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

Reduced phase space

In this chapter we discuss these topics:

• Global gauge reduction for strictly invariant functions (action functionals):

  • Derived critical loci inside Lie algebroids

  • Schouten bracket on Lie algebroids

• Local gauge reduction for weakly invariant local functions (Lagrangian densities):

  • Local antibracket

  • Local BV-BRST complex

  • Global BV-BRST complex

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 (def. ) if spacetime were compact, so that, by the principle of extremal action (prop. ), 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 below.

The genuine 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 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. below).

$\,$

derived critical loci inside Lie algebroids

By analogy with the algebraic formulation of smooth functions between Cartesian spaces (the embedding of Cartesian spaces into formal duals of R-algebras, prop. ) it is clear how to define a map (homomorphism) between Lie algebroids:

Definition

(homomorphism between Lie algebroids)

Given two derived Lie algebroids $\mathfrak{a}$, $\mathfrak{a}'$ (def. ), 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 “non-curved sh-map”.)

Example

(invariant functions in terms of Lie algebroids)

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

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

on a supermanifold $X$. Then there is a canonical homomorphism of Lie algebroids (def. )

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

from the manifold $X$ regarded as a Lie algebroid by example to the action Lie algebroid $X/\mathfrak{g}$ (example ), 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 . Then homomorphisms of Lie algebroids (def. ) 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 $p$ (1) 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

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 ), 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. ) then it should in fact be a section of the infinitesimal neighbourhood (example ) of the zero section inside the cotangent bundle, the infinitesimal cotangent bundle $T^\ast_{inf}(X/\mathfrak{g})$ of the Lie algebroid (def. ebelow).

To motivate the definition below of infinitesimal cotangent bundle of a Lie algebroid recall from example 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 vector fields are equivalently derivations of smooth functions (prop. ). This leads to the following definition:

Definition

(infinitesimal cotangent Lie algebroid)

Let $\mathfrak{a}$ be a Lie ∞-algebroid (def. ) 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. ) there is a canonical homomorphism of Lie algebroids (def. ) from the infinitesimal cotangent Lie algebroid down to the base Lie algebroid:

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

given dually by the identity on the original generators.

Example

(infinitesimal cotangent bundle of action Lie algebroid)

Let $X/\mathfrak{g}$ be an action Lie algebroid (def. ) whose Chevalley-Eilenberg differential is given in local coordinates by (?)

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

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

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

and

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

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

(5)$$\phi^\ddagger_a \;\coloneqq\; \frac{\partial}{\partial \phi^a} \phantom{AAAAA} c\ddagger_\alpha \;\coloneqq\; \frac{\partial}{\partial c^\alpha} \,.$$

so that the generator content then reads as follows:

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

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

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

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:

Proposition

(exterior differential of invariant function is section of infinitesimal cotangent bundle)

For $\mathfrak{a}$ a Lie ∞-algebroid (def. ) over some $X$; and $S \;\colon\;\mathfrak{a} \longrightarrow \mathbb{R}$ a invariant smooth function on it (example ) there is an induced section $d S$ of the infinitesimal cotangent Lie algebroid (def. ) bundle projection (2):

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

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;

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

3. all other derivations to zero.

Proof

We discuss the proof in the special case that $\mathfrak{a} = X/\mathfrak{g}$ is an action Lie algebroid (def. ) hence where $T^\ast_{inf}(\mathfrak{a}) = T^\ast_{inf}(X/\mathfrak{g})$ is as in example . 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 (4)

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

Notice that the left vertical map is indeed as shown, due to the invariance of $S$ (example ), 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 (3) and using the invariance of $S$ (example )

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

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 space that makes the following 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 general abstract way to express the equation $d S = 0$.

In this general abstract form the concept of critical locus generalizes to invariant functions on 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-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 Higher structures in Physics.

Definition

(derived critical locus of invariant function on Lie ∞-algebroid)

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

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

be an invariant function (example ) and consider the section of its infinitesimal cotangent bundle $T^\ast_{inf} \mathfrak{a}$ (def. ) corresponding to its exterior derivative via prop. :

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

Then the derived critical locus of $S$ is the derived Lie algebroid (def. ) 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. ) except for two changes:

1. all derivations are shifted down in degree by one;

   rephrased in terms of graded manifold (remark ) this means that the graded manifold underlying $\mathfrak{a}_{d S \simeq 0}$ is $T^\ast_{inf}[-1]\mathfrak{a}$;

2. 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. ) 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:

Example

(derived critical locus inside action Lie algebroid)

Consider an invariant function (def. ) on an action Lie algebroid (def. )

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

for the case that the underlying supermanifold $X$ is a super Cartesian space (def. ) with global coordinates $(\phi^a)$ as in example . Then the derived critical locus (def. )

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

is, in terms of its Chevalley-Eilenberg algebra $CE\left( (X/\mathfrak{g})_{d S \simeq 0} \right)$ (def. ) given as follows:

Its generators are those of $CE\left( T^\ast_{inf}(X/\mathfrak{g}) \right)$ as in (6), 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 ) this means that the graded manifold underlying the derived critical locus is the shifted infinitesimal cotangent bundle of the graded manifold $\mathfrak{g}[1] \times X$ (?) which underlies the action Lie algebroid (def. ):

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

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

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

which is just the expression for the differential (7) in $CE\left( T^\ast_{inf}(X/\mathfrak{g}) \right)$ from example , 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 theoretic version of the ordinary concept of critical locus $X_{d S = 0}$:

Example

(ordinary critical locus is cochain cohomology of derived critical locus in degree 0)

Let $X$ be an superpoint (def. ) or more generally the infinitesimal neighbourhood (example ) of a point in a super Cartesian space (def. ) 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 ) 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 degree vanishes. Instead, the cohomology of the Chevalley-Eilenberg algebra of a derived critical locus in 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 .

This issue we discuss in detail in the chapter Gauge fixing, see prop. below.

In order to generalize the statement of example 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:

Proposition

(Chevalley-Eilenberg algebra of derived critical locus is total complex of BV-BRST bicomplex)

Let $(X/\mathfrak{g})_{d S \simeq 0}$ be a derived critical locus inside an action Lie algebroid as in example . Then its Chevalley-Eilenberg differential (9) 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:

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

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

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 invariance of $S$ (example ):

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

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

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 .

Then if the vertical differential (prop. )

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

has vanishing cochain cohomology in negative $af$-degree

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

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 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

The first statement follows from the spectral sequence 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 is trivial, and hence the claim follows.

Now if $X$ is inside the infinitesimal neighbourhood of a point, then example 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 (11) the construction of a derived critical locus inside an action Lie algebroid provides a resolution of the space of those functions which are

1. restricted to the critical locus (a homotopy intersection);

2. invariant under the Lie algebra action (a homotopy quotient).

We apply this general mechanism below 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,

2. gauge invariant.

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

$\,$

Schouten bracket/antibracket

Since the infinitesimal cotangent Lie algebroid $T^\ast_{inf} \mathfrak{a}$ has function algebra given by tensor products of tangent vector fields/derivations, we expect that a graded analogue of the Lie bracket of ordinary tangent vector fields exists on the Chevalley-Eilenberg algebra $CE\left( T^\ast_{inf} \mathfrak{a}\right)$. This is indeed the case, and crucial for the theory:

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 .

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 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 (5) for the coordinate-derivations the Schouten bracket is equivalently given by

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

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), Henneaux-Teitelboim 92, section 15.5.2)

The power of the Schouten bracket/antibracket rests in the fact that it makes the Chevalley-Eilenberg differential on a derived critical locus $(X/\mathfrak{g})_{d S \simeq 0}$ become a Hamiltonian vector field, for “Hamiltonian” the sum of $S$ with the Chevalley-Eilenberg differential of $X/\mathfrak{g}$:

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 .

Then the CE-differential (9) of the derived critical locus $X/\mathfrak{g}\vert_{S \simeq 0}$ is simply the Schouten bracket/antibracket (def. ) with the sum

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

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 (?) and using the notation for derivations from (5) 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

This is a simple straightforward computation, but we spell it out for illustration of the general principle. The result is to be compared with (9):

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 (9).

As a corollary we obtain:

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 .

Then the Schouten bracket/antibracket (def. ) 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 $T^\ast[-1](X \times \mathfrak{g}[1])$ (8), then the 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 it is clear that this is best understood by first considering the refinement of the Schouten bracket/antibracket to this situation.

$\,$

local antibracket

If we think of the invariant function $S$ in def. as being the action functional (example ) of a Lagrangian field theory $(E,\mathbf{L})$ (def. ) over a 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. In that case one may still appeal to a construction on the space of field histories as in example by considering the action functional for all adiabatically switched $b \mathbf{L}$ Lagrangians, with $b \in C_{cp}^\infty(\Sigma)$. This approach is taken in (Fredenhagen-Rejzner 11a).

Here we instead consider now the “local lift” or “de-transgression” of the above construction from the space of field histories to the jet bundle of the field bundle of the theory, refining the BV-BRST complex (prop. ) to the local BV-BRST complex (prop. below), corresponding to the local BRST complex from example (Barnich-Brandt-Henneaux 00).

This requires a slight refinement of the construction that leads to example : In contrast to the action functional $S = \tau_\Sigma(g\mathbf{L})$ (example ), 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 ). 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. , which however here we do not explicitly, but only via passing to cochain cohomology as in example ).

This means that the Euler-Lagrange form $\delta_{EL} \mathbf{L}$ is, off-shell, not a section of the infinitesimal cotangent bundle (def. ) 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. below). To see the required twist, it is most convenient to make use of a local version of the antibracket (def. below), via local refinement of example . As a result we may form the local derived critical locus as in def. 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. 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 to the case where the base manifold is generalized to a field bundle (def. ) and the Lie algebra to a gauge parameter bundle (def. ):

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. ) over some spacetime $\Sigma$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a bundle of gauge parameters (def. ) which are closed (def. ), 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 ).

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 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 (8).

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 ) 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

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

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 $\phi^a$

• $c^\ddagger_\alpha$ 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. to the case where the base manifold is generalized to the jet bundle (def. ) field bundle (def. ) and the Lie algebra to the jet bundle of a gauge parameter bundle (def. ):

Definition

(local antibracket)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. ) over Minkowski spacetime $\Sigma$ (def. ), and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a bundle of gauge parameters (def. ) which are closed (def. ), inducing via example 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

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

of its underlying graded bundle from def. .

Then on the horizontal $p+1$-forms on this bundle (def. ) which in terms of the volume form may all be decomposed as (?)

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

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 (14) 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 (12) but with Euler-Lagrange derivatives (?) instead of partial derivatives,

The second version is this:

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

where again $\frac{\delta_{EL}}{\delta \phi^a}$ denotes the Euler-Lagrange variational derivative (?)

(Barnich-Henneaux 96 (2.9) and (2.12), reviewed in Barnich 10 (4.9))

Proposition

(basic properties of the local antibracket)

The local antibracket from def. satisfies the following properties:

1. The two versions differ by a total spacetime derivative (def. ):

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

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

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

3. 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:

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

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

4. the horizontally 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).

Proof

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

The third point requires some computation (Barnich-Henneaux 96 (B.9)).

Finally that $\{-,-\}'$ vanishes when at least one of its arguments is horizontally exact follows from the fact that already the Euler-Lagrange derivative vanishes on this argument (example ). 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. :

Remark

(local classical master equation)

The third item in prop. 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. ) with $\mathbf{K}$ is a differential

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

2. the local antibracket (def. ) of $\mathbf{K}$ with itself is a total spacetime derivative:

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

3. the other variant of the local antibracket (def. ) 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

With the local version of the antibracket in hand (def. ) it is now straightforward to refine the construction of a derived critical locus inside an action Lie algebroid (example ) to the “derived” shell (?) inside the formal dual of the local BRST complex (example ). The result is a derived Lie algebroid whose Chevalley-Eilenberg algebra is called the local BV-BRST complex. This is example below.

The following definition is the local refinement of def. :

Definition

(local infinitesimal cotangent Lie algebroid)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. ) over some spacetime $\Sigma$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a bundle of gauge parameters (def. ) which are closed (def. ), inducing via example 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. ) as well as the gauge parameter bundle $\mathcal{G} \overset{gb}{\to} \Sigma$ are trivial vector bundles (example ) over Minkowski spacetime $\Sigma$ (def. ) with field coordinates $(\phi^a)$ and gauge parameter coordinates $(c^\alpha)$.

Then the vertical infinitesimal cotangent Lie algebroid (def. ) has coordinates as in (6) 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

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

to be called the BRST 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 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 antibracket $\{-,-\}$ (16) with the BRST Lagrangian density (18)

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

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. is now this:

Proposition

(Euler-Lagrange form is section of local cotangent bundle of jet bundle gauge-action Lie algebroid)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. ) over some spacetime $\Sigma$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a gauge parameter bundle (def. ) which are closed (def. ), inducing via example the Lie algebroid $E / ( \mathcal{G} \times_\Sigma T \Sigma )$ and via def. its local cotangent Lie ∞-algebroid $T^\ast_{inf}_\Sigma(E / ( \mathcal{G} \times_\Sigma T \Sigma ))$.

Then the Euler-Lagrange variational derivative (prop. ) constitutes a section of the local cotangent Lie ∞-algebroid (def. )

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

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

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. below.

The local analog of def. is now the following definition of the “derived prolonged shell” of the theory (recall the ordinary prolonged shell $\mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E)$ from (?)):

Definition

(derived reduced prolonged shell)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. ) over some spacetime $\Sigma$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a bundle of closed irreducible gauge parameters (def. ), inducing via prop. 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. ). This is local refinement of example :

Example

(local BV-BRST complex)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. ) over Minkowski spacetime $\Sigma$, and let $\mathcal{G} \overset{gb}{\to} \Sigma$ be a gauge parameter bundle (def. ) which is closed (def. ). Assume that both are trivial vector bundles (example ) with field coordinates as in prop. .

Then the Chevalley-Eilenberg algebra of the derived prolonged shell $(E/( \mathcal{G} \times_\Sigma T \Sigma ))_{\delta_{EL}L \simeq 0}$ (def. ) 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 ) 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

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

which is the local antibracket (def. ) 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. ) with the BRST Lagrangian (18).

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.

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

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))

Proof

Due to the construction in def. 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 second theorem (prop. ).

Next we check

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

on generators. On the 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 (?) in the proof of the on-shell invariance of the equations of motion under infinitesimal symmetries of the Lagrangian (prop. )

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. . Under this identification, the fact that

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

is relation (?) in prop. .

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. ) with vanishing gauge parameter bundle (def. ) (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. ) is the plain local BV-complex of def. .

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

Example

(local BV-BRST complex of vacuum electromagnetism on Minkowski spacetime)

Consider the Lagrangian field theory of free electromagnetism on Minkowski spacetime (example ) with gauge parameter as in example . With the field and gauge parameter coordinates as chosen in these examples

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

then the local BV-BRST complex (prop. ) 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:

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 , with local BRST complex as in example , hence with BRST Lagrangian (18) 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 ) 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))

$\,$

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. ) 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:

Definition

(Weil algebra of a Lie algebroid)

Given a derived Lie algebroid $\mathfrak{a}$ over some $X$ (def. ), 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

Example

(classical Weil algebra)

Let $\mathfrak{g}$ be a Lie algebra with corresponding Lie algebroid $B \mathfrak{g}$ (example ). Then the Weil algebra (def. ) of $B \mathfrak{g}$ is the traditional Weil algebra of $\mathfrak{g}$ from classical Lie theory.

Definition

(variational BV-bicomplex)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. ) equipped with a gauge parameter bundle $\mathcal{G}$ (def. ) which is closed (def. ). Consider the Lie algebroid $E/(\mathcal{G} \times_\Sigma T \Sigma)$ from example , 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. ) has as differential the variational derivative (def. ) 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. ) has differential

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

Since $s$ is the BV-BRST differential (prop. ) this defines the “BV-BRST variational bicomplex”.

$\,$

global BV-BRST complex

Finally we may apply transgression of variational differential forms to turn the local BV-BRST complex on smooth functions on the jet bundle into a global BV-BRST complex on graded local observables on the graded space of field histories.

Definition

(global BV-BRST complex)

Let $(E,\mathbf{L})$ be a Lagrangian field theory (def. ) equipped with a gauge parameter bundle $\mathcal{G}$ (def. ) which is closed (def. ). Then on the local observables (def. ) on the space of field histories (def. ) of the graded field bundle

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

underlying the local BV-BRST complex (15), consider the linear map

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

where $\alpha, \beta \in \Omega^{p+1,0}_{\Sigma,cp}(E_{\text{BV-BRST}})$ (def. ), where $\tau_\Sigma$ denotes transgression of variational differential forms (def. ), and where on the right $\{-,-\}$ is the local antibracket (def. ).

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. ), and hence the transgression of the BV-BRST differential (19) 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 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

(21)$$\left\{ -S + S_{BRST} \;,\;- \right\} \;\coloneqq\; \left\{ -\tau_\Sigma \mathbf{L} + \tau_\Sigma \mathbf{L}_{BRST} \;,\, - \right\}$$

for this (global) BV-BRST differential.

This uniquely extends as a graded derivation to multilocal observables (def. ) and from there along the dense subspace inclusion (?)

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

to a differential on off-shell polynomial observables (def. ):

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

This differential graded-commutative superalgebra

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

is the global BV-BRST complex of the given Lagrangian field theory with the chosen gauge parameters.

Proof

We need to check that the global antibracket (20) is well defined:

By the last item of prop. 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 local observables is well defined, def. .

Example

(global BV-differential in components)

In the situation of def. , assume that the field bundles of all 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. ) is given by

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

where

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

   is the differential operator (?) from def. , corresponding to the Euler-Lagrange equations of motion.

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

   is the sum of the cohomological degree and of the super-degree of $\Phi^A$ (as in def. , def. ).

It follows that the cochain cohomology of the global BV-differential $\{-S',-\}$ (22) in $deg_{af} = 0$ is the space of on-shell polynomial observables:

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

Proof

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. ) and the Euler-Lagrange equations (?)

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

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. ):

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 (24) follows by (?) in theorem .

$\,$

This concludes our discussion of the reduced phase space of a Lagrangian field theory exhibited, dually by its local BV-BRST complex. In the next chapter we finally turn to the key implication of this construction: the gauge fixing of a Lagrangian gauge theory which makes the collection of fields and auxiliary fields (ghost fields and antifields) jointly have a (differential-graded) covariant phase space.