A first idea of quantum field theory -- Reduced phase space

## Reduced phase space
 {#ReducedPhaseSpace}

In this chapter we discuss these topics:

* Global gauge reduction for strictly [[invariant]] functions ([[action functionals]]):

  * _[Derived critical loci inside Lie algebroids](#DerivedCriticalLocusInsideLieAlgebroids)_

  * _[Schouten bracket on Lie algebroids](#SchoutenBracketAntibracket)_

* Local gauge reduction for weakly invariant local functions ([[Lagrangian densities]]):

  * _[Local antibracket](#LocalJetBundleAntibracket)_

  * _[Local BV-BRST complex](#DerivedCriticalLocusOnJetBundle)_

  * _[Global BV-BRST complex](#BVBRSTComplexGlobal)_

For a [[Lagrangian field theory]] with [[infinitesimal gauge symmetries]], the _[[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 _[[homotopy quotient]]_ by the [[gauge symmetries]] and its _[[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 _[[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 _[[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 _local_ incarnation of the redcuced phase space is modeled by the genuine _[[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 _global [[BV-BRST complex]]_ of the [[Lagrangian field theory]] (def. \ref{ComplexBVBRSTGlobal} below).

$\,$

**[[derived critical loci]] inside [[Lie algebroids]]**
 {#DerivedCriticalLocusInsideLieAlgebroids}

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]]:

+-- {: .num_defn #HomomorphismBetweenLieAlgebroids}
###### Definition
**([[homomorphism]] between [[Lie algebroids]])**

Given two [[derived Lie algebroids]] $\mathfrak{a}$, $\mathfrak{a}'$ (def. \ref{LInfinityAlgebroid}), then a [[homomorphism]]
between them

$$
  f \;\colon\; \mathfrak{a} \longrightarrow \mathfrak{a}'
$$

is a [[dg-algebra]]-[[homomorphism]] between their [[Chevalley-Eilenberg algebras]] going the other way around

$$
  CE(\mathfrak{a}) \longleftarrow CE(\mathfrak{a}') \;\colon\; f^\ast
$$

such that this covers an algebra homomorphism on the function algebras:

$$
  \array{
    CE(\mathfrak{a}) &\overset{f^\ast}{\longleftarrow}& CE(\mathfrak{a}')
    \\
    \downarrow && \downarrow
    \\
    C^\infty(X) &\underset{(f\vert_X)^\ast}{\longleftarrow}& C^\infty(Y)
  }
  \,.
$$

(This is also called a "[[curved sh-map|non-curved sh-map]]".)

=--

+-- {: .num_example #GaugeInvariantFunctionsIntermsOfLieAlgebroids}
###### Example
**([[invariant]] [[functions]] in terms of [[Lie algebroids]])**

Let $\mathfrak{g}$ be a [[super Lie algebra]] equipped with a [[Lie algebra action]] (def. \ref{InfinitesimalActionByLieAlgebra})

$$
  \array{
    \mathfrak{g} \times X && \overset{R}{\longrightarrow} && T X
    \\
    & {}_{\mathllap{pr_2}}\searrow && \swarrow_{\mathrlap{rb}}
    \\
    && X
  }
$$

on a [[supermanifold]] $X$. Then there is a canonical homomorphism of [[Lie algebroids]] (def. \ref{HomomorphismBetweenLieAlgebroids})

$$
  \label{ProjectionMapForActionLieAlgebroid}
  \array{
    X &&& CE(X) &=& C^\infty(X) &\oplus& 0
    \\
    \downarrow^{\mathrlap{p}}
      &\phantom{AAA}&&
    \uparrow^{\mathrlap{p^\ast}}
      &&
    \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{0}}
    \\
    X/\mathfrak{g}
      &&&
    CE(X/\mathfrak{g})
      &=&
    C^\infty(X)
      &\oplus&
    C^\infty(X) \otimes \wedge^\bullet \mathfrak{g}^\ast
  }
$$

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 _[[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

$$
  S \;\colon\; X/\mathfrak{g} \longrightarrow \mathbb{R}^1
  \,,
$$

hence _smooth functions on the Lie algebroid_, are equivalently

* ordinary [[smooth functions]] $S \;\colon\; X \longrightarrow \mathbb{R}^1$ on the underlying [[smooth manifold]],

* which are [[invariant]] under the Lie algebra action in that $R(-)(S) = 0$.

In terms of the canonical [[homotopy quotient]] [[coprojection]] map $p$ (eq:ProjectionMapForActionLieAlgebroid) this says that a smooth function on $X$ [[extension]] extends to the [[action Lie algebroid]] precisely if it is [[invariant]]:

$$
  \array{
    X &\overset{S}{\longrightarrow}& \mathbb{R}^1
    \\
    {}^{\mathllap{p}}\downarrow
      &
    \nearrow_{
      \mathrlap{
          \text{exists precisely if} \; S \; \text{is invariant}
      }
     }
    \\
    X/\mathfrak{g}
  }
$$

=--

+-- {: .proof}
###### Proof

An $\mathbb{R}$-algebra homomorphism

$$
  CE( X/\mathfrak{g} )
    \overset{S^\ast}{\longleftarrow}
  C^\infty(\mathbb{R}^1)
$$

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 $0 = d_{CE} S = R(-)(f)$:

$$
  \array{
     \left\{ S \right\}
       &\overset{S^\ast}{\longleftarrow}&
     \left\{ x \right\}
     \\
     {}^{\mathllap{d_{CE(X/\mathfrak{g})}}}\downarrow
       &&
     \downarrow^{\mathrlap{d_{CE(\mathbb{R}^1)} = 0 } }
     \\
     \left\{
       R(-)(S)
       =
       0
     \right\}
     &\underset{S^\ast}{\longleftarrow}&
     \left\{ 0 \right\}
  }
$$

=--

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 _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 _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:

+-- {: .num_defn #LieAlgebroidInfinitesimalCotangentBundle}
###### Definition
**([[automorphism ∞-Lie algebra|infinitesimal cotangent Lie algebroid]])**

Let $\mathfrak{a}$ be a [[Lie ∞-algebroid]] (def. \ref{LInfinityAlgebroid}) over some manifold $X$.
Then its _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 $CE(\mathfrak{a})$:

$$
 (T^\ast_{inf} \mathfrak{a})^\ast_\bullet
 \;\coloneqq\;
 \mathfrak{a}^\ast_\bullet \oplus Der(CE(\mathfrak{a}))_\bullet
$$

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):

$$
  \array{
    \mathllap{ d_{CE(T^\ast_{inf} \mathfrak{a})} }
      &\mathrlap{ \vert_{\mathfrak{a}^\ast} }&
      &
    \coloneqq & d_{CE(\mathfrak{a})}
  \\
  \mathllap{ d_{CE(T^\ast_{inf} \mathfrak{a})} }
     & \mathrlap{ \vert_{Der(\mathfrak{a})} } &
     \phantom{ \vert_{Der(\mathfrak{a})} }
     &
   \coloneqq & [d_{CE(\mathfrak{a})},-]
  }
$$

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:

$$
  \label{CotangentLieAlgebrpoidProjection}
  \array{
    T^\ast_{inf} \mathfrak{a}
       &\phantom{AAA}&&
    CE(T^\ast_{inf} \mathfrak{g})
     &=&
    CE(\mathfrak{a})
      &\oplus&
    \wedge^{\bullet \geq 1}_{CE(\mathfrak{a})} Der(\mathfrak{a})
    \\
    \downarrow^{\mathrlap{cb}} &&& \uparrow^{\mathrlap{cb^\ast}}
      && \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{0}}
    \\
    \mathfrak{a}
      &&&
    CE(\mathfrak{a}) &=& CE(\mathfrak{a}) &\oplus& 0
  }
$$

given dually by the identity on the original generators.

=--

+-- {: .num_example #CotangentBundleOfActionLieAlgebroid}
###### Example
**([[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 (eq:DifferentialOnActionLieAlgebroid)

$$
  d_{CE(X/\mathfrak{g})}
  \;=\;
  \tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma \frac{\partial}{\partial c^\alpha}
  +
  c^\alpha R_a^\alpha  \frac{\partial}{\partial \phi^a}
  \,.
$$

Then its infinitesimal cotangent Lie algebroid $T^\ast_{inf} (X/\mathfrak{g})$ (def. \ref{LieAlgebroidInfinitesimalCotangentBundle})
has the generators

$$
  \array{
    &
    \left(
      \frac{\partial}{\partial c^\alpha}
    \right)
    &
    \left(
      \phi^a
    \right)
    ,
    \left(
      \frac{\partial}{\partial \phi^a}
    \right)
    &
    \left(
      c^\alpha
    \right)
    \\
    deg =
    &
    -1
    &
    0
    &
    +1
  }
$$

and we find that CE-differential on the new derivation generators is given by

$$  \label{CotangentLieAlgebroidDifferentialForActionLieAlgebroidOnGhostFieldCoordinates}
  \begin{aligned}
    d_{CE(T^\ast_{inf}(X/\mathfrak{g}))}
    \left(
      \frac{\partial}{\partial c^\alpha}
    \right)
    & \coloneqq
    \left[d_{CE(X/\mathfrak{g})}, \frac{\partial}{\partial c^\alpha} \right]
    \\
     & =
     R_\alpha^a  \frac{\partial}{\partial \phi^a}
      +
     \gamma^\beta{}_{\alpha \gamma} c^\gamma \frac{\partial}{\partial c^\beta}
  \end{aligned}
$$

and

$$
  \label{CotangentLieAlgebroidDifferentialForActionLieAlgebroidOnFieldCoordinates}
  \begin{aligned}
    d_{CE(T^\ast_{inf}(X/\mathfrak{g}))}
    \left(
      \frac{\partial}{\partial \phi^a}
    \right)
    & \coloneqq
    \left[
      d_{CE(X/\mathfrak{g})},
      \frac{\partial}{\partial \phi^a}
    \right]
    \\
    & =
    -
    c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \frac{\partial}{\partial \phi^b}
  \end{aligned}
  \,.
$$

To amplify that the [[derivations]] _on_ $CE(X/\mathfrak{g})$, such as $\frac{\partial}{\partial \phi^a}$ and $\frac{\partial}{\partial c^\alpha}$, are now [[coordinate functions]] _in_ $CE(T^\ast_{inf}(X/\mathfrak{g}))$ one writes them as

$$
  \label{AntiNotationForDerivations}
  \phi^\ddagger_a \;\coloneqq\; \frac{\partial}{\partial \phi^a}
  \phantom{AAAAA}
  c\ddagger_\alpha \;\coloneqq\; \frac{\partial}{\partial c^\alpha}
  \,.
$$

so that the generator content then reads as follows:

$$
  \label{GeneratorsOfDerivedCriticalLocusInActionLieAlgebroid}
  \array{
    &
    \left(
      c^\ddagger_\alpha
    \right)
    &
      \left(
        \phi^a
      \right)
      ,
      \left(
        \phi^\ddagger_a
      \right)
    &
    \left(
      c^\alpha
    \right)
    \\
    deg =
    &
    -1
    &
    0
    &
    +1
  }
  \,.
$$

In this notation the full action of the CE-differential for $T^\ast_{inf}(X/\mathfrak{g})$ is therefore the following:

$$
  \label{CEDifferentialOnGeneratorsForInfinitesimalCotangentBundleOfActionLieAlgebroid}
  \array{
    & d_{CE(T^\ast_{inf}(X/\mathfrak{g}))}
    \\
    \phi^a
      &\mapsto&
    c^\alpha R^a_\alpha
    \\
    c^\alpha
      & \mapsto&
    \tfrac{1}{2} \gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
    \\
    \phi^\ddagger_a
      &\mapsto&
    - c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \phi^\ddagger_b
    \\
    c^\ddagger_\alpha
      &\mapsto&
    R_\alpha^a  \phi^\ddagger_a
      +
    \gamma^\beta{}_{\alpha \gamma} c^\gamma c^\ddagger_\beta
  }
$$

=--

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:

+-- {: .num_prop #ExteriorDifferentialOfGaugeInvariantFunctionIsSectionOfInfinitesimalCotangentLieAlgebroid}
###### Proposition
**([[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 (eq:CotangentLieAlgebrpoidProjection):

$$
  \array{
    && T^\ast_{inf} \mathfrak{a}
    \\
    & {}^{\mathllap{d S}}\nearrow & \downarrow^{\mathrlap{cb}}
    \\
    \mathfrak{a}
    &=&
    \mathfrak{a}
  }
  \,,
$$

given dually by the [[homomorphism]] of [[differential graded-commutative superalgebras]]

$$
  (d S)^\ast \;\colon\; CE(T^\ast_{inf} \mathfrak{a}) \longrightarrow CE(\mathfrak{a})
$$

which sends

1. the generators in $\mathfrak{a}^\ast$ to themselves;

1. a [[vector field]] $v$ on $X$, regarded as a degree-0 [[derivation]] to
   $d S(v) = v(S) \in C^\infty(X)$;

1. all other derivations to zero.

=--

+-- {: .proof}
###### 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 (eq:CotangentLieAlgebroidDifferentialForActionLieAlgebroidOnFieldCoordinates)

$$
  \array{
    \left\{
       \frac{\partial S}{\partial \phi^a}
    \right\}
      &\overset{(d S)^\ast}{\longleftarrow}&
    \left\{ \phi^\ddagger_a \right\}
    \\
    {}^{\mathllap{d_{CE(X/\mathfrak{g})}}}\downarrow
      &&
    \downarrow^{\mathrlap{d_{CE(T^\ast_{inf} (X/\mathfrak{g}))}}}
    \\
    \left\{
       -c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \frac{\partial S}{\partial \phi^b}
    \right\}
      &\underset{(d S)^\ast}{\longleftarrow}&
    \left\{
       -c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \phi^\ddagger_b
    \right\}
  }
$$

{#NoticeThatTheLeftVerticalMap} 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]]":

$$
  \begin{aligned}
    d_{CE(X/\mathfrak{g})}\left( \frac{\partial S}{\partial \phi_a} \right)
    & =
    c^\alpha R_\alpha^{b} \frac{\partial}{\partial \phi^b} \frac{\partial}{\partial \phi^a} S
    \\
    & =
    \frac{\partial}{\partial \phi^a}
    \left(
      c^\alpha
      \underset{
        = 0
      }{
      \underbrace{
        R_\alpha^b \frac{\partial S}{\partial \phi^b}
      }
      }
    \right)
    \;-\;
    c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a}
    \frac{\partial S}{\partial \phi^b}
  \end{aligned}
$$

Similarly, on the derivation $c^\ddagger_\alpha \coloneqq \frac{\partial}{\partial c^\alpha}$ we find from (eq:CotangentLieAlgebroidDifferentialForActionLieAlgebroidOnGhostFieldCoordinates) and using the invariance of $S$ (example \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids})

$$
  \array{
    \left\{
      0
    \right\}
      &\overset{(d S)^\ast}{\longleftarrow}&
    \left\{
      c^\ddagger_\alpha
    \right\}
    \\
    {}^{\mathllap{d_{CE(X/\mathfrak{g})}}}\downarrow
      &&
    \downarrow^{\mathrlap{d_{CE(T^\ast_{inf}(X/\mathfrak{g}))}}}
    \\
    \left\{
      0 = R_\alpha^a \frac{\partial S}{\partial \phi^a}
    \right\}
      &\underset{(d S)^\ast}{\longleftarrow}&
    \left\{
      R_\alpha^a  \phi^\ddagger_a
      +
      \gamma^\beta{}_{\alpha \gamma} c^\gamma
      c^\ddagger_\alpha
    \right\}
  }
  \,.
$$

This shows that the differentials are being respected.

=--

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]]

$$
  X_{d S = 0}
  \;=\;
  \left\{
     x \in X \;\vert\;
     d S(x) = 0
  \right\}
$$

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]]:

$$
  \array{
    X_{d S = 0}
      &\overset{\phantom{AAA}}{\hookrightarrow}&
    X
    \\
    \downarrow && \downarrow^{\mathrlap{0}}
    \\
    X &\underset{d S}{\longrightarrow}& T^\ast_{inf} X
  }
  \,.
$$

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  _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 _[[derived critical locus]]_. The following definition simply states what this comes down to in components. For a detailed derivation see at _[[derived critical locus]]_ and for general introduction to [[higher differential geometry]] and [[higher Lie theory]] see at _[[schreiber:Higher Structures|Higher structures in Physics]]_.

+-- {: .num_defn #DerivedCriticalLocusOfGaugeInvariantFunctionOnLieAlgebroid}
###### Definition
**([[derived critical locus]] of [[invariant]] function on [[Lie ∞-algebroid]])**

Let $\mathfrak{a}$ be a [[Lie ∞-algebroid]] (def. \ref{LInfinityAlgebroid}) over some $X$, let

$$
  S \;\colon\; \mathfrak{a} \longrightarrow \mathbb{R}
$$

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}:

$$
  \array{
    \mathfrak{a} && \overset{d S}{\longrightarrow} && T^\ast_{inf} \mathfrak{a}
    \\
    & {}_{\mathllap{id}}\searrow && \swarrow_{\mathrlap{cb}}
    \\
    && \mathfrak{a}
  }
$$

Then the _[[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]]:

$$
  \array{
    \mathfrak{a}_{d S \simeq 0}
      &\longrightarrow&
    \mathfrak{a}
    \\
    \downarrow &(pb)& \downarrow^{\mathrlap{0}}
    \\
    \mathfrak{a}
      &\underset{d S}{\longrightarrow}&
    T^\ast_{inf} \mathfrak{a}
  }
  \,.
$$

This means equivalently (details are at _[[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:

1. 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 $T^\ast_{inf}[-1]\mathfrak{a}$;

1. 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 $d S(v) = v(S)$.

=--

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:

+-- {: .num_example #ArchetypeOfBVBRSTComplex}
###### Example
**([[derived critical locus]] inside [[action Lie algebroid]])**

Consider an [[invariant]] function (def. \ref{GaugeInvariantFunctionsIntermsOfLieAlgebroids})
on an [[action Lie algebroid]] (def. \ref{ActionLieAlgebroid})

$$
  S \;\colon\; X/\mathfrak{g} \overset{\phantom{AAA}}{\longrightarrow} \mathbb{R}
$$

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})

$$
  (X/\mathfrak{g})_{d S \simeq 0}
$$

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 (eq:GeneratorsOfDerivedCriticalLocusInActionLieAlgebroid), except for a shift of degree of the [[derivation]]-generators down by one:

$$
  \array{
    &
    \left(
      c^\ddagger_{\alpha}
    \right)
    &
    \left(
      \phi^\ddagger_a
    \right)
    &
    \left(
      \phi^a
    \right)
    &
    \left(
      c^\alpha
    \right)
    \\
    deg = &
    -2
    &
    -1
    &
    0
    &
    +1
  }
$$

Rephrased in terms of [[graded manifold]] (remark \ref{dgManifolds}) this means that the [[graded manifold]] underlying the derived critical locus is the _shifted infinitesimal cotangent bundle_ of the graded manifold $\mathfrak{g}[1] \times X$ (eq:ActionLieAlgebroidGradedManifold) which underlies the [[action Lie algebroid]] (def. \ref{ActionLieAlgebroid}):

$$
  \label{ShiftedCotangentBundleForCriticalLocusInsideLieAlgebroid}
  (X/\mathfrak{g})_{d S \simeq 0}
    \;=_{grmfd}\;
  T^\ast_{inf}[-1]\left( \mathfrak{g}[1] \times X \right)
$$

and if $X = \mathbb{R}^{b\vert s}$ is a [[super Cartesian space]] this becomes more specifically

$$
  \begin{aligned}
    (\mathbb{R}^{p \vert q}/\mathfrak{g})_{d S \simeq 0}
      & =_{grmfd}
    T^\ast_{inf}[-1]\left( \mathfrak{g}[1] \times \mathbb{R}^{p \vert q} \right)
    \\
    & =_{\phantom{grmfd}}
    \underset{
       (c^\alpha)
    }{
    \underbrace{
     \mathfrak{g}[1]
    }}
      \times
    \underset{ (\phi^a) }{
    \underbrace{
      \mathbb{R}^{p\vert q}
    }}
      \times
    \underset{ (\phi^\ddagger_a) }{
    \underbrace{
      (\mathbb{R}^{p \vert q})^\ast_{inf}[-1]
    }}
      \times
    \underset{ (c^\ddagger_\alpha) }{
    \underbrace{
      \mathfrak{g}^\ast[-2]
    }}
  \end{aligned}
$$

Moreover, on these generators the CE-differential is given by

$$
  \label{ExplicitCEDifferentialInCotangentBundleOfActionLieAlgebroid}
  \array{
    & d_{CE\left((X/\mathfrak{g})_{d S \simeq 0}\right)}
    \\
    \phi^a
      &\mapsto&
    c^\alpha R^a_\alpha
    \\
    c^\alpha
      & \mapsto&
    \tfrac{1}{2} \gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
    \\
    \phi^\ddagger_a
      &\mapsto&
    \underset{
      new
    }{
    \underbrace{
      \frac{\partial S}{\partial \phi^a}
    }}
    -
    c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \phi^\ddagger_b
    \\
    c^\ddagger_\alpha
      &\mapsto&
    R_\alpha^a  \phi^\ddagger_a
      +
    \gamma^\beta{}_{\alpha \gamma} c^\gamma c^\ddagger_b
  }
$$

which is just the expression for the differential (eq: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.

=--

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]] $X_{d S = 0}$:

+-- {: .num_example #OrdinaryCriticalLocusAsCohomologyOfDerivedCriticalLocus}
###### Example
**(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

$$
  \begin{aligned}
    &
    &
    \left(
      \phi^\ddagger_a
    \right)
    &
    \left(
      \phi^a
    \right)
    &
    \\
    deg = &
    &
    -1
    &
    0
    &
  \end{aligned}
$$

and [[differential]] given by

$$
  \array{
     & d_{CE\left( X_{d S \simeq 0} \right)}
     \\
     \phi^a &\mapsto& 0
     \\
     \phi^\ddagger_a &\mapsto& \frac{\partial S}{\partial \phi^a}
  }
  \,.
$$

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 $\left( \frac{\partial S}{\partial \phi^a} \right)$

$$
  H^0\left(
     CE\left( X_{d S \simeq 0}  \right)
  \right)
  \;=\;
  C^\infty(X)/\left( \frac{\partial S}{\partial \phi^a} \right)
  \,.
$$

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]]:

$$
  H^0\left(
   CE\left( X_{d S \simeq 0} \right)
  \right)
  \;\simeq\;
  C^\infty\left( X_{d S = 0} \right)
  \,.
$$

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 _[Gauge fixing](#GaugeFixing)_, see prop. \ref{BVComplexIsHomologicalResolutionPreciselyIfNoNonTrivialImplicitGaugeSymmetres} below.

=--

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]]:

+-- {: .num_prop #DerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure}
###### Proposition
**([[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]] (eq:ExplicitCEDifferentialInCotangentBundleOfActionLieAlgebroid) may be decomposed as the sum of two anti-commuting differential

$$
  d_{CE\left( (X/\mathfrak{g})_{d S \simeq 0}  \right)}
  \;=\;
  s_{BRST} +  s_{BS}
$$

which are defined on the generators of the [[Chevalley-Eilenberg algebra]] as follows:

$$
  \label{ExplicitBVandBRSTDifferentialInCotangentBundleOfActionLieAlgebroid}
  \array{
    & s_{BV}
    \\
    \phi^a
      &\mapsto&
    0
    \\
    c^\alpha
      & \mapsto&
    0
    \\
    \phi^\ddagger_a
      &\mapsto&
     \frac{\partial S}{\partial \phi^a}
    \\
    c^\ddagger_\alpha
      &\mapsto&
    R_\alpha^a  \phi^\ddagger_a
    \\
    \phantom{A}
    \\
    & s_{BRST}
    \\
    \phi^a
      &\mapsto&
    c^\alpha R^a_\alpha
    \\
    c^\alpha
      & \mapsto&
    \tfrac{1}{2} \gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
    \\
    \phi^\ddagger_a
      &\mapsto&
    -
    c^\alpha \frac{\partial R_\alpha^b}{\partial \phi^a} \phi^\ddagger_b
    \\
    c^\ddagger_\alpha
      &\mapsto&
    \gamma^\beta{}_{\alpha \gamma} c^\gamma c^\ddagger_b
  }
$$

If we moreover decompose the degree of the generators into two degrees

$$
  \array{
    &
    \left(
      c^\ddagger_{\alpha}
    \right)
    &
    \left(
      \phi^\ddagger_a
    \right)
    &
    \left(
      \phi^a
    \right)
    &
    \left(
      c^\alpha
    \right)
    \\
    deg_{gh} =
    &
    0
    &
    0
    &
    0
    &
    +1
    \\
    deg_{af} =
    &
    -2
    &
    -1
    &
    0
    &
    0
  }
$$

then these two differentials constitute a [[bicomplex]]

$$
  \array{
     CE^{0,0}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
       &\overset{s_{BRST}}{\longrightarrow}&
     CE^{1,0}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
       &\overset{s_{BRST}}{\longrightarrow}&
     CE^{2,0}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
       &\overset{s_{BRST}}{\longrightarrow}&
     \cdots
     \\
     \big\uparrow^{\mathrlap{s_{BV}}}
       &&
     \big\uparrow^{\mathrlap{s_{BV}}}
       &&
     \big\uparrow^{\mathrlap{s_{BV}}}
       &&
     \\
     CE^{0,-1}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
       &\overset{s_{BRST}}{\longrightarrow}&
     CE^{1,-1}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
       &\overset{s_{BRST}}{\longrightarrow}&
     CE^{2,-1}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
       &\overset{s_{BRST}}{\longrightarrow}&
     \cdots
     \\
     \big\uparrow^{\mathrlap{s_{BV}}}
       &&
     \big\uparrow^{\mathrlap{s_{BV}}}
       &&
     \big\uparrow^{\mathrlap{s_{BV}}}
       &&
     \\
     CE^{0,-2}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
       &\overset{s_{BRST}}{\longrightarrow}&
     CE^{1,-2}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
       &\overset{s_{BRST}}{\longrightarrow}&
     CE^{2,-2}\left( (X/\mathfrak{g})_{d S \simeq 0}\right)
       &\overset{s_{BRST}}{\longrightarrow}&
     \cdots
     \\
     \big\uparrow^{\mathrlap{s_{BV}}}
       &&
     \big\uparrow^{\mathrlap{s_{BV}}}
       &&
     \big\uparrow^{\mathrlap{s_{BV}}}
       &&
     \\
     \vdots && \vdots && \vdots
  }
$$

whose [[total complex]] is the [[Chevalley-Eilenberg dg-algebra]] of the derived critical locus

$$
  \begin{aligned}
    CE\left(
      (X/\mathfrak{g})_{d S \simeq 0}
    \right)
    & =
    \underset{ gh, af }{\bigoplus}
    CE^{gh,af}\left(
       (X/\mathfrak{g})_{d S \simeq 0}
    \right)
    \\
    d_CE\left( (X/\mathfrak{g})_{d S \simeq 0}  \right){}
    & =
    s_{BV} + s_{BRST}
  \end{aligned}
  \,.
$$

=--

+-- {: .proof}
###### Proof

It is clear from the definition that the graded [[derivations]] $s_{BV}$ and $s_{BRST}$ have (i.e. increase) bidegree as follows:

$$
  \array{
     & s_{BRST} & s_{BV}
    \\
    deg_{gh} = & +1 & 0
    \\
    deg_{af} = & 0 & +1
  }
  \,.
$$

This implies that in

$$
  \begin{aligned}
    0 & =
    \left( d_{CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)} \right)^2
    \\
    & =
    \left( s_{BV} +  s_{BRST}\right)^2
    \\
    & =
    \underset{ = 0 }{
    \underbrace{
      \left( s_{BV}\right)^2
    }}
    +
    \underset{ = 0 }{
    \underbrace{
      \left( s_{BRST} \right)^2
    }}
    +
    \underset{ = 0 }{
    \underbrace{
      \left[ s_{BV}, s_{BRST} \right]
    }
    }
  \end{aligned}
$$

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}):

$$
  \begin{aligned}
    s_{BV} \left( s_{BV} \left( c^\ddagger_\alpha \right) \right)
    & =
    s_BV\left( R_\alpha^a \phi^\ddagger_a \right)
    \\
    & =
    R_\alpha^a \frac{\partial S}{\partial \phi^a}
    \\
    & = 0
  \end{aligned}
$$

=--

As a corollary of prop. \refDerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure{} we obtain the generalization of example \ref{OrdinaryCriticalLocusAsCohomologyOfDerivedCriticalLocus} to non-trivial $\mathfrak{g}$-actions:

+-- {: .num_prop #CochainCohomologyOfBVBRSTComplexInDegreeZero}
###### Proposition
**([[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})

$$
  \array{
    CE^{\bullet, \bullet+1}\left( (X/\mathfrak{g})_{d S \simeq 0} \right)
    \\
    \uparrow^{\mathrlap{s_{BV}}}
    \\
    CE^{\bullet, \bullet}\left( (X/\mathfrak{g})_{d S \simeq 0} \right)
  }
$$

has vanishing [[cochain cohomology]] in [[negative number|negative]] $af$-degree

$$
  \label{VanishingOfNaiveLieAlgebroidBVCohomlogyInNegativeDegree}
  H^{\bullet \leq 1}(s_{BV}) = 0
$$

then the [[cochain cohomology]] of the full [[Chevalley-Eilenberg dg-algebra]] is given by the cochain cohomology of $s_{BRST}$ on $H^0(s_{BV})$:

$$
  H^k\left(
    CE\left(
      (X/\mathfrak{g})_{d S \simeq 0}
    \right)
  \right)
  \;\simeq\;
  H^k\left(  H^0(s_{BV}), s_{BRST} \right)
  \,.
$$

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]]:

$$
  H^0
  \left(
    CE\left(
      (X/\mathfrak{g})_{d S \simeq 0}
    \right)
  \right)
  \;=\;
  \left\{
    X_{d S = 0}
      \overset{f}{\to}
    \mathbb{R}
     \;\vert\;
     \left(R_\alpha^a \frac{\partial f}{\partial \phi^a} = 0\right)
  \right\}
$$

=--

+-- {: .proof}
###### Proof

The first statement follows from the [[spectral sequence]] [[spectral sequence of a double complex|of the double complex]]

$$
  H^{gh}
  \left(
    H^{af}
    \left(
      CE\left(  (X/\mathfrak{g})_{d S \simeq 0} \right)
    \right)
  \right)
  \;\Rightarrow\;
  H^{gh + af}\left(
      CE\left(  (X/\mathfrak{g})_{d S \simeq 0} \right)
  \right)
  \,.
$$

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 [extension problem](spectral+sequence#ExtensionProblem) 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

$$
  \begin{aligned}
    H^0\left( CE\left( (X/\mathfrak{g})_{d S \simeq 0}\right) \right)
    & =
    ker(s_{BRST})\vert_{C^\infty\left( X_{d S = 0} \right) }
    \,/\,
    \underset{= 0}{
    \underbrace{
      im(s_{BRST})\vert_{C^\infty\left(  X_{d S = 0} \right)}
    }
    }
    \\
    & = ker(s_{BRST})\vert_{C^\infty\left( X_{d S = 0} \right) }
    \\
    & =
    \left\{
       X_{d S = 0} \overset{f}{\longrightarrow} \mathbb{R}
       \,\vert\,
       \left(
         R_\alpha^a \frac{\partial S}{\partial \phi^a}
         =
         0
       \right)
    \right\}
  \end{aligned}
$$

=--

This means that under condition (eq:VanishingOfNaiveLieAlgebroidBVCohomlogyInNegativeDegree) the construction of a [[derived critical locus]] inside an [[action Lie algebroid]] provides a [[resolution]] of the space of those functions which are

1. _[[restriction|restricted]]_ to the [[critical locus]] (a [[homotopy intersection]]);

1. _[[invariant]]_ under the [[Lie algebra action]] (a [[homotopy quotient]]).

We apply this general mechanism [below](#DerivedCriticalLocusOnJetBundle) to [[Lagrangian field theory]], where it serves to provide a [[resolution]] by the _[[BV-BRST complex]]_ of the space of [[observables]] which are

1. [[on-shell]],

1. _[[gauge invariance|gauge invariant]]_.

But in order to control this application, we first establish the tool of the _[[Schouten bracket]]/[[antibracket]]_.

$\,$

**[[Schouten bracket]]/[[antibracket]]**
 {#SchoutenBracketAntibracket}

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:

+-- {: .num_defn #SchoutenBracketAndAntibracket}
###### Definition
**([[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 $X/\mathfrak{g}$

$$
  [-,-]
  \;\colon\;
   Der(CE(X/\mathfrak{g}))
    \otimes
   Der(CE(X/\mathfrak{g}))
    \longrightarrow
   Der(CE(X/\mathfrak{g}))
$$

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]] $(X/\mathfrak{g})_{d S \simeq 0}$

$$
  \left\{ -,-\right\}
    \;\colon\;
   CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)
    \otimes
   C\left( (X/\mathfrak{g})_{d S \simeq 0} \right)
    \longrightarrow
   CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)
$$

such that this is a graded [[derivation]] in both arguments.

This is called the _[[Schouten bracket]]_.

There is an elegant way to rewrite this in terms of components: With the notation (eq:AntiNotationForDerivations) for the coordinate-derivations the [[Schouten bracket]] is equivalently given by

$$
  \label{Antibracket}
  \begin{aligned}
    \left\{ f,g \right\}
    & =
    \phantom{+}
    \frac{\overset{\leftarrow}{\partial} f}{\partial \phi^\ddagger_a}
    \frac{\overset{\rightarrow}{\partial} g}{\partial {\phi}^a}
    -
    \frac{\overset{\leftarrow}{\partial} f}{\partial \phi^a}
    \frac{\overset{\rightarrow}{\partial} g}{\partial \phi^\ddagger_a}
    \\
    &
    \phantom{=}
    +
    \frac{\overset{\leftarrow}{\partial} f}{\partial c^\ddagger_\alpha}
    \frac{\overset{\rightarrow}{\partial} g}{\partial {c}^{\alpha}}
    -
    \frac{\overset{\leftarrow}{\partial} f}{\partial c^{\alpha}}
    \frac{\overset{\rightarrow}{\partial} g}{\partial c^\ddagger_\alpha}
  \end{aligned}
  \,,
$$

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 _[[antibracket]]_.

=--

(e. g. [Henneaux 90, (53d)](antibracket#Henneaux90), [Henneaux-Teitelboim 92, section 15.5.2](antibracket#HenneauxTeitelboim92))

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 $X/\mathfrak{g}$:

+-- {: .num_example #ChevalleyEilenbergDifferentialOnDerivedCriticalLocusIsHamiltonianViaAntibracket}
###### Example
**([[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 (eq: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]]

$$
  \label{BVBRSTFunctionForActionLieAlgebroid}
  S_{\text{BV-BRST}}
  \;\coloneqq\;
  S - d_{CE(X/\mathfrak{g})}
$$

of the [[Chevalley-Eilenberg differential]] of $X/\mathfrak{g}$ and the function $-S$:

$$
  d_{CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right) }(-)
  \;=\;
  \left\{
    - S + d_{CE(X/\mathfrak{g})}
    \,,\,
    (-)
  \right\}
  \,.
$$

In coordinates, using the expression for $d_{CE(X/\mathfrak{g})}$ from (eq:DifferentialOnActionLieAlgebroid) and using the notation for derivations from (eq:AntiNotationForDerivations) this means that

$$
  d_{CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)}(-)
  \;=\;
  \left\{
     - S
     +
     c^\alpha R_\alpha^a \phi^\ddagger_a
      -
     \tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
     c^\ddagger_\alpha
     \,,\,
     (-)
  \right\}
  \,.
$$

=--

+-- {: .proof}
###### 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 (eq:ExplicitCEDifferentialInCotangentBundleOfActionLieAlgebroid):

for $\phi^a$:

$$
  \begin{aligned}
  \left\{
     - S
     +
     c^\alpha R_\alpha^{a'} \phi^\ddagger_{a'}
      -
     \tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
     c^\ddagger_\alpha
     \,,\,
     \phi^a
  \right\}
  &
  =
  \left\{
   c^\alpha R_\alpha^{a'} \phi^\ddagger_{a'} \,,\, \phi^a
  \right\}
  \\
  & =
  c^\alpha R_\alpha^{a'}
  \underset{
    \delta_{a'}^a
  }{
  \underbrace{
  \left\{
   \phi^\ddagger_{a'} \,,\, \phi^a
  \right\}
  }
  }
  \\
  & =
  c^\alpha R_\alpha^{a}
  \end{aligned}
$$

for $c^\alpha$:

$$
  \begin{aligned}
  \left\{
     - S
     +
     c^\alpha R_\alpha^{a} \phi^\ddagger_{a}
      -
     \tfrac{1}{2}\gamma^{\alpha'}{}_{\beta \gamma} c^\beta c^\gamma
     c^\ddagger_{\alpha'}
     \,,\,
     c^\alpha
  \right\}
  & =
  \left\{
     \tfrac{1}{2}\gamma^{\alpha'}{}_{\beta \gamma} c^\beta c^\gamma
     c^\ddagger_{\alpha'}
     \,,\,
     c^\alpha
  \right\}
  \\
  & =
  \tfrac{1}{2}\gamma^{\alpha'}{}_{\beta \gamma} c^\beta c^\gamma
  \underset{
    \delta_{\alpha'}^\alpha
  }{
  \underbrace{
  \left\{
     c^\ddagger_{\alpha'}
     \,,\,
     c^\alpha
  \right\}
  }
  }
  \\
  & =
  \tfrac{1}{2}\gamma^{\alpha}{}_{\beta \gamma} c^\beta c^\gamma
  \end{aligned}
$$

for $\phi^\ddagger_a$:

$$
  \begin{aligned}
  \left\{
     - S
     +
     c^\alpha R_\alpha^{a'} \phi^\ddagger_{a'}
      -
     \tfrac{1}{2}\gamma^{\alpha}{}_{\beta \gamma} c^\beta c^\gamma
     c^\ddagger_{\alpha}
     \,,\,
     \phi^\ddagger_a
  \right\}
  & =
  -
  \underset{
   = -\frac{\partial S}{\partial \phi^a}
  }{
  \underbrace{
  \left\{
    S \,,\, \phi^{\ddagger}_a
  \right\}
  }
  }
  +
  \left\{
    c^\alpha R_\alpha^{a'} \phi^\ddagger_{a'}
    \,,\,
    \phi^\ddagger_a
  \right\}
  \\
  & =
  \frac{\partial S}{\partial \phi^a}
   + c^\alpha
    \underset{
      = -\frac{\partial R_\alpha^{a'}}{\partial \phi^a}
    }{
    \underbrace{
      \left\{ R_\alpha^{a'} \,,\, \phi^\ddagger_a \right\}
    }
    }
   \phi^\ddagger_{a'}
  \\
  & =
   \frac{\partial S}{\partial \phi^a}
   -
   c^\alpha \frac{\partial R_\alpha^{a'}}{\partial \phi^a}
   \phi^\ddagger_{a'}
  \end{aligned}
$$

for $c^\ddagger_\alpha$:

$$
  \begin{aligned}
  \left\{
     - S
     +
     c^{\alpha'} R_{\alpha'}^{a} \phi^\ddagger_{a}
      -
     \tfrac{1}{2}\gamma^{\alpha'}{}_{\beta \gamma} c^\beta c^\gamma
     c^\ddagger_{\alpha'}
     \,,\,
     c^\ddagger_\alpha
  \right\}
  & =
  \left\{
    c^{\alpha'} R_{\alpha'}^a \phi^{\ddagger}_a
    \,,\,
    c^\ddagger_{\alpha}
  \right\}
  \;+\;
  \left\{
    \tfrac{1}{2} \gamma^{\alpha'}{}_{\beta \gamma} c^\beta c^\gamma
    c^\ddagger_{\alpha'}
    \,,\,
    c^\ddagger_\alpha
  \right\}
  \\
  & =
  \left\{
    c^{\alpha'}
    \,,\,
    c^\ddagger_{\alpha}
  \right\}
  R_{\alpha'}^a \phi^{\ddagger}_a
  \;+\;
  \tfrac{1}{2} \gamma^{\alpha'}{}_{\beta \gamma}
  \underset{ = - c^\beta \delta_{\alpha}^\gamma + \delta_{\alpha}^\beta c^\gamma}{
  \underbrace{
  \left\{
    c^\beta c^\gamma
    \,,\,
    c^\ddagger_\alpha
  \right\}
  }}
  c^\ddagger_{\alpha'}
  \\
  & =
  R_\alpha^a \phi^\ddagger_{a}
  +
  \gamma^{\alpha'}{}_{\alpha \gamma} c^\gamma c^\ddagger_{\alpha'}
  \end{aligned}
$$

Hence these values of the [[Schouten bracket]]/[[antibracket]] indeed all agree with the values of the CE-differential from (eq:ExplicitCEDifferentialInCotangentBundleOfActionLieAlgebroid).

=--

As a corollary we obtain:

+-- {: .num_prop #ClassicalMasterEquation}
###### Proposition
**([[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_{\text{BV-BRST}}

$$
  S_{\text{BV-BRST}}
  \;\coloneqq\;
  S - d_{CE\left( X/\mathfrak{g}\right)}
$$

with itself vanishes:

$$
  \left\{
    S_{\text{BV-BRST}}
    \,,\,
    S_{\text{BV-BRST}}
  \right\}
  \;=\;
  0
  \,.
$$

Conversely, given a shifted [[cotangent bundle]] of the form $T^\ast[-1](X \times \mathfrak{g}[1])$ (eq: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

$$
  \left\{
    S, S
  \right\}
  \;=\;
  0
  \,.
$$

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 _[[master equation]]_ and here specifically the
_[[classical master equation]]_.

=--

$\,$

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.

$\,$

**[[local BV-BRST complex|local]] [[antibracket]]**
 {#LocalJetBundleAntibracket}

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
_[[BV-BRST complex]]_ of the theory.

In applications of interest, the spacetime $\Sigma$ is _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 ([Fredenhagen-Rejzner 11a](BV-BRST+formalism#FredenhagenRejzner11a)).

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 _[[local BV-BRST complex]]_ (prop. \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm} below), corresponding to the [[local BRST complex]] from example \ref{LocalOffShellBRSTComplex} ([Barnich-Brandt-Henneaux 00](local+BRST+cohomology#BarnichBrandtHenneaux00)).

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 _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 _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 _[[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}):

+-- {: .num_defn #InfinitesimalCotangentBundleOfFieldAndGaugeParameterBundle}
###### Definition
**([[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]]

$$
  E / ( \mathcal{G} \times_\Sigma T \Sigma )
  \;=\;
  \left(
    J^\infty_\Sigma( E \times_\Sigma (\mathcal{G}[1]) )
    ,
    s_{BRST} )
  \right)
$$

whose [[Chevalley-Eilenberg algebra]] is the _[[local BRST complex]]_ of the field theory (example \ref{LocalOffShellBRSTComplex}).

Then we write

$$
  T^\ast_{\Sigma,inf}\left(
    E \times_\Sigma (\mathcal{G}[1])
  \right)
  \,,
  \phantom{AAA}
  T^\ast_{\Sigma,inf}[-1]\left(
    E \times_\Sigma (\mathcal{G}[1])
  \right)
$$

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 (eq: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

$$
  \label{coordslocalOnInfinitesimalCotangentOfFieldBundleTimesGaugeParameterBundle}
  \array{
      T^\ast_{\Sigma,inf}\left(
    E \times_\Sigma (\mathcal{G}[1])
  \right)
  & \phantom{AAAAA}&
  T^\ast_{\Sigma,inf}[-1]\left(
    E \times_\Sigma (\mathcal{G}[1])
  \right)
  \\
  & \phantom{A}
  \\
  \array{
    & (c^\ddagger_\alpha), & (\phi^\ddagger_a),(\phi^a), &  (c^\alpha)
    \\
    deg = & -1 & 0 & 1
  }
  & \phantom{AA}&
  \array{
    & (c^\ddagger_\alpha), & (\phi^\ddagger_a)\, & (\phi^a), &  (c^\alpha)
    \\
    deg = & -2 & -1 &  0 & 1
  }
  }
$$

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 _[[antifields]]_:

* $\phi^\ddagger_a$ is _[[antifield]]_ to the [[field (physics)|field]] $\phi^a$

* $c^\ddagger_\alpha$ is _[[antifield]]_ to the [[ghost field]] $c^\alpha$.

=--

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}):

+-- {: .num_defn #LocalAntibracket}
###### Definition
**([[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]]

$$
  E / ( \mathcal{G} \times_\Sigma T \Sigma )
  \;=\;
  \left(
    J^\infty_\Sigma( E \times_\Sigma (\mathcal{G}[1]) )
    ,
    s_{BRST} )
  \right)
$$

whose [[Chevalley-Eilenberg algebra]] is the _[[local BRST complex]]_ of the field theory with shifted infinitesimal [[vertical cotangent bundle]]

$$
  \label{BVBRSTGradedFieldBundle}
  E_{\text{BV-BRST}}
  \;\coloneqq\;
  T^\ast_{\Sigma,inf}[-1]\left(
    E \times_\Sigma (\mathcal{G}[1])
  \right)
$$

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 (eq:LagrangianFunctionViaVolumeForm)

$$
  H
    \;=\;
  h \, dvol_\Sigma
  \;\in\;
  \Omega^{p+1}_\Sigma\left( \,T^\ast_{\Sigma,inf}[-1]\left( E \times_\Sigma (\mathcal{G}[1]) \right) \, \right)
$$

the _[[local antibrackets]]_

$$
  \{-,-\}'
  ,
  \{-,-\}
  \;\colon\;
  \Omega^{p+1,0}_\Sigma( \, T^\ast_{\Sigma,inf}[-1](E \times_\Sigma \mathcal{G}[1]) \, )
    \,\otimes\,
  \Omega^{p+1,0}_\Sigma( \, T^\ast_{\Sigma,inf}[-1](E \times_\Sigma \mathcal{G}[1]) \, )
    \longrightarrow
  \Omega^{p+1,0}_\Sigma( \, T^\ast_{\Sigma,inf}[-1](E \times_\Sigma \mathcal{G}[1]) \, )
$$

are the functions which are given in the [[local coordinates]] (eq:coordslocalOnInfinitesimalCotangentOfFieldBundleTimesGaugeParameterBundle) as follows:

The first version is

$$
  \begin{aligned}
    \left\{ f\, dvol_\Sigma \,,\,g \, dvol_\Sigma \right\}'
    & \coloneqq
    \phantom{+}
    \left(
    \frac{\overset{\leftarrow}{\delta}_{EL} f }{\delta \phi^\ddagger_a}
    \frac{\overset{\rightarrow}{\delta}_{EL} g}{\delta {\phi^a}^{\phantom{\ddagger}}}
    -
    \frac{\overset{\leftarrow}{\delta}_{EL}}{\delta {\phi^a}^{\phantom{\ddagger}}}
    \frac{\overset{\rightarrow}{\delta}_{EL} g}{\delta \phi^\ddagger_a}
    \right)
    dvol_\Sigma
    \\
    &
    \phantom{=} +
    \left(
    \frac{\overset{\leftarrow}{\delta}_{EL} f}{\delta c^\ddagger_\alpha}
    \frac{\overset{\rightarrow}{\delta}_{EL} g}{\delta {c^\alpha}^{\phantom{\ddagger}}}
    -
    \frac{\overset{\leftarrow}{\delta}_{EL}}{\delta {c^\alpha}^{\phantom{\ddagger}}}
    \frac{\overset{\rightarrow}{\delta}_{EL} g}{\delta c^\ddagger_\alpha}
    \right)
    dvol_\Sigma
    \,.
  \end{aligned}
$$

This is of the form of the [[Schouten bracket]] (eq:Antibracket) but with [[Euler-Lagrange derivatives]] (eq:EulerLagrangeEquationGeneral) instead of [[partial derivatives]],

The second version is this:

$$
  \label{LocalCommutatorOfDerivationsOnJetBundle}
  \begin{aligned}
    \left\{
      f \, dvol_\Sigma,
      g \, dvol_\Sigma
    \right\}
    & \coloneqq
    \phantom{+}
    \left(
    \left(
      \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
      \left(
        \frac{\overset{\leftarrow}{\delta}_{EL} f}{\delta \phi^a}
      \right)
    \right)
    \left(
      \frac{\overset{\rightarrow}{\partial} g}{\partial {\phi}^\ddagger_{a,\mu_1 \cdots \mu_k}}
    \right)
    -
    \left(
      \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
      \left(
        \frac{\overset{\leftarrow}{\delta}_{EL} f}{\delta \phi^\ddagger_a}
      \right)
    \right)
    \left(
      \frac{\overset{\rightarrow}{\partial} g}{\partial \phi^a_{,\mu_1 \cdots \mu_k}}
    \right)
    \right) \, dvol_\Sigma
    \\
    &
    \phantom{\coloneqq}
    +
    \left(
    \left(
      \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
      \left(
        \frac{\overset{\leftarrow}{\delta}_{EL} f}{\delta c^\alpha}
      \right)
    \right)
    \left(
      \frac{\overset{\rightarrow}{\partial} g}{\partial {c}^\ddagger_{\alpha,\mu_1 \cdots \mu_k}}
    \right)
    -
    \left(
      \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
      \left(
        \frac{\overset{\leftarrow}{\delta}_{EL} f}{\delta c^\ddagger_\alpha}
      \right)
    \right)
    \left(
      \frac{\overset{\rightarrow}{\partial} g}{\partial c^\alpha_{,\mu_1 \cdots \mu_k}}
    \right)
    \right) \, dvol_\Sigma
  \end{aligned}
$$

where again $\frac{\delta_{EL}}{\delta \phi^a}$ denotes the [[Euler-Lagrange variational derivative]] (eq:EulerLagrangeEquationGeneral)

=--

([Barnich-Henneaux 96 (2.9) and (2.12)](local+BRST+cohomology#BarnichHenneaux96), reviewed in [Barnich 10 (4.9)](BRST+complex#Barnich10))

+-- {: .num_prop #BasicPropertiesOfTheLocalAntibracket}
###### Proposition
**(basic properties of the [[local antibracket]])**

The [[local antibracket]] from def. \ref{LocalAntibracket} satisfies the following properties:

1. The two versions differ by a [[total spacetime derivative]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}):

   $$
     \{f,g\}
     =
     \{f,g\}' + d(...)
     \,.
   $$

1. The primed version is strictly graded skew-symmetric:

   $$
     \left\{f \, dvol_\Sigma \,,\, g\, dvol_\Sigma \right\}'
     \;=\;
     - (-1)^{deg(f) deg(g)} \, \left\{g \, dvol_\Sigma \,,\, f\, dvol_\Sigma \right\}
   $$

1. 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:

   $$
     \label{LocalAntibracketGradedDerivationInSecondArgument}
     \left\{ f\, dvol_\Sigma, \left\{ g\, dvol_\Sigma \,,\, h\, dvol_\Sigma \right\}\right\}
     \;=\;
       \underset{
         =
         \left\{
           \left\{ f\, dvol_\Sigma \,,\, g\, dvol_\Sigma  \right\}'
           \,,
           h\, dvol_\Sigma
         \right\}
        }{
        \underbrace{
          \left\{
          \left\{
             f\, dvol_\Sigma
             \,,\,
             g\, dvol_\Sigma
          \right\}
          \,,\,
          h\, dvol_\Sigma
          \right\}
        }
     }
     \;+\;
     (-1)^{(deg(f)+1) deg(g)}
     \left\{
       g\, dvol_\Sigma
       \,,\,
       \left\{
         f\, dvol_\Sigma
         \,,\,
         h\, dvol_\Sigma
       \right\}
     \right\}
   $$

   and the first term on the right is equivalently given by the primed bracket, as shown under the brace;

1. 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

   $$
     \{ f dvol_\Sigma, g \, dvol_\Sigma \}' = 0
     \phantom{AAA}
     \{ f dvol_\Sigma, g \, dvol_\Sigma \} = d(\cdots)
   $$

for all $f$, $g$ of homogeneous degree $deg(f)$ and $deg(g)$, respectively.

=--

([Barnich-Henneaux 96 (B.6) and footnote 9](local+BRST+cohomology#BarnichHenneaux96)).

+-- {: .proof}
###### Proof

That the two expressions differ by a horizontally exact terms follows by the very definition of the [[Euler-Lagrange derivative]] (eq:EulerLagrangeEquationGeneral). Also the graded skew symmetry of the primed bracket is manifest.

The third point requires some computation ([Barnich-Henneaux 96 (B.9)](local+BRST+cohomology#BarnichHenneaux96)).

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.

=--

The following is the local refinement of prop. \ref{ClassicalMasterEquation}:

+-- {: .num_defn #ClassicalMasterEquationLocal}
###### Remark
**(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

$$
  \mathbf{K} = K\, dvol_\Sigma
  \,,
  \phantom{AAA}
  deg(L) \in 2 \mathbb{Z}
$$

are equivalent:

1. forming the [[local antibracket]] (def.  \ref{LocalAntibracket}) with $\mathbf{K}$ is a [[differential]]

   $$
     \left(\left\{ \mathbf{K},-\right\}\right)^2 = 0
     \,,
   $$

1. the [[local antibracket]] (def. \ref{LocalAntibracket}) of $\mathbf{K}$ with itself is a [[total spacetime derivative]]:

   $$
     \left\{ \mathbf{K}, \mathbf{K}\right\} = d(...)
   $$

1. the other variant of the [[local antibracket]] (def. \ref{LocalAntibracket}) of $\mathbf{K}$ with itself is a [[total spacetime derivative]]:

   $$
     \left\{ \mathbf{K}, \mathbf{K}\right\}' = d(...)
   $$

This condition is also called the _local [[classical master equation]]_.

=--

$\,$

**[[derived critical locus]] on [[jet bundle]] -- the [[local BV-BRST complex]]**
  {#DerivedCriticalLocusOnJetBundle}

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]] (eq: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 _[[local BV-BRST complex]]_.
This is example \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm} below.

The following definition \ref{LocalInfinitesimalCotangentLieAlgebroid} is the local refinement of
def. \ref{LieAlgebroidInfinitesimalCotangentBundle}:

+-- {: .num_defn #LocalInfinitesimalCotangentLieAlgebroid}
###### Definition
**(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]]

$$
  E / ( \mathcal{G} \times_\Sigma T \Sigma )
  \;=\;
  \left(
    J^\infty_\Sigma( E \times_\Sigma (\mathcal{G}[1]) )
    ,
    s_{BRST} )
  \right)
$$

whose [[Chevalley-Eilenberg algebra]] is the _[[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 (eq:GeneratorsOfDerivedCriticalLocusInActionLieAlgebroid) as well as all the corresponding jets
and including also the
horizontal differentials:

$$
  \array{
    &
    \left(
      c^\ddagger_{\alpha,\mu_1 \cdots \mu_k}
    \right)
    &
      \left(
        \phi^a_{,\mu_1 \cdots \mu_k}
      \right)
      ,
      \left(
        \phi^\ddagger_{a,\mu_1 \cdots \mu_k}
      \right)
    &
    \left(
      c^\alpha_{,\mu_1 \cdots \mu_k}
    \right),
    \left(
      d x^\mu
    \right)
    \\
    deg =
    &
    -1
    &
    0
    &
    +1
  }
  \,.
$$

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

$$
  \label{BRSTFunctionForClosed}
  L_{BRST}
  \;=\;
  \left(
    \underset{k \in \mathbb{N}}{\sum}
    c^\alpha_{,\mu_1 \cdots \mu_k}
    R_\alpha^{a \mu_1 \cdots \mu_k}
  \right)
  \phi^\ddagger_a
  \;+\;
  \tfrac{1}{2}
  \gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
  c^\ddagger_\alpha
  \;\in\;
  C^\infty\left(
    T^\ast_{\Sigma,inf}( E \times_\Sigma \mathcal{G}[1] )
  \right)
  \,,
$$

to be called the _[[BRST complex|BRST]] [[Lagrangian function]]_ and the product with the [[spacetime]] [[volume form]]

$$
  L_{BRST} \, dvol_\Sigma
  \;\in\;
  \Omega^{p+1,0}_\Sigma(E \times_\Sigma \mathcal{G}[1])
$$

as the _[[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]] $\{-,-\}$ (eq:LocalCommutatorOfDerivationsOnJetBundle)
with the BRST Lagrangian density (eq:BRSTFunctionForClosed)

$$
  d_{CE(T^\ast_{\Sigma,inf}( E/(\mathcal{G} \times_\Sigma T \Sigma) ))}
  \;\coloneqq\;
  \left\{
    L_{BRST} dvol_\Sigma, -
  \right\}
$$

This defines an $L_\infty$-algebroid to be denoted

$$
  T^\ast_{\Sigma,inf}( E/(\mathcal{G} \times_\Sigma T \Sigma) )
  \,.
$$

=--

The local refinement of prop. \ref{ExteriorDifferentialOfGaugeInvariantFunctionIsSectionOfInfinitesimalCotangentLieAlgebroid} is now this:

+-- {: .num_prop #EulerLagrangeFormIsSectionOfLocalCotangentBundleOfJetBundleGaugeActionLieAlgebroid}
###### Proposition
**([[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})

$$
  \array{
    && T^\ast_{\Sigma,inf}\left( E/(\mathcal{G} \times_\Sigma T \Sigma) \right)
    \\
    & {}^{\mathllap{ \delta_{EL} \mathbf{L} }}\nearrow & \downarrow^{\mathrlap{cb}}
    \\
    E/(\mathcal{G} \times_\Sigma T \Sigma)
    &=&
    E/(\mathcal{G} \times_\Sigma T \Sigma)
  }
$$

given dually

$$
  CE(E/(\mathcal{G} \times_\Sigma T\Sigma))
    \overset{(\delta_{EL}\mathbf{L})^\ast}{\longleftarrow}
  CE(T^\ast_{inf}(E/(\mathcal{G}\times_\Sigma T \Sigma)))
$$

by

$$
  \array{
    \left\{
      \phi^a_{,\mu_1 \cdots \mu_k}
    \right\}
      &\longleftarrow&
    \left\{
      \phi^a_{,\mu_1 \cdots \mu_k}
    \right\}
    \\
    \left\{
      c^\alpha_{,\mu_1 \cdots \mu_k}
    \right\}
      &\longleftarrow&
    \left\{
      c^\alpha_{,\mu_1 \cdots \mu_k}
    \right\}
    \\
    \left\{
       \frac{d^k}{ d x^{\mu_1} \cdots d x^{\mu_k}}
       \left(
         \frac{\delta_{EL} L}{\delta \phi^a}
       \right)
    \right\}
      &\longleftarrow&
    \left\{
      \phi^\ddagger_{a,\mu_1 \cdots \mu_k}
    \right\}
    \\
    \left\{
      0
    \right\}
      &\longleftarrow&
    \left\{
      c^\ddagger_{\alpha,\mu_1 \cdots \mu_k}
    \right\}
  }
$$

=--

+-- {: .proof}
###### 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.

=--

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 (eq:ProlongedShellInJetBundle)):

+-- {: .num_defn #DerivedProlongedShell}
###### Definition
**(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 _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:

$$
  \array{
    (E/( \mathcal{G} \times_\Sigma T \Sigma ))_{\delta_{EL}L \simeq 0}
      &\longrightarrow&
    E/( \mathcal{G} \times_\Sigma T \Sigma )
    \\
    \downarrow
      &(pb)&
    \downarrow^{\mathrlap{0}}
    \\
    E/(\mathcal{G} \times_\Sigma T \Sigma)
     &\underset{\delta_{EL} L}{\longrightarrow}&
    T^\ast_{\Sigma,inf}
    \left(
      E/( \mathcal{G} \times_\Sigma T \Sigma )
    \right)
  }
$$

=--

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}:

+-- {: .num_example #LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm}
###### Example
**([[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

$$
  CE\left(
    (E/( \mathcal{G} \times_\Sigma T \Sigma ))_{\delta_{EL}L \simeq 0}
  \right)
  \;=\;
  \left(
    C^\infty\left(
      T^\ast_{\Sigma,inf}( E \times_\Sigma \mathcal{G}[1] \times_\Sigma T^\ast \Sigma[1] )
    \right)
    \,,\,
    \underset{
       = s
    }{
    \underbrace{
    \left\{
       \left(- L + L_{BRST}\right) dvol_\Sigma
       \,,
       (-)
    \right\}
    }
    }
    \;+\;
    d
  \right)
$$

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 _[[BV-BRST differential]]_

$$
  \label{LocalAntibracketVersionOfBVBRSTDifferential}
  s
  \;\coloneqq\;
    \left\{
       \left(- L + L_{BRST}\right) dvol_\Sigma
       \,,
       (-)
    \right\}
$$

which is the [[local antibracket]] (def. \ref{LocalAntibracket}) with the _[[BV-BRST Lagrangian density]]_

$$
  \left( -L + L_{BRST}\right)
  \;\in\;
  \Omega^{p+1,0}_\Sigma\left( T^\ast_{\Sigma,inf}[-1]\left( E \times_\Sigma \mathcal{G}[1] \right)\right)
$$

which itself is the sum of (minus) the given [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) with the BRST Lagrangian (eq:BRSTFunctionForClosed).

The action of the [[BV-BRST differential]] on the generators is as follows:

$$
  \array{
    & & \array{ \text{BV-BRST differential} \\ s } &
    \\
    \text{field}
    &
    \phi^a
      &\mapsto&
    \underset{ = s_{BRST}(\phi^a) }{
    \underbrace{
    \left(
      \underset{k \in \mathbb{N}}{\sum} c^\alpha_{,\mu_1 \cdots \mu_k} R_\alpha^{a \mu_1 \cdots \mu_k}
    \right)
    }
    }
    &
    \text{gauge symmetry}
    \\
    \text{
      ghost field
    }
    &
    c^\alpha
      &\mapsto&
    \underset{ = s_{BRST}(c^\alpha) }{
    \underbrace{
      \tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
    }
    }
    &
    \text{Lie bracket}
    \\
    \text{antifield}
    &
    \phi^\ddagger_\alpha
      &\mapsto&
    \phantom{-}
    \underset{ = s_{BV}(\phi^\ddagger_a) }{
    \underbrace{
      \frac{\delta_{EL} L}{\delta \phi^a}
    }}
    &
    \text{equations of motion}
    \\
    &&&
    \underset{ = s_{BRST}(\phi^\ddagger_a) }{
    \underbrace{
    -
    \left(
      \underset{k \in \mathbb{N}}{\sum}
      \frac{\delta_{EL}}{\delta \phi^a}
      \left(
        c^\alpha_{,\mu_1 \cdots \mu_k}
        R_\alpha^{b \mu_1 \cdots \mu_k}
        \phi^\ddagger_b
      \right)
    \right)
    }
    }
    &
    \\
    \array{
      \text{antifield of}
      \\
      \text{ghost field}
    }
    &
    c^\ddagger_\alpha
      &\mapsto&
    \underset{ = s_{BV}(c^\ddagger_\alpha) }{
    \underbrace{
    -
    \underset{k \in \mathbb{N}}{\sum}
      (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
      \left(
        R_\alpha^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a
      \right)
    }
    }
    &
    \text{Noether identities}
    \\
    &&& +
    \underset{ = s_{BRST}(c^\ddagger_\alpha) }{
    \underbrace{
      \gamma^{\alpha'}{}_{ \alpha \beta} c^\beta c^\ddagger_{\alpha'}
    }
    }
  }
$$

and this extends to jets of generator by $s \circ d + d \circ s = 0$.

This is called the _[[local BV-BRST complex]]_.

By introducing a bigrading as in prop. \ref{DerivedCriticalLocusOfActionLiAlgebroidBicomplexStructure}

$$
  \array{
    &
    \left(
      c^\ddagger_{\alpha, \mu_1 \cdots \mu_k}
    \right)
    &
    \left(
      \phi^\ddagger_{a, \mu_1 \cdots \mu_k}
    \right)
    &
    \left(
      \phi^a_{,\mu_1 \cdots \mu_k}
    \right)
    &
    \left(
      c^\alpha_{,\mu_1 \cdots \mu_k}
    \right)
    \\
    deg_{gh} =
    &
    0
    &
    0
    &
    0
    &
    +1
    \\
    deg_{af} =
    &
    -2
    &
    -1
    &
    0
    &
    0
  }
$$

this splits into the [[total complex]] of a [[bicomplex]] with

$$
  s \;=\; s_{BV} + s_{BRST}
$$

with

$$
  \array{
     & s_{BRST} & s_{BV}
     \\
     deg_{gh} = & +1 & 0
     \\
     deg_{af} = & 0 & +1
  }
$$

as shown in the above table. Under this decomposition, the _[[classical master equation]]_

$$
  s^2 = 0
  \phantom{AAAA}
  \Leftrightarrow
  \phantom{AAAA}
  \left\{ \left( -L + L_{BRST}\right) dvol_\Sigma \,,\, \left( -L + L_{BRST}\right) dvol_\Sigma \right\} = 0
$$

is equivalent to three conditions:

$$
  \array{
    \left( s_{BV} \right)^2 = 0 && \text{Noether's second theorem}
    \\
    \left( s_{BRST} \right)^2 = 0 && \text{closure of gauge symmetry}
    \\
    \left[ s_{BV}, s_{BRST} \right] = 0 &&
    \left\{
    \array{
       \text{ gauge symmetry preserves the shell },
       \\
       \text{ gauge symmetry acts on Noether identities }
    }
    \right.
  }
$$

=--

(e.q. [Barnich 10 (4.10)](BRST+complex#Barnich10))

+-- {: .proof}
###### Proof

Due to the construction in def. \ref{DerivedProlongedShell} the [[BRST differential]] by itself is already assumed to square to the

$$
  \left(s_{BRST}\right)^2 = 0
$$

The remaining conditions we may check on 0-jet generators.

The condition

$$
  \left( s_{BV} \right)^2 = 0
$$

is non-trivial only on the [[antifields]] of the [[ghost fields]]. Here we obtain

$$
  \begin{aligned}
    s_{BV} s_{BV} c^\ddagger_\alpha
    & =
    -\underset{k \in \mathbb{N}}{\sum}
    (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
    \left(
      R_\alpha^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a
    \right)
    \\
    & =
    -\underset{k \in \mathbb{N}}{\sum}
    (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
    \left(
      R_\alpha^{a \mu_1 \cdots \mu_k} \frac{\delta_{EL} L}{\delta \phi^a}
    \right)
  \end{aligned}
$$

That this vanishes is the statement of [[Noether's theorem|Noether's second theorem]] (prop. \ref{NoetherIdentities}).

Next we check

$$
  s_{BV} \circ s_{BRST} + s_{BRST} \circ s_{BV}
  =
  0
$$

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

$$
  \begin{aligned}
    s_{BRST} s_{BV} \phi^{\ddagger}_a
    & =
    s_{BRST} \frac{\delta_{EL} L}{\delta \phi^a}
    \\
    & =
    \underset{k}{\sum}
    \underset{q}{\sum}
    \frac{d^q}{d x^{\nu_1} \cdots d x^{\nu_q}}
    \left(
      c^\alpha_{,\mu_1 \cdots \mu_k}
      R_\alpha^{b \mu_1 \cdots \mu_k}
    \right)
    \frac{\partial}{\partial \phi^b_{,\nu_1 \cdots \nu_q}}
    \frac{\delta_{EL} L}{\delta \phi^a}
  \end{aligned}
$$

and on the other hand

$$
  \begin{aligned}
    s_{BV} s_{BRST} \phi^\ddagger_a
    & =
    -
    s_{BV}
    \underset{k}{\sum}
    \frac{\delta_{EL}}{\delta \phi^a}
    \left(
      c^\alpha_{,\mu_1 \cdots \mu_k}
      R_\alpha^{b \mu_1 \cdots \mu_k}
      \phi^\ddagger_b
    \right)
    \\
    & =
    +
    \underset{k}{\sum}
    \underset{q}{\sum}
    (-1)^q
    \frac{d^q}{d x^{\nu_1} \cdots d x^{\nu_q}}
    \left(
      \frac{\partial}{\partial \phi^a_{,\mu_1 \cdots \mu_q}}
      \left(
        c^\alpha_{,\mu_1 \cdots \mu_k}
        R_\alpha^{b \mu_1 \cdots \mu_k}
      \right)
      \frac{\delta_{EL} L}{\delta \phi^b}
    \right)
  \end{aligned}
$$

That the sum of these two terms indeed vanishes is equation (eq: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

$$
  \begin{aligned}
    s_{BV} s_{BRST} c^\ddagger_\alpha
    & =
    s_{BV} \gamma^{\alpha'}{}_{\alpha \beta} c^\beta c^\ddagger_{\alpha'}
    \\
    & =
    -
    \gamma^{\alpha'}{}_{\alpha \beta} c^\beta
      \underset{k \in \mathbb{N}}{\sum}
        (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
        \left(
          R_{\alpha'}^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a
        \right)
  \end{aligned}
$$

as well as

$$
  \begin{aligned}
    s_{BRST} s_{BV} c^\ddagger_\alpha
    & =
    s_{BRST}
    \left(
        \underset{k \in \mathbb{N}}{\sum}
          (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
          \left(
            R_\alpha^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a
          \right)
    \right)
    \\
    & =
    R
    \left(
        \underset{k \in \mathbb{N}}{\sum}
          (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
          \left(
            R_\alpha^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a
          \right)
    \right)
    \;-\;
    \left(
        \underset{k \in \mathbb{N}}{\sum}
          (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
          \left(
            R_\alpha^{a \mu_1 \cdots \mu_k}
    \left(
      \underset{q \in \mathbb{N}}{\sum}
      \frac{\delta_{EL}}{\delta \phi^a}
      \left(
        c^{\alpha'}_{,\nu_1 \cdots \nu_q}
        R_{\alpha'}^{b \nu_1 \cdots \nu_q}
        \phi^\ddagger_b
      \right)
    \right)
          \right)
    \right)
    \\
    & +
    R
    \left(
        \underset{k \in \mathbb{N}}{\sum}
          (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
          \left(
            R_\alpha^{a \mu_1 \cdots \mu_k} \phi^\ddagger_a
          \right)
    \right)
    \;-\;
    \left(
        \underset{k \in \mathbb{N}}{\sum}
          (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}}
          \left(
            R_\alpha^{a \mu_1 \cdots \mu_k}
    \left(
      \underset{q,r \in \mathbb{N}}{\sum}
      (-1)^{r}
      \frac{d^r}{d x^{\rho_1} \cdots d x^{\rho_r}}
      \left(
        c^{\alpha'}_{,\nu_1 \cdots \nu_q}
        \frac{\partial R_{\alpha'}^{b \nu_1 \cdots \nu_q}}{\partial \phi^a_{,\rho_1 \cdots \rho_r}}
        \phi^\ddagger_b
      \right)
    \right)
          \right)
    \right)
    \\
    & = (R \cdot N_R)_a^b (\phi^\ddagger_b)
  \end{aligned}
$$

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

$$
  \left(
    s_{BRST}s_{BV}
    +
    s_{BV} s_{BRST}
  \right)
  c^\ddagger_\alpha
  =
  0
$$

is relation (eq:LieActionOnNoetherOperatorGivesLieBracketUnderNoetherTheorem) in prop. \ref{LieAlgebraActionOfInfinitesimalSymmetriesOfTheLagrangianOnNoetherOperators}.

=--

+-- {: .num_example #DerivedProlongedShellInAbsenceOfExplicitGaugeSymmetries}
###### Example
**(derived prolonged shell in the absence of explicit gauge symmetry -- the [[local BV-complex]])**

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \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}.

$$
  s = s_{BV}
    \,.
$$

=--

+-- {: .num_example #LocalBVComplexOfVacuumElectromagnetismOnMinkowskiSpacetime}
###### Example
**([[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

$$
  \left(
    (a_\mu), c
  \right)
$$

then the [[local BV-BRST complex]] (prop. \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm})
has generators

$$
  \array{
    & c^\ddagger & (a^\ddagger)^\mu & a_\mu & c
    \\
    deg =
    &
    -2 & -1 & 0 & 1
  }
$$

together with their [[total spacetime derivatives]],
and the local BV-BRST differential $s$ acts on these generators as follows:

$$
  s
  \;\colon\;
  \left\{
  \array{
    (a^\dagger)^\mu &\mapsto&  f^{\nu \mu}_{,\nu}  & \text{(equations of Motion -- vacuum Maxwell equations)}
    \\
    c^\ddagger &\mapsto& (a^\ddagger)^\mu_{,\mu} & \text{(Noether identity)}
    \\
    a_\mu &\mapsto& c_{,\mu} & \text{(infinitesimal gauge transformation)}
  }
  \right.
$$

=--

More generally:

+-- {: .num_example #LocalBVBRSTComplexOfYangMillsTheory}
###### Example
**([[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]] (eq:BRSTFunctionForClosed) given by

$$
  L_{BRST}
  =
  \left(
    c^\alpha_{,\mu}
      -
    \gamma^\alpha{}_{\beta \gamma}c^\beta a^\gamma_\mu
  \right)
  (a^\ddagger)_\alpha^\mu
  \;+\;
  \tfrac{1}{2}
  \gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma c^\ddagger_\alpha
  \,.
$$

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:

$$
  \array{
    & & s &
    \\
    \text{field}
    &
    a_\mu^\alpha
       &\mapsto&
    c^\alpha_{,\mu}
      -
    \gamma^\alpha{}_{\beta \gamma}c^\beta a^\gamma_\mu
    &
    \text{gauge symmetry}
    \\
    \text{
      ghost field
    }
    &
    c^\alpha
      &\mapsto&
    \tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma
    &
    \text{Lie bracket}
    \\
    \text{antifield}
    &
    (a^\ddagger)^\mu_\alpha
      &\mapsto&
    \phantom{-}
    \left(
      \frac{d}{d x^\mu} f^{\mu \nu \alpha'}
        +
      \gamma^{\alpha'}{}_{\beta' \gamma} a_\mu^{\beta'} f^{\mu \nu \gamma}
    \right)
    k_{\alpha' \alpha}
    &
    \text{equations of motion}
    \\
    &&&
    -
    \gamma^{\alpha'}{}_{\beta \alpha}c^\beta
    (a^\ddagger)_{\alpha'}^\mu
    &
    \\
    \text{anti ghostfield}
    &
    c^\ddagger_\alpha
      &\mapsto&
     \gamma^{\alpha'}{}_{\alpha \gamma} a^\gamma_\mu (a^\ddagger)^\mu_{\alpha'}
     +  \frac{d}{d x^\mu} (a^\ddagger)^\mu_\alpha
    &
    \text{Noether identities}
    \\
    &&& + \gamma^{\alpha'}{}_{ \alpha \beta} c^\beta c^\ddagger_{\alpha'}
  }
$$

=--

(e.g. [Barnich-Brandt-Henneaux 00 (2.8)](local+BRST+cohomology#BarnichBrandtHenneaux00))

$\,$

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.

$\,$

**(derived variational bicomplex)**

The analog of the [[de Rham complex]] of a [[derived Lie algebroid]] is called the _[[Weil algebra]]_:

+-- {: .num_defn #WeilAlgebra}
###### Definition
**([[Weil algebra]] of a [[Lie algebroid]])**

Given a [[derived Lie algebroid]] $\mathfrak{a}$ over some $X$ (def. \ref{LInfinityAlgebroid}), its [[Weil algebra]] is

$$
  W(\mathfrak{a})
    \;\coloneqq\;
  \left(
    Sym_{C^\infty(X)}( \Gamma(T^\ast_{inf} X) \oplus \mathfrak{a}_\bullet \oplus \mathfrak{a}[1]_\bullet )
     \;,\;
    \mathbf{d}_W \coloneqq \mathbf{d} + d_{CE}
  \right)
  \,,
$$

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.

=--

| [[smooth manifolds]] | [[derived Lie algebroids]] |
|----------------------|----------------------------|
| [[algebra of functions]] | [[Chevalley-Eilenberg algebra]] |
| algebra of [[differential forms]] | [[Weil algebra]] |

+-- {: .num_example #ClassicalWeilAlgebra}
###### Example
**(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]].

=--

+-- {: .num_defn #BVVariationalBicomplex}
###### Definition
**([[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]]

$$
  \begin{aligned}
    d_{W}
    & = \mathbf{d} - (d - s_{BRST})
    \\
    & =
    \delta + s_{BRST}
  \end{aligned}
  \,.
$$

Therefore we speak of the _[[variational BRST-bicomplex]]_ and write

$$
  \Omega^\bullet_\Sigma( E/(\mathcal{G} \times_\Sigma T \Sigma) )
  \,.
$$

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

$$
  \begin{aligned}
    d_W
      & =  \mathbf{d} - (d - s)
    \\
    & = \delta + s
  \end{aligned}
  \,.
$$

Since $s$ is the [[BV-BRST differential]] (prop. \ref{LocalBVBRSTComplexIsDerivedCriticalLocusOfEulerLagrangeForm})
this defines the "BV-BRST [[variational bicomplex]]".

=--

$\,$

**global [[BV-BRST complex]]**
 {#BVBRSTComplexGlobal}

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]].

+-- {: .num_defn #ComplexBVBRSTGlobal}
###### Definition
**(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]]

$$
  E_{\text{BV-BRST}}
  =
  T^\ast_{\Sigma,inf}[-1](E \times_\Sigma \mathcal{G}[1])
$$

underlying the [[local BV-BRST complex]] (eq:BVBRSTGradedFieldBundle), consider the [[linear map]]

$$
  \label{LocalAntibracketTransgressed}
  \array{
    LocObs(E_{\text{BV-BRST}})
      \otimes
    LocObs(E_{\text{BV-BRST}})
      &\overset{\{-,-\}}{\longrightarrow}&
    LocObs(E_{\text{BV-BRST}})
    \\
    \tau_\Sigma(\alpha), \tau_\Sigma(\beta)
    &\mapsto&
    \tau_\Sigma( \{\alpha, \beta\} )
  }
$$

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) _[[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]] (eq:LocalAntibracketVersionOfBVBRSTDifferential) is a well-defined
[[differential]] on the graded [[local observables]]

$$
  \left\{
    -\tau_\Sigma \mathbf{L}
    + \tau_\Sigma \mathbf{L}_{BRST}
    \;,\, -
  \right\}
  \;\colon\;
  LocObs(E_{\text{BV-BRST}})
  \longrightarrow
  LocObs(E_{\text{BV-BRST}})
  \,,
$$

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

$$
  \label{GlobalBVBRSTDifferential}
  \left\{
    -S + S_{BRST}
    \;,\;-
  \right\}
  \;\coloneqq\;
  \left\{
    -\tau_\Sigma \mathbf{L}
    + \tau_\Sigma \mathbf{L}_{BRST}
    \;,\, -
  \right\}
$$

for this (global) _[[BV-BRST differential]]_.

This uniquely extends as a graded [[derivation]] to [[multilocal observables]] (def. \ref{LocalObservables}) and from there along the [[dense subspace]] inclusion  (eq:InclusionOfPolynomialLocalObservablesIntoPolynomialObservables)

$$
  PolyMultiLocObs(E_{\text{BV-BRST}})
    \overset{\text{dense}}{\hookrightarrow}
  PolyObs(E_{\text{BV-BRST}})
$$

to a differential on [[off-shell]] [[polynomial observables]] (def. \ref{PolynomialObservables}):

$$
  \{-S' + S'_{BRST}\}
   \;\colon\;
  PolyObs(E_{\text{BV-BRST}})
    \longrightarrow
  PolyObs(E_{\text{BV-BRST}})
$$

This [[differential graded-commutative superalgebra]]

$$
  \label{GlobalBVComplexdgAlgebra}
  \left(
    \left(
      \underset{
        \text{vector space}
      }{
      \underbrace{
        PolyObs(E_{\text{BV-BRST}})
      }}
      ,
      \underset{
        \text{product}
      }{
      \underbrace{
        (-)\cdot(-)
      }}
    \right)
    ,
    \underset{
      \text{differential}
    }{
    \underbrace{
      \{-S' + S'_{BRST}, -\}
    }}
  \right)
$$

is the _global [[BV-BRST complex]]_ of the given [[Lagrangian field theory]] with the chosen [[gauge parameters]].

=--

+-- {: .proof}
###### Proof

We need to check that the global [[antibracket]] (eq: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}.

=--

+-- {: .num_example #BVDifferentialGlobal}
###### Example
**(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

$$
  \label{ComponentsOfGlobalBVDifferential}
  \begin{aligned}
    \left\{ -S', -\right\}
    & =
    \int_\Sigma
    j^{\infty}\left(\mathbf{\Phi}\right)^\ast
    \left(
      \frac{\overset{\leftarrow}{\delta}_{EL} L}{\delta \phi^A}
    \right)(x)
    \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
    \,
    dvol_\Sigma(x)
    \\
    & =
    \underset{A}{\sum}
      (-1)^{deg(\phi^A)}
    \int_\Sigma
      (P_{A B}\mathbf{\Phi}^A)(x)
      \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
    \,
    dvol_\Sigma(x)
  \end{aligned}
$$

where

1. $P \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(E^\ast)$

   is the [[differential operator]] (eq:DifferentialOperatorEulerLagrangeDerivative) from def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime}, corresponding to the [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]].

1. $deg(\phi^A) \coloneqq n_{(\Phi^A)} + \sigma_{\Phi^A} \;\in\; \mathbb{Z}/2$

   is the sum of the cohomological degree and of the super-degree of $\Phi^A$ (as in def. \ref{differentialgradedcommutativeSuperalgebra}, def. \ref{A+first+idea+of+quantum+field+theoryDifferentialFormOnSuperCartesianSpaces}).

It follows that the [[cochain cohomology]] of the global [[BV-differential]] $\{-S',-\}$ (eq:GlobalBVComplexdgAlgebra) in $deg_{af} = 0$ is the space of [[on-shell]] [[polynomial observables]]:

$$
  \label{OnShellPolynomialObservablesAsBVCohomology}
  \underset{
    \text{off-shell}
  }{
  \underbrace{
    PolyObs(E_{\text{BV-BRST}})_{def(af = 0)}
  }}/im(\{-S',-\})
  \;\simeq\;
  \underset{
    \text{on-shell}
  }{
  \underbrace{
    PolyObs(E_{\text{BV-BRST}}, \mathbf{L}')
  }}
  \,.
$$

=--

+-- {: .proof}
###### 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

$$
  \left\{ -\mathbf{L}',-\right\}
  \;=\;
  \frac{\overset{\leftarrow}{\delta}_{EL} \mathbf{L}'}{\delta \phi^A}
  \frac{\delta}{\delta \phi^{\ddagger}_A}
$$

With this the expression for $\{-S',-\}$ follows directly from the definition of the global antibracket (def. \ref{ComplexBVBRSTGlobal}) and the [[Euler-Lagrange equations]] (eq:DifferentialOperatorEulerLagrangeDerivative)

$$
  (P \Phi)_A = j^\infty_\Sigma(\Phi)\left( \frac{\delta_{EL} L}{\delta \phi^A} \right)
  \,.
$$

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

$$
  \begin{aligned}
    (-1)^{
      \left(
        n_{(\phi^A)} n_{(\phi^A)}
          +
        \sigma_{(\phi^A)} \sigma_{(\phi^A)}
      \right)
    }
    & =
    (-1)^{
      \left(
        n_{(\phi^A)} + \sigma_{(\phi^A)}
      \right)
    }
    \\
    & =
    (-1)^{deg(\phi^A)}
  \end{aligned}
  \,.
$$

From this the identification (eq:OnShellPolynomialObservablesAsBVCohomology) follows by (eq:PolynomialOnShellObservablesArePolynomialOffShellobservableModuloTheEquationsOfMotion) in theorem \ref{LinearObservablesForGreeFreeFieldTheoryAreDistributionalSolutionsToTheEquationsOfMotion}.

=--

$\,$

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 [next chapter](#GaugeFixing) 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]].