrak{g})_{d S \simeq 0}$ inside an [[action Lie algebroid]] $X/\mathfrak{g}$ as in example \ref{ArchetypeOfBVBRSTComplex}. Then the graded [[commutator]] of graded [[derivations]] of the [[Chevalley-Eilenberg algebra]] of $X/\mathfrak{g}$ \begin{displaymath} [-,-] \;\colon\; Der(CE(X/\mathfrak{g})) \otimes Der(CE(X/\mathfrak{g})) \longrightarrow Der(CE(X/\mathfrak{g})) \end{displaymath} uniquely [[extension|extends]], by the graded [[Leibniz rule]], to a graded bracket of degree $(1,even)$ on the CE-algebra of the [[derived critical locus]] $(X/\mathfrak{g})_{d S \simeq 0}$ \begin{displaymath} \left\{ -,-\right\} \;\colon\; CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right) \otimes C\left( (X/\mathfrak{g})_{d S \simeq 0} \right) \longrightarrow CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right) \end{displaymath} such that this is a graded [[derivation]] in both arguments. This is called the \emph{[[Schouten bracket]]}. There is an elegant way to rewrite this in terms of components: With the notation \eqref{AntiNotationForDerivations} for the coordinate-derivations the [[Schouten bracket]] is equivalently given by \begin{equation} \begin{aligned} \left\{ f,g \right\} & = \phantom{+} \frac{\overset{\leftarrow}{\partial} f}{\partial \phi^\ddagger_a} \frac{\overset{\rightarrow}{\partial} g}{\partial {\phi}^a} - \frac{\overset{\leftarrow}{\partial} f}{\partial \phi^a} \frac{\overset{\rightarrow}{\partial} g}{\partial \phi^\ddagger_a} \\ & \phantom{=} + \frac{\overset{\leftarrow}{\partial} f}{\partial c^\ddagger_\alpha} \frac{\overset{\rightarrow}{\partial} g}{\partial {c}^{\alpha}} - \frac{\overset{\leftarrow}{\partial} f}{\partial c^{\alpha}} \frac{\overset{\rightarrow}{\partial} g}{\partial c^\ddagger_\alpha} \end{aligned} \,, \label{Antibracket}\end{equation} where the arrow over the [[partial derivative]] indicates that we we pick up signs via the [[Leibniz rule]] either as usual, going through products from left to right (for $\overset{\rightarrow}{\partial}$) or by going through the products from right to left (for $\overset{\leftarrow}{\partial}$). In this form the [[Schouten bracket]] is called the \emph{[[antibracket]]}. \end{defn} (e. g. \href{antibracket#Henneaux90}{Henneaux 90, (53d)}, \href{antibracket#HenneauxTeitelboim92}{Henneaux-Teitelboim 92, section 15.5.2}) The power of the [[Schouten bracket]]/[[antibracket]] rests in the fact that it makes the [[Chevalley-Eilenberg differential]] on a [[derived critical locus]] $(X/\mathfrak{g})_{d S \simeq 0}$ become a [[Hamiltonian vector field]], for ``[[Hamiltonian]]'' the sum of $S$ with the [[Chevalley-Eilenberg differential]] of $X/\mathfrak{g}$: \begin{example} \label{ChevalleyEilenbergDifferentialOnDerivedCriticalLocusIsHamiltonianViaAntibracket}\hypertarget{ChevalleyEilenbergDifferentialOnDerivedCriticalLocusIsHamiltonianViaAntibracket}{} \textbf{([[Chevalley-Eilenberg differential]] of [[derived critical locus]] is [[Hamiltonian vector field]] for the [[Schouten bracket]]/[[antibracket]])} Let $(X/\mathfrak{g})_{d S \simeq 0}$ be a [[derived critical locus]] inside an [[action Lie algebroid]] as in example \ref{ArchetypeOfBVBRSTComplex}. Then the CE-differential \eqref{ExplicitCEDifferentialInCotangentBundleOfActionLieAlgebroid} of the [[derived critical locus]] $X/\mathfrak{g}\vert_{S \simeq 0}$ is simply the [[Schouten bracket]]/[[antibracket]] (def. \ref{SchoutenBracketAndAntibracket}) with the [[sum]] \begin{equation} S_{\text{BV-BRST}} \;\coloneqq\; S - d_{CE(X/\mathfrak{g})} \label{BVBRSTFunctionForActionLieAlgebroid}\end{equation} of the [[Chevalley-Eilenberg differential]] of $X/\mathfrak{g}$ and the function $-S$: \begin{displaymath} d_{CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right) }(-) \;=\; \left\{ - S + d_{CE(X/\mathfrak{g})} \,,\, (-) \right\} \,. \end{displaymath} In coordinates, using the expression for $d_{CE(X/\mathfrak{g})}$ from \eqref{DifferentialOnActionLieAlgebroid} and using the notation for derivations from \eqref{AntiNotationForDerivations} this means that \begin{displaymath} d_{CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)}(-) \;=\; \left\{ - S + c^\alpha R_\alpha^a \phi^\ddagger_a - \tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma c^\ddagger_\alpha \,,\, (-) \right\} \,. \end{displaymath} \end{example} \begin{proof} This is a simple straightforward computation, but we spell it out for illustration of the general principle. The result is to be compared with \eqref{ExplicitCEDifferentialInCotangentBundleOfActionLieAlgebroid}: for $\phi^a$: \begin{displaymath} \begin{aligned} \left\{ - S + c^\alpha R_\alpha^{a'} \phi^\ddagger_{a'} - \tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma c^\ddagger_\alpha \,,\, \phi^a \right\} & = \left\{ c^\alpha R_\alpha^{a'} \phi^\ddagger_{a'} \,,\, \phi^a \right\} \\ & = c^\alpha R_\alpha^{a'} \underset{ \delta_{a'}^a }{ \underbrace{ \left\{ \phi^\ddagger_{a'} \,,\, \phi^a \right\} } } \\ & = c^\alpha R_\alpha^{a} \end{aligned} \end{displaymath} for $c^\alpha$: \begin{displaymath} \begin{aligned} \left\{ - S + c^\alpha R_\alpha^{a} \phi^\ddagger_{a} - \tfrac{1}{2}\gamma^{\alpha'}{}_{\beta \gamma} c^\beta c^\gamma c^\ddagger_{\alpha'} \,,\, c^\alpha \right\} & = \left\{ \tfrac{1}{2}\gamma^{\alpha'}{}_{\beta \gamma} c^\beta c^\gamma c^\ddagger_{\alpha'} \,,\, c^\alpha \right\} \\ & = \tfrac{1}{2}\gamma^{\alpha'}{}_{\beta \gamma} c^\beta c^\gamma \underset{ \delta_{\alpha'}^\alpha }{ \underbrace{ \left\{ c^\ddagger_{\alpha'} \,,\, c^\alpha \right\} } } \\ & = \tfrac{1}{2}\gamma^{\alpha}{}_{\beta \gamma} c^\beta c^\gamma \end{aligned} \end{displaymath} for $\phi^\ddagger_a$: \begin{displaymath} \begin{aligned} \left\{ - S + c^\alpha R_\alpha^{a'} \phi^\ddagger_{a'} - \tfrac{1}{2}\gamma^{\alpha}{}_{\beta \gamma} c^\beta c^\gamma c^\ddagger_{\alpha} \,,\, \phi^\ddagger_a \right\} & = - \underset{ = -\frac{\partial S}{\partial \phi^a} }{ \underbrace{ \left\{ S \,,\, \phi^{\ddagger}_a \right\} } } + \left\{ c^\alpha R_\alpha^{a'} \phi^\ddagger_{a'} \,,\, \phi^\ddagger_a \right\} \\ & = \frac{\partial S}{\partial \phi^a} + c^\alpha \underset{ = -\frac{\partial R_\alpha^{a'}}{\partial \phi^a} }{ \underbrace{ \left\{ R_\alpha^{a'} \,,\, \phi^\ddagger_a \right\} } } \phi^\ddagger_{a'} \\ & = \frac{\partial S}{\partial \phi^a} - c^\alpha \frac{\partial R_\alpha^{a'}}{\partial \phi^a} \phi^\ddagger_{a'} \end{aligned} \end{displaymath} for $c^\ddagger_\alpha$: \begin{displaymath} \begin{aligned} \left\{ - S + c^{\alpha'} R_{\alpha'}^{a} \phi^\ddagger_{a} - \tfrac{1}{2}\gamma^{\alpha'}{}_{\beta \gamma} c^\beta c^\gamma c^\ddagger_{\alpha'} \,,\, c^\ddagger_\alpha \right\} & = \left\{ c^{\alpha'} R_{\alpha'}^a \phi^{\ddagger}_a \,,\, c^\ddagger_{\alpha} \right\} \;+\; \left\{ \tfrac{1}{2} \gamma^{\alpha'}{}_{\beta \gamma} c^\beta c^\gamma c^\ddagger_{\alpha'} \,,\, c^\ddagger_\alpha \right\} \\ & = \left\{ c^{\alpha'} \,,\, c^\ddagger_{\alpha} \right\} R_{\alpha'}^a \phi^{\ddagger}_a \;+\; \tfrac{1}{2} \gamma^{\alpha'}{}_{\beta \gamma} \underset{ = - c^\beta \delta_{\alpha}^\gamma + \delta_{\alpha}^\beta c^\gamma}{ \underbrace{ \left\{ c^\beta c^\gamma \,,\, c^\ddagger_\alpha \right\} }} c^\ddagger_{\alpha'} \\ & = R_\alpha^a \phi^\ddagger_{a} + \gamma^{\alpha'}{}_{\alpha \gamma} c^\gamma c^\ddagger_{\alpha'} \end{aligned} \end{displaymath} Hence these values of the [[Schouten bracket]]/[[antibracket]] indeed all agree with the values of the CE-differential from \eqref{ExplicitCEDifferentialInCotangentBundleOfActionLieAlgebroid}. \end{proof} As a corollary we obtain: \begin{prop} \label{ClassicalMasterEquation}\hypertarget{ClassicalMasterEquation}{} \textbf{([[classical master equation]])} Let $(X/\mathfrak{g})_{d S \simeq 0}$ be a [[derived critical locus]] inside an [[action Lie algebroid]] as in example \ref{ArchetypeOfBVBRSTComplex}. Then the [[Schouten bracket]]/[[antibracket]] (def. \ref{SchoutenBracketAndAntibracket}) of the function $S_{\text{BV-BRST}}$ S\_\{$\backslash$text\{BV-BRST\}\} \begin{displaymath} S_{\text{BV-BRST}} \;\coloneqq\; S - d_{CE\left( X/\mathfrak{g}\right)} \end{displaymath} with itself vanishes: \begin{displaymath} \left\{ S_{\text{BV-BRST}} \,,\, S_{\text{BV-BRST}} \right\} \;=\; 0 \,. \end{displaymath} Conversely, given a shifted [[cotangent bundle]] of the form $T^\ast[-1](X \times \mathfrak{g}[1])$ \eqref{ShiftedCotangentBundleForCriticalLocusInsideLieAlgebroid}, then the [[mathematical structure|struture]] of a [[differential]] of degree +1 on its [[algebra of functions]] is equivalent to a degree-0 element $S \in C^\infty(T^\ast[-1](X \times \mathfrak{g}[1]))$ such that \begin{displaymath} \left\{ S, S \right\} \;=\; 0 \,. \end{displaymath} Since therefore this equation controls the structure of [[derived critical loci]] once the underlying manifold $X$ and [[Lie algebra]] $\mathfrak{g}$ is specified, it is also called the \emph{[[master equation]]} and here specifically the \emph{[[classical master equation]]}. \end{prop} $\,$ This concludes our discussion of plain [[derived critical loci]] inside [[Lie algebroids]]. Now we turn to applying these considerations about to [[Lagrangian densities]] on a [[jet bundle]], which are [[invariant]] under [[infinitesimal gauge symmetries]] generally only up to a [[total spacetime derivative]]. By example \ref{ChevalleyEilenbergDifferentialOnDerivedCriticalLocusIsHamiltonianViaAntibracket} it is clear that this is best understood by first considering the refinement of the [[Schouten bracket]]/[[antibracket]] to this situation. $\,$ \textbf{[[local BV-BRST complex|local]] [[antibracket]]} If we think of the invariant function $S$ in def. \ref{DerivedCriticalLocusOfGaugeInvariantFunctionOnLieAlgebroid} as being the [[action functional]] (example \ref{ActionFunctional}) of a [[Lagrangian field theory]] $(E,\mathbf{L})$ (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) over a [[compact space|compact]] [[spacetime]] $\Sigma$, with $X$ the [[space of field histories]] (or rather an [[infinitesimal neighbourhood]] therein), hence with $\mathfrak{g}$ a Lie algebra of [[gauge symmetries]] acting on the field histories, then the [[Chevalley-Eilenberg algebra]] $CE\left((X/\mathfrak{g})_{d S \simeq 0}\right)$ of the [[derived critical locus]] of $S$ is called the \emph{[[BV-BRST complex]]} of the theory. In applications of interest, the spacetime $\Sigma$ is \emph{not} [[compact space|compact]]. In that case one may still appeal to a construction on the [[space of field histories]] as in example \ref{ArchetypeOfBVBRSTComplex} by considering the action functional for all [[adiabatic switching|adiabatically switched]] $b \mathbf{L}$ Lagrangians, with $b \in C_{cp}^\infty(\Sigma)$. This approach is taken in (\href{BV-BRST+formalism#FredenhagenRejzner11a}{Fredenhagen-Rejzner 11a}). Here we instead consider now the ``local lift'' or ``de-transgression'' of the above construction from the [[space of field histories]] to the [[jet bundle]] of the field bundle of the theory, refining the [[BV-BRST complex]] (prop. \ref{DerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure}) to the \emph{[[local BV-BRST complex]]} (prop. \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm} below), corresponding to the [[local BRST complex]] from example \ref{LocalOffShellBRSTComplex} (\href{local+BRST+cohomology#BarnichBrandtHenneaux00}{Barnich-Brandt-Henneaux 00}). This requires a slight refinement of the construction that leads to example \ref{ArchetypeOfBVBRSTComplex}: In contrast to the [[action functional]] $S = \tau_\Sigma(g\mathbf{L})$ (example \ref{ActionFunctional}), the [[Lagrangian density]] $\mathbf{L}$ is not strictly \emph{invariant} under [[infinitesimal gauge transformations]], in general, rather it may change up to a horizontally exact term (by the very definition \ref{GaugeParameters}). The same is then true, in general, for its [[Euler-Lagrange variational derivative]] $\delta_{EL} \mathbf{L}$ (unless we have already restricted to the [[shell]], by prop. \ref{InfinitesimalSymmetriesOfLagrangianAreAlsoSymmetriesOfTheEquationsOfMotion}, which however here we do not explicitly, but only via passing to [[cochain cohomology]] as in example \ref{OrdinaryCriticalLocusAsCohomologyOfDerivedCriticalLocus}). This means that the [[Euler-Lagrange form]] $\delta_{EL} \mathbf{L}$ is, [[off-shell]], not a section of the infinitesimal cotangent bundle (def. \ref{LieAlgebroidInfinitesimalCotangentBundle}) of the gauge action Lie algebroid on the jet bundle. But it turns out that it still is a section of local refinement of the cotangent bundle, which is twisted by horizontally exact terms (prop. \ref{EulerLagrangeFormIsSectionOfLocalCotangentBundleOfJetBundleGaugeActionLieAlgebroid} below). To see the required twist, it is most convenient to make use of a local version of the [[antibracket]] (def. \ref{LocalAntibracket} below), via local refinement of example \ref{ChevalleyEilenbergDifferentialOnDerivedCriticalLocusIsHamiltonianViaAntibracket}. As a result we may form the \emph{local} [[derived critical locus]] as in def. \ref{DerivedCriticalLocusOfGaugeInvariantFunctionOnLieAlgebroid} but now with the invariance of the [[Lagrangian density]] only up to [[total spacetime derivatives]] taken into account. Its [[Chevalley-Eilenberg algebra]] is called the \emph{[[local BV-BRST complex]]} (prop. \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm} below). The following is the direct refinement of the concept of the underlying [[graded manifold]] of the infinitesimal [[cotangent bundle]] of an [[action Lie algebroid]] in example \ref{CotangentBundleOfActionLieAlgebroid} to the case where the base manifold is generalized to a [[field bundle]] (def. \ref{FieldsAndFieldBundles}) and the [[Lie algebra]] to a [[gauge parameter bundle]] (def. \ref{GaugeParameters}): \begin{defn} \label{InfinitesimalCotangentBundleOfFieldAndGaugeParameterBundle}\hypertarget{InfinitesimalCotangentBundleOfFieldAndGaugeParameterBundle}{} \textbf{([[infinitesimal neighbourhood]] of [[zero section]] in [[cotangent bundle]] of [[fiber product]] of [[field bundle]] with shifted [[gauge parameter bundle]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) over some [[spacetime]] $\Sigma$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a bundle of [[gauge parameters]] (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) which are closed (def. \ref{GaugeParametersClosed}), inducing the [[Lie algebroid]] \begin{displaymath} E / ( \mathcal{G} \times_\Sigma T \Sigma ) \;=\; \left( J^\infty_\Sigma( E \times_\Sigma (\mathcal{G}[1]) ) , s_{BRST} ) \right) \end{displaymath} whose [[Chevalley-Eilenberg algebra]] is the \emph{[[local BRST complex]]} of the field theory (example \ref{LocalOffShellBRSTComplex}). Then we write \begin{displaymath} T^\ast_{\Sigma,inf}\left( E \times_\Sigma (\mathcal{G}[1]) \right) \,, \phantom{AAA} T^\ast_{\Sigma,inf}[-1]\left( E \times_\Sigma (\mathcal{G}[1]) \right) \end{displaymath} for, on the left, the [[infinitesimal neighbourhood]] of the [[zero section]] of the [[vertical cotangent bundle]] of the [[graded manifold|graded]] [[fiber product]] of the [[field bundle]] with the fiber-wise shifted [[gauge parameter bundle]], as well as its shifted version on the right, as in \eqref{ShiftedCotangentBundleForCriticalLocusInsideLieAlgebroid}. In [[local coordinates]] this means the following: Assuming that the [[field bundle]] $E$ and the [[gauge parameter bundle]] $\mathcal{G}$ are [[trivial vector bundles]] (example \ref{TrivialVectorBundleAsAFieldBundle}) with fiber coordinates $(\phi^a)$ and $(c^\alpha)$, respectively, then $T^\ast_{\Sigma,inf}\left(E \times_\Sigma (\mathcal{G}[1])\right)$ is the trivial graded vector bundle with fiber coordinates \begin{equation} \itexarray{ T^\ast_{\Sigma,inf}\left( E \times_\Sigma (\mathcal{G}[1]) \right) & \phantom{AAAAA}& T^\ast_{\Sigma,inf}[-1]\left( E \times_\Sigma (\mathcal{G}[1]) \right) \\ & \phantom{A} \\ \itexarray{ & (c^\ddagger_\alpha), & (\phi^\ddagger_a),(\phi^a), & (c^\alpha) \\ deg = & -1 & 0 & 1 } & \phantom{AA}& \itexarray{ & (c^\ddagger_\alpha), & (\phi^\ddagger_a)\, & (\phi^a), & (c^\alpha) \\ deg = & -2 & -1 & 0 & 1 } } \label{coordslocalOnInfinitesimalCotangentOfFieldBundleTimesGaugeParameterBundle}\end{equation} and such that smooth functions on $T^\ast_{\Sigma,inf}\left(E \times_\Sigma (\mathcal{G}[1])\right)$ are [[formal power series]] in $c^\ddagger_\alpha$ (necessarily due to degree reasons) and in $\phi^\ddagger_a$ (reflecting the [[infinitesimal neighbourhood]] of the [[zero section]]). Here the shifted cotangents to the fields are called the \emph{[[antifields]]}: \begin{itemize}% \item $\phi^\ddagger_a$ is \emph{[[antifield]]} to the [[field (physics)|field]] $\phi^a$ \item $c^\ddagger_\alpha$ is \emph{[[antifield]]} to the [[ghost field]] $c^\alpha$. \end{itemize} \end{defn} The following is the direct refinement of the concept of the [[Schouten bracket]] on an [[action Lie algebroid]] from def. \ref{SchoutenBracketAndAntibracket} to the case where the base manifold is generalized to the [[jet bundle]] (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) [[field bundle]] (def. \ref{FieldsAndFieldBundles}) and the [[Lie algebra]] to the [[jet bundle]] of a [[gauge parameter bundle]] (def. \ref{GaugeParameters}): \begin{defn} \label{LocalAntibracket}\hypertarget{LocalAntibracket}{} \textbf{([[local antibracket]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) over [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}), and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a bundle of [[gauge parameters]] (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) which are closed (def. \ref{GaugeParametersClosed}), inducing via example \ref{LocalOffShellBRSTComplex} the [[Lie algebroid]] \begin{displaymath} E / ( \mathcal{G} \times_\Sigma T \Sigma ) \;=\; \left( J^\infty_\Sigma( E \times_\Sigma (\mathcal{G}[1]) ) , s_{BRST} ) \right) \end{displaymath} whose [[Chevalley-Eilenberg algebra]] is the \emph{[[local BRST complex]]} of the field theory with shifted infinitesimal [[vertical cotangent bundle]] \begin{equation} E_{\text{BV-BRST}} \;\coloneqq\; T^\ast_{\Sigma,inf}[-1]\left( E \times_\Sigma (\mathcal{G}[1]) \right) \label{BVBRSTGradedFieldBundle}\end{equation} of its underlying graded bundle from def. \ref{InfinitesimalCotangentBundleOfFieldAndGaugeParameterBundle}. Then on the horizontal $p+1$-forms on this bundle (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}) which in terms of the [[volume form]] may all be decomposed as \eqref{LagrangianFunctionViaVolumeForm} \begin{displaymath} H \;=\; h \, dvol_\Sigma \;\in\; \Omega^{p+1}_\Sigma\left( \,T^\ast_{\Sigma,inf}[-1]\left( E \times_\Sigma (\mathcal{G}[1]) \right) \, \right) \end{displaymath} the \emph{[[local antibrackets]]} \begin{displaymath} \{-,-\}' , \{-,-\} \;\colon\; \Omega^{p+1,0}_\Sigma( \, T^\ast_{\Sigma,inf}[-1](E \times_\Sigma \mathcal{G}[1]) \, ) \,\otimes\, \Omega^{p+1,0}_\Sigma( \, T^\ast_{\Sigma,inf}[-1](E \times_\Sigma \mathcal{G}[1]) \, ) \longrightarrow \Omega^{p+1,0}_\Sigma( \, T^\ast_{\Sigma,inf}[-1](E \times_\Sigma \mathcal{G}[1]) \, ) \end{displaymath} are the functions which are given in the [[local coordinates]] \eqref{coordslocalOnInfinitesimalCotangentOfFieldBundleTimesGaugeParameterBundle} as follows: The first version is \begin{displaymath} \begin{aligned} \left\{ f\, dvol_\Sigma \,,\,g \, dvol_\Sigma \right\}' & \coloneqq \phantom{+} \left( \frac{\overset{\leftarrow}{\delta}_{EL} f }{\delta \phi^\ddagger_a} \frac{\overset{\rightarrow}{\delta}_{EL} g}{\delta {\phi^a}^{\phantom{\ddagger}}} - \frac{\overset{\leftarrow}{\delta}_{EL}}{\delta {\phi^a}^{\phantom{\ddagger}}} \frac{\overset{\rightarrow}{\delta}_{EL} g}{\delta \phi^\ddagger_a} \right) dvol_\Sigma \\ & \phantom{=} + \left( \frac{\overset{\leftarrow}{\delta}_{EL} f}{\delta c^\ddagger_\alpha} \frac{\overset{\rightarrow}{\delta}_{EL} g}{\delta {c^\alpha}^{\phantom{\ddagger}}} - \frac{\overset{\leftarrow}{\delta}_{EL}}{\delta {c^\alpha}^{\phantom{\ddagger}}} \frac{\overset{\rightarrow}{\delta}_{EL} g}{\delta c^\ddagger_\alpha} \right) dvol_\Sigma \,. \end{aligned} \end{displaymath} This is of the form of the [[Schouten bracket]] \eqref{Antibracket} but with [[Euler-Lagrange derivatives]] \eqref{EulerLagrangeEquationGeneral} instead of [[partial derivatives]], The second version is this: \begin{equation} \begin{aligned} \left\{ f \, dvol_\Sigma, g \, dvol_\Sigma \right\} & \coloneqq \phantom{+} \left( \left( \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \left( \frac{\overset{\leftarrow}{\delta}_{EL} f}{\delta \phi^a} \right) \right) \left( \frac{\overset{\rightarrow}{\partial} g}{\partial {\phi}^\ddagger_{a,\mu_1 \cdots \mu_k}} \right) - \left( \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \left( \frac{\overset{\leftarrow}{\delta}_{EL} f}{\delta \phi^\ddagger_a} \right) \right) \left( \frac{\overset{\rightarrow}{\partial} g}{\partial \phi^a_{,\mu_1 \cdots \mu_k}} \right) \right) \, dvol_\Sigma \\ & \phantom{\coloneqq} + \left( \left( \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \left( \frac{\overset{\leftarrow}{\delta}_{EL} f}{\delta c^\alpha} \right) \right) \left( \frac{\overset{\rightarrow}{\partial} g}{\partial {c}^\ddagger_{\alpha,\mu_1 \cdots \mu_k}} \right) - \left( \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \left( \frac{\overset{\leftarrow}{\delta}_{EL} f}{\delta c^\ddagger_\alpha} \right) \right) \left( \frac{\overset{\rightarrow}{\partial} g}{\partial c^\alpha_{,\mu_1 \cdots \mu_k}} \right) \right) \, dvol_\Sigma \end{aligned} \label{LocalCommutatorOfDerivationsOnJetBundle}\end{equation} where again $\frac{\delta_{EL}}{\delta \phi^a}$ denotes the [[Euler-Lagrange variational derivative]] \eqref{EulerLagrangeEquationGeneral} \end{defn} (\href{local+BRST+cohomology#BarnichHenneaux96}{Barnich-Henneaux 96 (2.9) and (2.12)}, reviewed in \href{BRST+complex#Barnich10}{Barnich 10 (4.9)}) \begin{prop} \label{BasicPropertiesOfTheLocalAntibracket}\hypertarget{BasicPropertiesOfTheLocalAntibracket}{} \textbf{(basic properties of the [[local antibracket]])} The [[local antibracket]] from def. \ref{LocalAntibracket} satisfies the following properties: \begin{enumerate}% \item The two versions differ by a [[total spacetime derivative]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}): \begin{displaymath} \{f,g\} = \{f,g\}' + d(...) \,. \end{displaymath} \item The primed version is strictly graded skew-symmetric: \begin{displaymath} \left\{f \, dvol_\Sigma \,,\, g\, dvol_\Sigma \right\}' \;=\; - (-1)^{deg(f) deg(g)} \, \left\{g \, dvol_\Sigma \,,\, f\, dvol_\Sigma \right\} \end{displaymath} \item The unprimed version $\{-,-\}$ strictly satisfies the graded [[Jacobi identity]]; in that it is a graded [[derivation]] in the second argument, of degree one more than the degree of the first argument: \begin{equation} \left\{ f\, dvol_\Sigma, \left\{ g\, dvol_\Sigma \,,\, h\, dvol_\Sigma \right\}\right\} \;=\; \underset{ = \left\{ \left\{ f\, dvol_\Sigma \,,\, g\, dvol_\Sigma \right\}' \,, h\, dvol_\Sigma \right\} }{ \underbrace{ \left\{ \left\{ f\, dvol_\Sigma \,,\, g\, dvol_\Sigma \right\} \,,\, h\, dvol_\Sigma \right\} } } \;+\; (-1)^{(deg(f)+1) deg(g)} \left\{ g\, dvol_\Sigma \,,\, \left\{ f\, dvol_\Sigma \,,\, h\, dvol_\Sigma \right\} \right\} \label{LocalAntibracketGradedDerivationInSecondArgument}\end{equation} and the first term on the right is equivalently given by the primed bracket, as shown under the brace; \item the [[horizontal derivative|horizontally]] [[exact differential form|exact]] [[horizontal differential forms]] are an [[ideal]] for either bracket, in that for $f dvol_\Sigma = d(\cdots)$ or $g dvol_\Sigma = d(\cdots)$ we have \begin{displaymath} \{ f dvol_\Sigma, g \, dvol_\Sigma \}' = 0 \phantom{AAA} \{ f dvol_\Sigma, g \, dvol_\Sigma \} = d(\cdots) \end{displaymath} \end{enumerate} for all $f$, $g$ of homogeneous degree $deg(f)$ and $deg(g)$, respectively. \end{prop} (\href{local+BRST+cohomology#BarnichHenneaux96}{Barnich-Henneaux 96 (B.6) and footnote 9}). \begin{proof} That the two expressions differ by a horizontally exact terms follows by the very definition of the [[Euler-Lagrange derivative]] \eqref{EulerLagrangeEquationGeneral}. Also the graded skew symmetry of the primed bracket is manifest. The third point requires some computation (\href{local+BRST+cohomology#BarnichHenneaux96}{Barnich-Henneaux 96 (B.9)}). Finally that $\{-,-\}'$ vanishes when at least one of its arguments is horizontally exact follows from the fact that already the [[Euler-Lagrange derivative]] vanishes on this argument (example \ref{TrivialLagrangianDensities}). This implies that $\{-,-\}$ is horizontally exact when at least one of its arguments is so, by the first item. \end{proof} The following is the local refinement of prop. \ref{ClassicalMasterEquation}: \begin{defn} \label{ClassicalMasterEquationLocal}\hypertarget{ClassicalMasterEquationLocal}{} \textbf{(local [[classical master equation]])} The third item in prop. \ref{BasicPropertiesOfTheLocalAntibracket} implies that the following conditions on a [[Lagrangian density]] $\mathbf{K} \in \Omega^{p+1}_\Sigma( T^\ast_{\Sigma,inf}( E \times_\Sigma \mathcal{G}[1] ) )$ whose degree is even \begin{displaymath} \mathbf{K} = K\, dvol_\Sigma \,, \phantom{AAA} deg(L) \in 2 \mathbb{Z} \end{displaymath} are equivalent: \begin{enumerate}% \item forming the [[local antibracket]] (def. \ref{LocalAntibracket}) with $\mathbf{K}$ is a [[differential]] \begin{displaymath} \left(\left\{ \mathbf{K},-\right\}\right)^2 = 0 \,, \end{displaymath} \item the [[local antibracket]] (def. \ref{LocalAntibracket}) of $\mathbf{K}$ with itself is a [[total spacetime derivative]]: \begin{displaymath} \left\{ \mathbf{K}, \mathbf{K}\right\} = d(...) \end{displaymath} \item the other variant of the [[local antibracket]] (def. \ref{LocalAntibracket}) of $\mathbf{K}$ with itself is a [[total spacetime derivative]]: \begin{displaymath} \left\{ \mathbf{K}, \mathbf{K}\right\}' = d(...) \end{displaymath} \end{enumerate} This condition is also called the \emph{local [[classical master equation]]}. \end{defn} $\,$ \textbf{[[derived critical locus]] on [[jet bundle]] -- the [[local BV-BRST complex]]} With the local version of the [[antibracket]] in hand (def. \ref{LocalAntibracket}) it is now straightforward to refine the construction of a [[derived critical locus]] inside an [[action Lie algebroid]] (example \ref{ArchetypeOfBVBRSTComplex}) to the ``derived'' [[shell]] \eqref{ShellInJetBundle} inside the formal dual of the [[local BRST complex]] (example \ref{LocalOffShellBRSTComplex}). The result is a [[derived Lie algebroid]] whose [[Chevalley-Eilenberg algebra]] is called the \emph{[[local BV-BRST complex]]}. This is example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm} below. The following definition \ref{LocalInfinitesimalCotangentLieAlgebroid} is the local refinement of def. \ref{LieAlgebroidInfinitesimalCotangentBundle}: \begin{defn} \label{LocalInfinitesimalCotangentLieAlgebroid}\hypertarget{LocalInfinitesimalCotangentLieAlgebroid}{} \textbf{(local infinitesimal cotangent Lie algebroid)} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) over some [[spacetime]] $\Sigma$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a bundle of [[gauge parameters]] (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) which are closed (def. \ref{GaugeParametersClosed}), inducing via example \ref{LocalOffShellBRSTComplex} the [[Lie algebroid]] \begin{displaymath} E / ( \mathcal{G} \times_\Sigma T \Sigma ) \;=\; \left( J^\infty_\Sigma( E \times_\Sigma (\mathcal{G}[1]) ) , s_{BRST} ) \right) \end{displaymath} whose [[Chevalley-Eilenberg algebra]] is the \emph{[[local BRST complex]]} of the field theory. Consider the case that both the [[field bundle]] $E \overset{fb}{\to} \Sigma$ (def. \ref{FieldsAndFieldBundles}) as well as the [[gauge parameter]] bundle $\mathcal{G} \overset{gb}{\to} \Sigma$ are [[trivial vector bundles]] (example \ref{TrivialVectorBundleAsAFieldBundle}) over [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}) with [[field (physics)|field]] coordinates $(\phi^a)$ and [[gauge parameter]] coordinates $(c^\alpha)$. Then the vertical infinitesimal cotangent Lie algebroid (def. \ref{LieAlgebroidInfinitesimalCotangentBundle}) has coordinates as in \eqref{GeneratorsOfDerivedCriticalLocusInActionLieAlgebroid} as well as all the corresponding jets and including also the horizontal differentials: \begin{displaymath} \itexarray{ & \left( c^\ddagger_{\alpha,\mu_1 \cdots \mu_k} \right) & \left( \phi^a_{,\mu_1 \cdots \mu_k} \right) , \left( \phi^\ddagger_{a,\mu_1 \cdots \mu_k} \right) & \left( c^\alpha_{,\mu_1 \cdots \mu_k} \right), \left( d x^\mu \right) \\ deg = & -1 & 0 & +1 } \,. \end{displaymath} In terms of these coordinates [[BRST differential]] $s_{BRST}$, thought of as a prolonged [[evolutionary vector field]] on $E \times_\Sigma \mathcal{G}$, corresponds to the smooth function on the shifted cotangent bundle given by \begin{equation} L_{BRST} \;=\; \left( \underset{k \in \mathbb{N}}{\sum} c^\alpha_{,\mu_1 \cdots \mu_k} R_\alpha^{a \mu_1 \cdots \mu_k} \right) \phi^\ddagger_a \;+\; \tfrac{1}{2} \gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma c^\ddagger_\alpha \;\in\; C^\infty\left( T^\ast_{\Sigma,inf}( E \times_\Sigma \mathcal{G}[1] ) \right) \,, \label{BRSTFunctionForClosed}\end{equation} to be called the \emph{[[BRST complex|BRST]] [[Lagrangian function]]} and the product with the [[spacetime]] [[volume form]] \begin{displaymath} L_{BRST} \, dvol_\Sigma \;\in\; \Omega^{p+1,0}_\Sigma(E \times_\Sigma \mathcal{G}[1]) \end{displaymath} as the \emph{[[BRST complex|BRST]] [[Lagrangian density]]}. We now define the [[Chevalley-Eilenberg differential]] on smooth functions on $T^\ast_{inf}( E/(\mathcal{G} \times_\Sigma T \Sigma) )$ to be given by the [[local BV-BRST complex|local]] [[antibracket]] $\{-,-\}$ \eqref{LocalCommutatorOfDerivationsOnJetBundle} with the BRST Lagrangian density \eqref{BRSTFunctionForClosed} \begin{displaymath} d_{CE(T^\ast_{\Sigma,inf}( E/(\mathcal{G} \times_\Sigma T \Sigma) ))} \;\coloneqq\; \left\{ L_{BRST} dvol_\Sigma, - \right\} \end{displaymath} This defines an $L_\infty$-algebroid to be denoted \begin{displaymath} T^\ast_{\Sigma,inf}( E/(\mathcal{G} \times_\Sigma T \Sigma) ) \,. \end{displaymath} \end{defn} The local refinement of prop. \ref{ExteriorDifferentialOfGaugeInvariantFunctionIsSectionOfInfinitesimalCotangentLieAlgebroid} is now this: \begin{prop} \label{EulerLagrangeFormIsSectionOfLocalCotangentBundleOfJetBundleGaugeActionLieAlgebroid}\hypertarget{EulerLagrangeFormIsSectionOfLocalCotangentBundleOfJetBundleGaugeActionLieAlgebroid}{} \textbf{([[Euler-Lagrange form]] is [[section]] of local cotangent bundle of [[jet bundle]] [[gauge symmetry|gauge]]-[[action Lie algebroid]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) over some [[spacetime]] $\Sigma$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a [[gauge parameter bundle]] (def. \ref{GaugeParameters}) which are closed (def. \ref{GaugeParametersClosed}), inducing via example \ref{LocalOffShellBRSTComplex} the [[Lie algebroid]] $E / ( \mathcal{G} \times_\Sigma T \Sigma )$ and via def. \ref{LocalInfinitesimalCotangentLieAlgebroid} its local cotangent [[Lie ∞-algebroid]] $T^\ast_{inf}_\Sigma(E / ( \mathcal{G} \times_\Sigma T \Sigma ))$. Then the [[Euler-Lagrange variational derivative]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) constitutes a [[section]] of the local cotangent Lie ∞-algebroid (def. \ref{LocalInfinitesimalCotangentLieAlgebroid}) \begin{displaymath} \itexarray{ && T^\ast_{\Sigma,inf}\left( E/(\mathcal{G} \times_\Sigma T \Sigma) \right) \\ & {}^{\mathllap{ \delta_{EL} \mathbf{L} }}\nearrow & \downarrow^{\mathrlap{cb}} \\ E/(\mathcal{G} \times_\Sigma T \Sigma) &=& E/(\mathcal{G} \times_\Sigma T \Sigma) } \end{displaymath} given dually \begin{displaymath} CE(E/(\mathcal{G} \times_\Sigma T\Sigma)) \overset{(\delta_{EL}\mathbf{L})^\ast}{\longleftarrow} CE(T^\ast_{inf}(E/(\mathcal{G}\times_\Sigma T \Sigma))) \end{displaymath} by \begin{displaymath} \itexarray{ \left\{ \phi^a_{,\mu_1 \cdots \mu_k} \right\} &\longleftarrow& \left\{ \phi^a_{,\mu_1 \cdots \mu_k} \right\} \\ \left\{ c^\alpha_{,\mu_1 \cdots \mu_k} \right\} &\longleftarrow& \left\{ c^\alpha_{,\mu_1 \cdots \mu_k} \right\} \\ \left\{ \frac{d^k}{ d x^{\mu_1} \cdots d x^{\mu_k}} \left( \frac{\delta_{EL} L}{\delta \phi^a} \right) \right\} &\longleftarrow& \left\{ \phi^\ddagger_{a,\mu_1 \cdots \mu_k} \right\} \\ \left\{ 0 \right\} &\longleftarrow& \left\{ c^\ddagger_{\alpha,\mu_1 \cdots \mu_k} \right\} } \end{displaymath} \end{prop} \begin{proof} The proof of this proposition is a special case of the observation that the differentials involved are part of the local BV-BRST differential; this will be a direct consequence of the proof of prop. \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm} below. \end{proof} The local analog of def. \ref{DerivedCriticalLocusOfGaugeInvariantFunctionOnLieAlgebroid} is now the following definition \ref{DerivedProlongedShell} of the ``derived prolonged shell'' of the theory (recall the ordinary [[prolonged shell]] $\mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E)$ from \eqref{ProlongedShellInJetBundle}): \begin{defn} \label{DerivedProlongedShell}\hypertarget{DerivedProlongedShell}{} \textbf{(derived reduced [[prolonged shell]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) over some [[spacetime]] $\Sigma$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a bundle of closed irreducible [[gauge parameters]] (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}), inducing via prop. \ref{EulerLagrangeFormIsSectionOfLocalCotangentBundleOfJetBundleGaugeActionLieAlgebroid} a section $\delta_{EL} L$ of the local cotangent Lie algebroid of the jet bundle gauge-action Lie algebroid. Then the \emph{derived prolonged shell} $(E/( \mathcal{G} \times_\Sigma T \Sigma ))_{\delta_{EL}L \simeq 0}$ is the [[derived critical locus]] of $\delta_{EL} L$, hence the [[homotopy pullback]] of $\delta_{EL} L$ along the zero section of the local cotangent Lie $\infty$-algebroid: \begin{displaymath} \itexarray{ (E/( \mathcal{G} \times_\Sigma T \Sigma ))_{\delta_{EL}L \simeq 0} &\longrightarrow& E/( \mathcal{G} \times_\Sigma T \Sigma ) \\ \downarrow &(pb)& \downarrow^{\mathrlap{0}} \\ E/(\mathcal{G} \times_\Sigma T \Sigma) &\underset{\delta_{EL} L}{\longrightarrow}& T^\ast_{\Sigma,inf} \left( E/( \mathcal{G} \times_\Sigma T \Sigma ) \right) } \end{displaymath} \end{defn} As before, for the purpose of our running examples the reader may take the following example as the definition of the derived reduced prolonged shell (def. \ref{DerivedProlongedShell}). This is local refinement of example \ref{ArchetypeOfBVBRSTComplex}: \begin{example} \label{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}\hypertarget{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}{} \textbf{([[local BV-BRST complex]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) over [[Minkowski spacetime]] $\Sigma$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a [[gauge parameter bundle]] (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) which is closed (def. \ref{GaugeParametersClosed}). Assume that both are [[trivial vector bundles]] (example \ref{TrivialVectorBundleAsAFieldBundle}) with field coordinates as in prop. \ref{EulerLagrangeFormIsSectionOfLocalCotangentBundleOfJetBundleGaugeActionLieAlgebroid}. Then the [[Chevalley-Eilenberg algebra]] of the derived prolonged shell $(E/( \mathcal{G} \times_\Sigma T \Sigma ))_{\delta_{EL}L \simeq 0}$ (def. \ref{DerivedProlongedShell}) is \begin{displaymath} CE\left( (E/( \mathcal{G} \times_\Sigma T \Sigma ))_{\delta_{EL}L \simeq 0} \right) \;=\; \left( C^\infty\left( T^\ast_{\Sigma,inf}( E \times_\Sigma \mathcal{G}[1] \times_\Sigma T^\ast \Sigma[1] ) \right) \,,\, \underset{ = s }{ \underbrace{ \left\{ \left(- L + L_{BRST}\right) dvol_\Sigma \,, (-) \right\} } } \;+\; d \right) \end{displaymath} where the underlying graded algebra is the [[algebra of functions]] on the (-1)-shifted [[vertical cotangent bundle]] of the [[fiber product]] of the [[field bundle]] with the (+1)-shifted [[gauge parameter bundle]] (as in example \ref{ArchetypeOfBVBRSTComplex}) and the shifted cotangent bundle of $\Sigma$, and where the [[Chevalley-Eilenberg differential]] is the sum of the [[horizontal derivative]] $d$ with the \emph{[[BV-BRST differential]]} \begin{equation} s \;\coloneqq\; \left\{ \left(- L + L_{BRST}\right) dvol_\Sigma \,, (-) \right\} \label{LocalAntibracketVersionOfBVBRSTDifferential}\end{equation} which is the [[local antibracket]] (def. \ref{LocalAntibracket}) with the \emph{[[BV-BRST Lagrangian density]]} \begin{displaymath} \left( -L + L_{BRST}\right) \;\in\; \Omega^{p+1,0}_\Sigma\left( T^\ast_{\Sigma,inf}[-1]\left( E \times_\Sigma \mathcal{G}[1] \right)\right) \end{displaymath} which itself is the sum of (minus) the given [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) with the BRST Lagrangian \eqref{BRSTFunctionForClosed}. The action of the [[BV-BRST differential]] on the generators is as follows: \begin{displaymath} \itexarray{ & & \itexarray{ \text{BV-BRST differential} \\ s } & \\ \text{field} & \phi^a &\mapsto& \underset{ = s_{BRST}(\phi^a) }{ \underbrace{ \left( \underset{k \in \mathbb{N}}{\sum} c^\alpha_{,\mu_1 \cdots \mu_k} R_\alpha^{a \mu_1 \cdots \mu_k} \right) } } & \text{gauge symmetry} \\ \text{ ghost field } & c^\alpha &\mapsto& \underset{ = s_{BRST}(c^\alpha) }{ \underbrace{ \tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma } } & \text{Lie bracket} \\ \text{antifield} & \phi^\ddagger_\alpha &\mapsto& \phantom{-} \underset{ = s_{BV}(\phi^\ddagger_a) }{ \underbrace{ \frac{\delta_{EL} L}{\delta \phi^a} }} & \text{equations of motion} \\ &&& \underset{ = s_{BRST}(\phi^\ddagger_a) }{ \underbrace{ - \left( \underset{k \in \mathbb{N}}{\sum} \frac{\delta_{EL}}{\delta \phi^a} \left( c^\alpha_{,\mu_1 \cdots \mu_k} R_\alpha^{b \mu_1 \cdots \mu_k} \phi^\ddagger_b \right) \right) } } & \\ \itexarray{ \text{antifield of} \\ \text{ghost field} } & c^\ddagger_\alpha &\mapsto& \underset{ = s_{BV}(c^\ddagger_\alpha) }{ \underbrace{ - \underset{k \in \mathbb{N}}{\sum} (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \left( R_\alpha^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a \right) } } & \text{Noether identities} \\ &&& + \underset{ = s_{BRST}(c^\ddagger_\alpha) }{ \underbrace{ \gamma^{\alpha'}{}_{ \alpha \beta} c^\beta c^\ddagger_{\alpha'} } } } \end{displaymath} and this extends to jets of generator by $s \circ d + d \circ s = 0$. This is called the \emph{[[local BV-BRST complex]]}. By introducing a bigrading as in prop. \ref{DerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure} \begin{displaymath} \itexarray{ & \left( c^\ddagger_{\alpha, \mu_1 \cdots \mu_k} \right) & \left( \phi^\ddagger_{a, \mu_1 \cdots \mu_k} \right) & \left( \phi^a_{,\mu_1 \cdots \mu_k} \right) & \left( c^\alpha_{,\mu_1 \cdots \mu_k} \right) \\ deg_{gh} = & 0 & 0 & 0 & +1 \\ deg_{af} = & -2 & -1 & 0 & 0 } \end{displaymath} this splits into the [[total complex]] of a [[bicomplex]] with \begin{displaymath} s \;=\; s_{BV} + s_{BRST} \end{displaymath} with \begin{displaymath} \itexarray{ & s_{BRST} & s_{BV} \\ deg_{gh} = & +1 & 0 \\ deg_{af} = & 0 & +1 } \end{displaymath} as shown in the above table. Under this decomposition, the \emph{[[classical master equation]]} \begin{displaymath} s^2 = 0 \phantom{AAAA} \Leftrightarrow \phantom{AAAA} \left\{ \left( -L + L_{BRST}\right) dvol_\Sigma \,,\, \left( -L + L_{BRST}\right) dvol_\Sigma \right\} = 0 \end{displaymath} is equivalent to three conditions: \begin{displaymath} \itexarray{ \left( s_{BV} \right)^2 = 0 && \text{Noether's second theorem} \\ \left( s_{BRST} \right)^2 = 0 && \text{closure of gauge symmetry} \\ \left[ s_{BV}, s_{BRST} \right] = 0 && \left\{ \itexarray{ \text{ gauge symmetry preserves the shell }, \\ \text{ gauge symmetry acts on Noether identities } } \right. } \end{displaymath} \end{example} (e.q. \href{BRST+complex#Barnich10}{Barnich 10 (4.10)}) \begin{proof} Due to the construction in def. \ref{DerivedProlongedShell} the [[BRST differential]] by itself is already assumed to square to the \begin{displaymath} \left(s_{BRST}\right)^2 = 0 \end{displaymath} The remaining conditions we may check on 0-jet generators. The condition \begin{displaymath} \left( s_{BV} \right)^2 = 0 \end{displaymath} is non-trivial only on the [[antifields]] of the [[ghost fields]]. Here we obtain \begin{displaymath} \begin{aligned} s_{BV} s_{BV} c^\ddagger_\alpha & = -\underset{k \in \mathbb{N}}{\sum} (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \left( R_\alpha^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a \right) \\ & = -\underset{k \in \mathbb{N}}{\sum} (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \left( R_\alpha^{a \mu_1 \cdots \mu_k} \frac{\delta_{EL} L}{\delta \phi^a} \right) \end{aligned} \end{displaymath} That this vanishes is the statement of [[Noether's theorem|Noether's second theorem]] (prop. \ref{NoetherIdentities}). Next we check \begin{displaymath} s_{BV} \circ s_{BRST} + s_{BRST} \circ s_{BV} = 0 \end{displaymath} on generators. On the [[field (physics)|fields]] $\phi^a$ and the [[ghost fields]] $c^\alpha$ this is trivial (both summands vanish separately). On the [[antifields]] we get on the one hand \begin{displaymath} \begin{aligned} s_{BRST} s_{BV} \phi^{\ddagger}_a & = s_{BRST} \frac{\delta_{EL} L}{\delta \phi^a} \\ & = \underset{k}{\sum} \underset{q}{\sum} \frac{d^q}{d x^{\nu_1} \cdots d x^{\nu_q}} \left( c^\alpha_{,\mu_1 \cdots \mu_k} R_\alpha^{b \mu_1 \cdots \mu_k} \right) \frac{\partial}{\partial \phi^b_{,\nu_1 \cdots \nu_q}} \frac{\delta_{EL} L}{\delta \phi^a} \end{aligned} \end{displaymath} and on the other hand \begin{displaymath} \begin{aligned} s_{BV} s_{BRST} \phi^\ddagger_a & = - s_{BV} \underset{k}{\sum} \frac{\delta_{EL}}{\delta \phi^a} \left( c^\alpha_{,\mu_1 \cdots \mu_k} R_\alpha^{b \mu_1 \cdots \mu_k} \phi^\ddagger_b \right) \\ & = + \underset{k}{\sum} \underset{q}{\sum} (-1)^q \frac{d^q}{d x^{\nu_1} \cdots d x^{\nu_q}} \left( \frac{\partial}{\partial \phi^a_{,\mu_1 \cdots \mu_q}} \left( c^\alpha_{,\mu_1 \cdots \mu_k} R_\alpha^{b \mu_1 \cdots \mu_k} \right) \frac{\delta_{EL} L}{\delta \phi^b} \right) \end{aligned} \end{displaymath} That the sum of these two terms indeed vanishes is equation \eqref{TowardsProofThatSymmetriesPreserveTheShell} in the proof of the on-shell invariance of the [[equations of motion]] under [[infinitesimal symmetries of the Lagrangian]] (prop. \ref{InfinitesimalSymmetriesOfLagrangianAreAlsoSymmetriesOfTheEquationsOfMotion}) Finally, on antifields of ghostfields we get \begin{displaymath} \begin{aligned} s_{BV} s_{BRST} c^\ddagger_\alpha & = s_{BV} \gamma^{\alpha'}{}_{\alpha \beta} c^\beta c^\ddagger_{\alpha'} \\ & = - \gamma^{\alpha'}{}_{\alpha \beta} c^\beta \underset{k \in \mathbb{N}}{\sum} (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \left( R_{\alpha'}^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a \right) \end{aligned} \end{displaymath} as well as \begin{displaymath} \begin{aligned} s_{BRST} s_{BV} c^\ddagger_\alpha & = s_{BRST} \left( \underset{k \in \mathbb{N}}{\sum} (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \left( R_\alpha^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a \right) \right) \\ & = R \left( \underset{k \in \mathbb{N}}{\sum} (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \left( R_\alpha^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a \right) \right) \;-\; \left( \underset{k \in \mathbb{N}}{\sum} (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \left( R_\alpha^{a \mu_1 \cdots \mu_k} \left( \underset{q \in \mathbb{N}}{\sum} \frac{\delta_{EL}}{\delta \phi^a} \left( c^{\alpha'}_{,\nu_1 \cdots \nu_q} R_{\alpha'}^{b \nu_1 \cdots \nu_q} \phi^\ddagger_b \right) \right) \right) \right) \\ & + R \left( \underset{k \in \mathbb{N}}{\sum} (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \left( R_\alpha^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a \right) \right) \;-\; \left( \underset{k \in \mathbb{N}}{\sum} (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \left( R_\alpha^{a \mu_1 \cdots \mu_k} \left( \underset{q,r \in \mathbb{N}}{\sum} (-1)^{r} \frac{d^r}{d x^{\rho_1} \cdots d x^{\rho_r}} \left( c^{\alpha'}_{,\nu_1 \cdots \nu_q} \frac{\partial R_{\alpha'}^{b \nu_1 \cdots \nu_q}}{\partial \phi^a_{,\rho_1 \cdots \rho_r}} \phi^\ddagger_b \right) \right) \right) \right) \\ & = (R \cdot N_R)_a^b (\phi^\ddagger_b) \end{aligned} \end{displaymath} where in the last line we identified the [[Lie algebra action]] of [[infinitesimal symmetries of the Lagrangian]] on [[Noether operators]] from def. \ref{NoetherOperator}. Under this identification, the fact that \begin{displaymath} \left( s_{BRST}s_{BV} + s_{BV} s_{BRST} \right) c^\ddagger_\alpha = 0 \end{displaymath} is relation \eqref{LieActionOnNoetherOperatorGivesLieBracketUnderNoetherTheorem} in prop. \ref{LieAlgebraActionOfInfinitesimalSymmetriesOfTheLagrangianOnNoetherOperators}. \end{proof} \begin{example} \label{DerivedProlongedShellInAbsenceOfExplicitGaugeSymmetries}\hypertarget{DerivedProlongedShellInAbsenceOfExplicitGaugeSymmetries}{} \textbf{(derived prolonged shell in the absence of explicit gauge symmetry -- the [[local BV-complex]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. $\backslash$ref\{LocalLagrangianDensityOnSecondOrde rJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime\}) with vanishing [[gauge parameter bundle]] (def. \ref{GaugeParameters}) (possibly because there are no non-trivial [[infinitesimal gauge symmetries]], such as for the [[scalar field]], or because none were chose), hence with no [[ghost fields]] introduced. Then the local [[derived critical locus]] of its [[Lagrangian density]] (def. \ref{DerivedProlongedShell}) is the plain [[local BV-complex]] of def. \ref{BVComplexOfOrdinaryLagrangianDensity}. \begin{displaymath} s = s_{BV} \,. \end{displaymath} \end{example} \begin{example} \label{LocalBVComplexOfVacuumElectromagnetismOnMinkowskiSpacetime}\hypertarget{LocalBVComplexOfVacuumElectromagnetismOnMinkowskiSpacetime}{} \textbf{([[local BV-BRST complex]] of [[vacuum]] [[electromagnetism]] on [[Minkowski spacetime]])} Consider the [[Lagrangian field theory]] of [[free field theory|free]] [[electromagnetism]] on [[Minkowski spacetime]] (example \ref{ElectromagnetismLagrangianDensity}) with [[gauge parameter]] as in example \ref{InfinitesimalGaugeSymmetryElectromagnetism}. With the [[field (physics)|field]] and [[gauge parameter]] coordinates as chosen in these examples \begin{displaymath} \left( (a_\mu), c \right) \end{displaymath} then the [[local BV-BRST complex]] (prop. \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}) has generators \begin{displaymath} \itexarray{ & c^\ddagger & (a^\ddagger)^\mu & a_\mu & c \\ deg = & -2 & -1 & 0 & 1 } \end{displaymath} together with their [[total spacetime derivatives]], and the local BV-BRST differential $s$ acts on these generators as follows: \begin{displaymath} s \;\colon\; \left\{ \itexarray{ (a^\dagger)^\mu &\mapsto& f^{\nu \mu}_{,\nu} & \text{(equations of Motion -- vacuum Maxwell equations)} \\ c^\ddagger &\mapsto& (a^\ddagger)^\mu_{,\mu} & \text{(Noether identity)} \\ a_\mu &\mapsto& c_{,\mu} & \text{(infinitesimal gauge transformation)} } \right. \end{displaymath} \end{example} More generally: \begin{example} \label{LocalBVBRSTComplexOfYangMillsTheory}\hypertarget{LocalBVBRSTComplexOfYangMillsTheory}{} \textbf{([[local BV-BRST complex]] of [[Yang-Mills theory]])} For $\mathfrak{g}$ a [[semisimple Lie algebra]], consider $\mathfrak{g}$-[[Yang-Mills theory]] on [[Minkowski spacetime]] from example \ref{YangMillsLagrangian}, with [[local BRST complex]] as in example \ref{YangMillsLocalBRSTComplex}, hence with [[BRST Lagrangian]] \eqref{BRSTFunctionForClosed} given by \begin{displaymath} L_{BRST} = \left( c^\alpha_{,\mu} - \gamma^\alpha{}_{\beta \gamma}c^\beta a^\gamma_\mu \right) (a^\ddagger)_\alpha^\mu \;+\; \tfrac{1}{2} \gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma c^\ddagger_\alpha \,. \end{displaymath} Then its [[local BV-BRST complex]] (example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}) has [[BV-BRST differential]] $s = \left\{ -L + L_{BRST} \,,\, - \right\}$ given on 0-jets as follows: \begin{displaymath} \itexarray{ & & s & \\ \text{field} & a_\mu^\alpha &\mapsto& c^\alpha_{,\mu} - \gamma^\alpha{}_{\beta \gamma}c^\beta a^\gamma_\mu & \text{gauge symmetry} \\ \text{ ghost field } & c^\alpha &\mapsto& \tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma & \text{Lie bracket} \\ \text{antifield} & (a^\ddagger)^\mu_\alpha &\mapsto& \phantom{-} \left( \frac{d}{d x^\mu} f^{\mu \nu \alpha'} + \gamma^{\alpha'}{}_{\beta' \gamma} a_\mu^{\beta'} f^{\mu \nu \gamma} \right) k_{\alpha' \alpha} & \text{equations of motion} \\ &&& - \gamma^{\alpha'}{}_{\beta \alpha}c^\beta (a^\ddagger)_{\alpha'}^\mu & \\ \text{anti ghostfield} & c^\ddagger_\alpha &\mapsto& \gamma^{\alpha'}{}_{\alpha \gamma} a^\gamma_\mu (a^\ddagger)^\mu_{\alpha'} + \frac{d}{d x^\mu} (a^\ddagger)^\mu_\alpha & \text{Noether identities} \\ &&& + \gamma^{\alpha'}{}_{ \alpha \beta} c^\beta c^\ddagger_{\alpha'} } \end{displaymath} \end{example} (e.g. \href{local+BRST+cohomology#BarnichBrandtHenneaux00}{Barnich-Brandt-Henneaux 00 (2.8)}) $\,$ So far the discussion yields just the [[algebra of functions]] on the derived reduced prolonged shell. We now discuss the derived analog of the full [[variational bicomplex]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}) to the derived reduced shell. $\,$ \textbf{(derived variational bicomplex)} The analog of the [[de Rham complex]] of a [[derived Lie algebroid]] is called the \emph{[[Weil algebra]]}: \begin{defn} \label{WeilAlgebra}\hypertarget{WeilAlgebra}{} \textbf{([[Weil algebra]] of a [[Lie algebroid]])} Given a [[derived Lie algebroid]] $\mathfrak{a}$ over some $X$ (def. \ref{LInfinityAlgebroid}), its [[Weil algebra]] is \begin{displaymath} W(\mathfrak{a}) \;\coloneqq\; \left( Sym_{C^\infty(X)}( \Gamma(T^\ast_{inf} X) \oplus \mathfrak{a}_\bullet \oplus \mathfrak{a}[1]_\bullet ) \;,\; \mathbf{d}_W \coloneqq \mathbf{d} + d_{CE} \right) \,, \end{displaymath} where $\mathbf{d}$ acts as the de Rham differential $\mathbf{d} \colon C^\infty(X) \to \Gamma(T^\ast_{inf} X)$ on functions, and as the degree shift operator $\mathbf{d} \colon \mathfrak{a}_\bullet \to \mathfrak{a}[1]_\bullet$ on the graded elements. \end{defn} \begin{tabular}{l|l} [[smooth manifolds]]&[[derived Lie algebroids]]\\ \hline [[algebra of functions]]&[[Chevalley-Eilenberg algebra]]\\ algebra of [[differential forms]]&[[Weil algebra]]\\ \end{tabular} \begin{example} \label{ClassicalWeilAlgebra}\hypertarget{ClassicalWeilAlgebra}{} \textbf{(classical [[Weil algebra]])} Let $\mathfrak{g}$ be a [[Lie algebra]] with corresponding [[Lie algebroid]] $B \mathfrak{g}$ (example \ref{BasicExamplesOfLieAlgebroids}). Then the Weil algebra (def. \ref{WeilAlgebra}) of $B \mathfrak{g}$ is the traditional Weil algebra of $\mathfrak{g}$ from classical [[Lie theory]]. \end{example} \begin{defn} \label{BVVariationalBicomplex}\hypertarget{BVVariationalBicomplex}{} \textbf{([[variational BV-bicomplex]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) equipped with a [[gauge parameter bundle]] $\mathcal{G}$ (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) which is closed (def. \ref{GaugeParametersClosed}). Consider the [[Lie algebroid]] $E/(\mathcal{G} \times_\Sigma T \Sigma)$ from example \ref{LocalOffShellBRSTComplex}, whose [[Chevalley-Eilenberg algebra]] is the [[local BRST complex]] of the theory. Then its [[Weil algebra]] $W(E/(\mathcal{G} \times_\Sigma T \Sigma))$ (def. \ref{WeilAlgebra}) has as differential the [[variational derivative]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}) plus the [[BRST differential]] \begin{displaymath} \begin{aligned} d_{W} & = \mathbf{d} - (d - s_{BRST}) \\ & = \delta + s_{BRST} \end{aligned} \,. \end{displaymath} Therefore we speak of the \emph{[[variational BRST-bicomplex]]} and write \begin{displaymath} \Omega^\bullet_\Sigma( E/(\mathcal{G} \times_\Sigma T \Sigma) ) \,. \end{displaymath} Similarly, the Weil algebra of the derived prolonged shell $(E/( \mathcal{G} \times_\Sigma T \Sigma ))_{\delta_{EL}L \simeq 0}$ (def. \ref{DerivedProlongedShell}) has differential \begin{displaymath} \begin{aligned} d_W & = \mathbf{d} - (d - s) \\ & = \delta + s \end{aligned} \,. \end{displaymath} Since $s$ is the [[BV-BRST differential]] (prop. \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}) this defines the ``BV-BRST [[variational bicomplex]]''. \end{defn} $\,$ \textbf{global [[BV-BRST complex]]} Finally we may apply [[transgression of variational differential forms]] to turn the [[local BV-BRST complex]] on smooth functions on the [[jet bundle]] into a global [[BV-BRST complex]] on graded [[local observables]] on the graded [[space of field histories]]. \begin{defn} \label{ComplexBVBRSTGlobal}\hypertarget{ComplexBVBRSTGlobal}{} \textbf{(global [[BV-BRST complex]])} Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) equipped with a [[gauge parameter bundle]] $\mathcal{G}$ (def. \ref{GaugeParametrizedInfinitesimalGaugeTransformation}) which is closed (def. \ref{GaugeParametersClosed}). Then on the [[local observables]] (def. \ref{LocalObservables}) on the [[space of field histories]] (def. \ref{FieldsAndFieldBundles}) of the [[graded manifold|graded]] [[field bundle]] \begin{displaymath} E_{\text{BV-BRST}} = T^\ast_{\Sigma,inf}[-1](E \times_\Sigma \mathcal{G}[1]) \end{displaymath} underlying the [[local BV-BRST complex]] \eqref{BVBRSTGradedFieldBundle}, consider the [[linear map]] \begin{equation} \itexarray{ LocObs(E_{\text{BV-BRST}}) \otimes LocObs(E_{\text{BV-BRST}}) &\overset{\{-,-\}}{\longrightarrow}& LocObs(E_{\text{BV-BRST}}) \\ \tau_\Sigma(\alpha), \tau_\Sigma(\beta) &\mapsto& \tau_\Sigma( \{\alpha, \beta\} ) } \label{LocalAntibracketTransgressed}\end{equation} where $\alpha, \beta \in \Omega^{p+1,0}_{\Sigma,cp}(E_{\text{BV-BRST}})$ (def. \ref{SpacetimeSupport}), where $\tau_\Sigma$ denotes [[transgression of variational differential forms]] (def. \ref{TransgressionOfVariationalDifferentialFormsToConfigrationSpaces}), and where on the right $\{-,-\}$ is the [[local antibracket]] (def. \ref{LocalAntibracket}). This is well-defined, in that this formula indeed depends on the [[horizontal differential forms]] $\alpha$ and $\beta$ only through the [[local observables]] $\tau_\Sigma(\alpha), \tau_\Sigma(\beta)$ which they induce. The resulting bracket is called the (global) \emph{[[antibracket]]}. Indeed the formula makes sense already if at least one of $\alpha, \beta$ have compact spacetime support (def. \ref{SpacetimeSupport}), and hence the [[transgression]] of the [[BV-BRST differential]] \eqref{LocalAntibracketVersionOfBVBRSTDifferential} is a well-defined [[differential]] on the graded [[local observables]] \begin{displaymath} \left\{ -\tau_\Sigma \mathbf{L} + \tau_\Sigma \mathbf{L}_{BRST} \;,\, - \right\} \;\colon\; LocObs(E_{\text{BV-BRST}}) \longrightarrow LocObs(E_{\text{BV-BRST}}) \,, \end{displaymath} where by example \ref{ActionFunctional} we may think of the first argument on the left as the BV-BRST [[action functional]] without [[adiabatic switching]], which makes sense inside the [[antibracket]] when acting on functionals with compact spacetime support. Hence we may suggestively write \begin{equation} \left\{ -S + S_{BRST} \;,\;- \right\} \;\coloneqq\; \left\{ -\tau_\Sigma \mathbf{L} + \tau_\Sigma \mathbf{L}_{BRST} \;,\, - \right\} \label{GlobalBVBRSTDifferential}\end{equation} for this (global) \emph{[[BV-BRST differential]]}. This uniquely extends as a graded [[derivation]] to [[multilocal observables]] (def. \ref{LocalObservables}) and from there along the [[dense subspace]] inclusion \eqref{InclusionOfPolynomialLocalObservablesIntoPolynomialObservables} \begin{displaymath} PolyMultiLocObs(E_{\text{BV-BRST}}) \overset{\text{dense}}{\hookrightarrow} PolyObs(E_{\text{BV-BRST}}) \end{displaymath} to a differential on [[off-shell]] [[polynomial observables]] (def. \ref{PolynomialObservables}): \begin{displaymath} \{-S' + S'_{BRST}\} \;\colon\; PolyObs(E_{\text{BV-BRST}}) \longrightarrow PolyObs(E_{\text{BV-BRST}}) \end{displaymath} This [[differential graded-commutative superalgebra]] \begin{equation} \left( \left( \underset{ \text{vector space} }{ \underbrace{ PolyObs(E_{\text{BV-BRST}}) }} , \underset{ \text{product} }{ \underbrace{ (-)\cdot(-) }} \right) , \underset{ \text{differential} }{ \underbrace{ \{-S' + S'_{BRST}, -\} }} \right) \label{GlobalBVComplexdgAlgebra}\end{equation} is the \emph{global [[BV-BRST complex]]} of the given [[Lagrangian field theory]] with the chosen [[gauge parameters]]. \end{defn} \begin{proof} We need to check that the global [[antibracket]] \eqref{LocalAntibracketTransgressed} is well defined: By the last item of prop. \ref{BasicPropertiesOfTheLocalAntibracket} the horizontally exact horizontal differential forms form a ``[[Lie ideal]]'' for the [[local antibracket]]. With this the proof that the transgressed bracket is well defined is the same as the proof that the global [[Poisson bracket]] on the [[Hamiltonian differential form|Hamiltonian]] [[local observables]] is well defined, def. \ref{PoissonBracketOnHamiltonianLocalObservables}. \end{proof} \begin{example} \label{BVDifferentialGlobal}\hypertarget{BVDifferentialGlobal}{} \textbf{(global BV-differential in components)} In the situation of def. \ref{ComplexBVBRSTGlobal}, assume that the [[field bundles]] of all [[field (physics)|fields]], [[ghost fields]] and [[auxiliary fields]] are [[trivial vector bundles]], with field/ghost-field/auxiliary-field coordinates on their [[fiber product]] bundle collectively denoted $(\phi^A)$. Then the first summand of the global BV-BRST differential (def. \ref{ComplexBVBRSTGlobal}) is given by \begin{equation} \begin{aligned} \left\{ -S', -\right\} & = \int_\Sigma j^{\infty}\left(\mathbf{\Phi}\right)^\ast \left( \frac{\overset{\leftarrow}{\delta}_{EL} L}{\delta \phi^A} \right)(x) \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)} \, dvol_\Sigma(x) \\ & = \underset{A}{\sum} (-1)^{deg(\phi^A)} \int_\Sigma (P_{A B}\mathbf{\Phi}^A)(x) \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)} \, dvol_\Sigma(x) \end{aligned} \label{ComponentsOfGlobalBVDifferential}\end{equation} where \begin{enumerate}% \item $P \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(E^\ast)$ is the [[differential operator]] \eqref{DifferentialOperatorEulerLagrangeDerivative} from def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}, corresponding to the [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]]. \item $deg(\phi^A) \coloneqq n_{(\Phi^A)} + \sigma_{\Phi^A} \;\in\; \mathbb{Z}/2$ is the sum of the cohomological degree and of the super-degree of $\Phi^A$ (as in def. \ref{differentialgradedcommutativeSuperalgebra}, def. $\backslash$ref\{A+first+idea+of+quantum+field+theoryDifferentialFormOnSuperCartesianSpaces\}). \end{enumerate} It follows that the [[cochain cohomology]] of the global [[BV-differential]] $\{-S',-\}$ \eqref{GlobalBVComplexdgAlgebra} in $deg_{af} = 0$ is the space of [[on-shell]] [[polynomial observables]]: \begin{equation} \underset{ \text{off-shell} }{ \underbrace{ PolyObs(E_{\text{BV-BRST}})_{def(af = 0)} }}/im(\{-S',-\}) \;\simeq\; \underset{ \text{on-shell} }{ \underbrace{ PolyObs(E_{\text{BV-BRST}}, \mathbf{L}') }} \,. \label{OnShellPolynomialObservablesAsBVCohomology}\end{equation} \end{example} \begin{proof} By definition, the part $\mathbf{L}'$ of the gauge fields Lagrangian density is independent of [[antifields]], so that the [[local antibracket]] with $\mathbf{L}'$ reduces to \begin{displaymath} \left\{ -\mathbf{L}',-\right\} \;=\; \frac{\overset{\leftarrow}{\delta}_{EL} \mathbf{L}'}{\delta \phi^A} \frac{\delta}{\delta \phi^{\ddagger}_A} \end{displaymath} With this the expression for $\{-S',-\}$ follows directly from the definition of the global antibracket (def. \ref{ComplexBVBRSTGlobal}) and the [[Euler-Lagrange equations]] \eqref{DifferentialOperatorEulerLagrangeDerivative} \begin{displaymath} (P \Phi)_A = j^\infty_\Sigma(\Phi)\left( \frac{\delta_{EL} L}{\delta \phi^A} \right) \,. \end{displaymath} where the sign $(-1)^{deg(\phi^A)}$ is the relative sign between $\frac{\delta_{EL} L}{\delta \phi^A} = \frac{\overset{\rightarrow}{\delta}_{EL} L'}{\delta \phi^A}$ and $\frac{\overset{\leftarrow}{\delta}_{EL} L'}{\delta \phi^A}$ (def. \ref{SchoutenBracketAndAntibracket}): By the assumption that $L'$ defines a [[free field theory]], $\mathbf{L}'$ is quadratic in the fields, so that from $deg(\mathbf{L}) = 0$ it follows that the derivations from the left and from the right differ by the relative sign \begin{displaymath} \begin{aligned} (-1)^{ \left( n_{(\phi^A)} n_{(\phi^A)} + \sigma_{(\phi^A)} \sigma_{(\phi^A)} \right) } & = (-1)^{ \left( n_{(\phi^A)} + \sigma_{(\phi^A)} \right) } \\ & = (-1)^{deg(\phi^A)} \end{aligned} \,. \end{displaymath} From this the identification \eqref{OnShellPolynomialObservablesAsBVCohomology} follows by \eqref{PolynomialOnShellObservablesArePolynomialOffShellobservableModuloTheEquationsOfMotion} in theorem \ref{LinearObservablesForGreeFreeFieldTheoryAreDistributionalSolutionsToTheEquationsOfMotion}. \end{proof} $\,$ This concludes our discussion of the [[reduced phase space]] of a [[Lagrangian field theory]] exhibited, [[formal dual|dually]] by its [[local BV-BRST complex]]. In the \hyperlink{GaugeFixing}{next chapter} we finally turn to the key implication of this construction: the [[gauge fixing]] of a [[Lagrangian field theory|Lagrangian]] [[gauge theory]] which makes the collection of [[field (physics)|fields]] and [[auxiliary fields]] ([[ghost fields]] and [[antifields]]) jointly have a (differential-graded) [[covariant phase space]]. \end{document}