BV-BRST formalism

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#### Variational calculus
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#### Physics
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#### $\infty$-Lie theory
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#### Symplectic geometry
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#Contents#
* table of contents
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## Idea

In [[physics]] and specifically in [[field theory]], the _BV-BRST formalism_ is a tool in [[homological algebra]], [[higher differential geometry]] and [[derived geometry]] to handle the [[intersection]]- and [[quotient]]-constructions that appear

1. in the construction of [[reduced phase space|reduced]] [[phase spaces]] of [[Lagrangian field theories]], in particular including [[gauge theories]]; ("Lagrangian BV")

1. in [[symplectic reduction]] of [[phase spaces]] ("Hamiltonian BV")

In either case the  _BRST-BV_ complex $C^\infty(P^{BV})$ is a model in [[dg-geometry]] of a joint [[homotopy intersection]] and [[homotopy quotient]], hence of an [[(∞,1)-colimit]] and [[(∞,1)-limit]], of  a [[space]] in [[higher differential geometry]]/[[derived geometry]].

Accordingly, the BRST-BV complex is built from two main pieces:

1. it contains in positive degree a [[BRST-complex]]: the [[Chevalley-Eilenberg algebra]] of the [[∞-Lie algebroid]] which is the homotopy [[quotient]] ([[action Lie algebroid]]) of the [[gauge group]] (in Lagrangian BV) or of the group of flows generated by the constraints (in Hamiltonian BFV) -- which is in general an [[∞-group]] in either case -- acting on configuration space $C$;

* it contains in negative degree a [[Koszul-Tate resolution]] of the [[critical locus]] of the [[action functional]] (for Lagrangian BV) or of the constraint surface (in Hamiltonian BFV).

## Lagrangian BV
 {#LagrangianBV}

### Classical BV as homological resolution of reduced phase space
 {#ClassicalBVAsHomologicalResolutionOfReducedPhaseSpace}

The _classical Lagrangian BV-BRST complex_ of a [[Lagrangian field theory]] is, under suitable conditions, a [[homological resolution]] of the [[homotopy intersection]] with the [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]] (this is the BV part) of the [[homotopy quotient]] by the [[infinitesimal symmetries of the Lagrangian]] (this is the BRST part), and hence a homological model of the [[reduced phase space]] of the Lagrangian field theory.

A detailed introduction to the classical Lagrangian BV-BRST formalism is at

* _[[A first idea of quantum field theory]]_, chapter _[11. Reduced phase space](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#ReducedPhaseSpace)

### Quantum BV as homological (path-)integration
 {#HomologicalIntegration}

We discuss here the interpretation of the _quantum BV-complex_ as a homological implementation of [[integration]] thought of as [[path integral]]-[[quantization]] (in [[perturbative quantum field theory]]).

We indicate how on a finite dimensional smooth manifold the [[BV-algebra]] appearing in Lagrangian BV-formalism is the dual of the [[de Rham complex]] of [[configuration space]] in the presence of a [[volume form]] and how, by extention, this allows to interpret the BV-complex as a means for defining ([[path integral|path]]-)[[integration]] over general [[configuration spaces]] of [[field (physics)|fields]] by passing to BV-[[cochain cohomology]].

(The interpretation of the BV-differential as the dual de Rham differential necessary for this is due to ([Witten 90](#Witten90)) ([Schwarz 92](#Schwarz92)). A particularly clear-sighted account of the general relation is in [Gwilliam 2013](#Gwilliam) ).

Further [below](#PoincareDualityOnHochschild) we discuss the generalization of these relation in terms of [[Poincaré duality]] on [[Hochschild cohomology|Hochschild (co)homology]].

1. [The idea of path integral quantization](#IdeaOfPathIntegralQuantization)

1. [Multivector fields dual to differential forms](#MultivectorFieldsDualToDifferentialForms)

1. [The quantum master equation: the path integral measure is a closed form](#TheQuantumMasterEquationAsClosureOfIntegralMeasure)

1. [Integration over manifolds by BV cohomology](#IntegrationOverManifoldsByBVCohomology)

1. [BV-quantization](#BVQuantization)

1. [Path integration and quantum observables by BV-cohomology](#PathIntegrationAndQuantumObservablesByBVCohomology)

#### The idea of path integal quantization
 {#IdeaOfPathIntegralQuantization}

The [[path integral]] in [[quantum field theory]] is supposed to be the [[integral]] over a [[configuration space]] $X$ of [[field (physics)|fields]] $\phi$ using a [[measure]] $\mu_S$ which is thought of in the form

$$
  \mu_S(\phi)
    \coloneqq
  \exp\left(\frac{i}{\hbar} S\left(\phi\right)\right) \cdot \mu(\phi)
  \;\;\;\;
  \phi \in X
  \,,
$$

for $\mu$ some other measure and $S : X \to \mathbb{R}$ the _[[action functional]]_ of the [[theory (physics)|theory]].

For $f$ a [[smooth function]] on the space of [[field (physics)|fields]] its value as an [[observable]] of the system is supposed to be what would be the [[expectation value]]

$$
  \langle f \rangle_S
  =
  \frac{\int_{\phi \in Fields} f(\phi) \cdot \mu(\phi)}{\int_{\phi \in Fields} \mu(\phi) }
$$

if the measure existed. Of course this does not make sense in terms of the usual notion of [[integration]] against [[measures]] since such measures do not exists except in the simplest situation. But there is a [[cohomology|cohomological]] notion of integration where instead of actually performing an integral, we identify its value, if it exists, with a cohomology class and generally interpret that cohomology class as the expectation value, even if an actual integral against a measure does not exist. This is what BV formalism achieves, which we discuss after some preliminaries below in _[Integration over manifolds by BV cohomology](#IntegrationOverManifoldsByBVCohomology)_.

#### Multivector fields dual to differential forms
 {#MultivectorFieldsDualToDifferentialForms}

If one thinks of $X$ as an ordinary $(d \lt \infty)$-[[dimension|dimensional]] [[smooth manifold]], then $\mu_S$ will be given by a [[volume form]], $\mu_S \in \Omega^d(X)$. By contraction of [[multivector fields]] with [[differential forms]], every choice of volume form on $X$ induces an [[isomorphism]] between [[differential forms]] and [[polyvector fields]]

$$
  \mu
    \colon
  \Omega^\bullet(X)
    \stackrel{\simeq}{\longrightarrow}
  \wedge^{d-\bullet} \Gamma(T X)
  \,,
$$

which is usefully thought of as reversing degrees. Under this isomorphism the [[deRham differential]] maps to a [[divergence]] operator, the _[[BV-operator]]_, conventionally denoted

$$
  \Delta
  \;\coloneqq\;
  \mu \circ d_{dR} \circ \mu^{-1}

$$

which interacts naturally with the canonical bracket on multivector fields: the [[Schouten bracket]]. (See at _[[polyvector field]]_ for more details.)

+-- {: .num_defn #TheDualBVComplexOfTheDeRhamComplexOnAManifolds}
###### Definition

For $X$ an [[orientation|oriented]] [[smooth manifold]] of [[dimension]] $n \in \mathbb{N}$ and for $\mu \in \Omega^n(X)$a [[volume form]], write

$$
  BV(X, \mu)
  \coloneqq
  (\wedge^\bullet \Gamma(T X), \Delta_\mu)
$$

for the [[cochain complex]] induced on [[multivector fields]] by dualizing the [[de Rham differential]] with $\mu$.

=--

+-- {: .num_remark #BVComplexOfManifoldIsPoisson0}
###### Remark

The [[Schouten bracket]] on $BV(X,\mu)$ makes this cochain complex a [[Poisson 0-algebra]].

=--

For more see at _[[relation between BV and BD]]_.

#### The quantum master equation: the path integral measure is a closed form
 {#TheQuantumMasterEquationAsClosureOfIntegralMeasure}

Observe that

* if we think of

  * the measure $\mu$ as some closed reference differential form on $X$;

  *  the exponentiated action functional $exp\left(\frac{i}{\hbar}S\left(-\right)\right)$ as a [[multivector field]] on $X$;

  * the expression $exp(\frac{i}{\hbar}S(-)) \mu$ as the contraction of this multivector field with $\mu$

* then the **BV quantum master equaton**
$\Delta \exp(\frac{i}{\hbar}S) = 0$ says nothing but that
$exp(\frac{i}{\hbar}S(-)) \mu$ is a _closed differential form_.

* If we furthermore take into account that in the presence of gauge symmetries the space $X$ is not a plain manifold but the $L_\infty$-[[Lie infinity-algebroid|algebroid]] of the gauge symmetries acting on the space of fields, hence an [[NQ-supermanifold]] (whose Chevalley-Eilenberg algebra is the **BRST complex**), then this just says that $\exp(\frac{i}{\hbar}S) \mu$ is an [[integration over supermanifolds|integrable form]] in the sense of [[integration over supermanifolds|integration theory of supermanifolds]].

This means that Lagrangian BV formalism is nothing but a way of describing closed differential forms on [[Lie infinity-algebroid]] in terms of multivectors contracted into a reference differention form. The multivectors dual to degree 0 elements in the $L_\infty$-[[Lie infinity-algebroid|algebroid]] are the so-called "**anti-fields**", while those dual to the higher degree elements are the so-called "**anti-ghosts**".

#### Integration over manifolds by BV-cohomology
 {#IntegrationOverManifoldsByBVCohomology}

The following proposition about integration of differential $n$-forms is the archetype for interpreting cohomology in BV-complexes in terms of [[integration]].

+-- {: .num_example}
###### Example

On the [[open ball]] of [[dimension]] $n$, the [[integration of differential forms]] of compact support $\int \;\colon\; \Omega^n_{cp} \to \mathbb{R}$ is equivalently given by the projection onto the [[quotient]] by the exact forms, hence by passing to [[cochain cohomology]] in the truncated [[de Rham complex]] $C^\infty(B^n) \to \cdots \to \Omega^{n-1}(B^n) \to \Omega^n(B^n)$.

=--

See at _[[kernel of integration is the exact differential forms]]_ for details.

This "integration without integration" is discussed in more detail at _[[Lie integration]]_.

Let $X$ be a [[closed manifold|closed]] [[orientation|oriented]] [[smooth manifold]] of [[finite number]] [[dimension]] $n$ and let $\mu_S \in \Omega^n(X)$ be any [[volume form]]. Let again

$$
  BV(X,\mu_S) \coloneqq( \wedge^\bullet \Gamma(T X), \Delta_{\mu_S} )
$$

be the corresponding dual [[cochain complex]] of the [[de Rham complex]] by def. \ref{TheDualBVComplexOfTheDeRhamComplexOnAManifolds} above.

+-- {: .num_defn #ExpectationValueOfFunctionOnManifold}
###### Definition

For $f \in C^\infty(X)$ a [[smooth function]], its **[[expectation value]]** with respect to $\mu_S$ is

$$
  \langle f\rangle_{\mu_S}
  \coloneqq
  \frac{ \int_X f \cdot \mu_S }{\int_X \mu_S }
  \,.
$$

=--

Write $[-]_{BV}$ for the [[cochain cohomology]] classes in the BV complex $BV(X, \mu_S)$.

+-- {: .num_prop #ExpectationValueOfFunctionOnManifoldByBVCohomology}
###### Proposition

For $f \in BV(X,\mu_S)_0 \simeq C^\infty(X)$ the cohomology class of $f$ in the BV complex is the [[expectation value]] of $f$, def. \ref{ExpectationValueOfFunctionOnManifold} times the cohomology class of the unit function 1:

$$
  [f]_{BV}
  =
  \langle f\rangle_{\mu_S}
  [1]_{BV}
  \,.
$$

=--

See ([Gwilliam 13, lemma 2.2.2](#Gwilliam)).

#### BV quantization
 {#BVQuantization}

Let $X$ be a [[closed manifold]] as above and write $BV(X, \mu)$ for the BV-complex def. \ref{TheDualBVComplexOfTheDeRhamComplexOnAManifolds}, induced by a given [[volume form]] $\mu \in \Omega^n(X)$.

+-- {: .num_prop #ShiftInBVDifferentialOnManifoldDueToFunctional}
###### Proposition

If $S \in C^\infty(X)$ then the BV-complex induced via def. \ref{TheDualBVComplexOfTheDeRhamComplexOnAManifolds} by the volume form

$$
  \mu_S \coloneqq \exp\left(\frac{1}{\hbar} S\right) \cdot \mu
$$

(for any constant $\hbar$ to be read as [[Planck's constant]]) has BV-differential related to that of $\mu$ itself by

$$
  \Delta_{\mu_S}
  =
  \Delta_\mu + \frac{1}{\hbar}\iota_{d S}
  \,,
$$

where $\iota_{d S} : \wedge^\bullet \Gamma(T X) \to \wedge^{\bullet-1} \Gamma(T X)$ is the operation of acting with a [[vector field]] on $S$ by [[differentiation]], extended as a graded [[derivation]] to [[multivector fields]].

=--

+-- {: .num_prop #ClassicalBVComplexOnManifoldAsDerivedCriticalLocus}
###### Proposition

The complex

$$
  BV_{cl}(X, S)
  \coloneqq
  (\wedge^\bullet \Gamma(T X), \iota_{d S})

$$

is the [[derived critical locus]] of the function $S$.

=--

By the discussion at _[[derived critical locus]]_.

+-- {: .num_remark #ClassicalAndQuantumBVComplexOverManifold}
###### Remark

Prop. \ref{ShiftInBVDifferentialOnManifoldDueToFunctional} and prop. \ref{ClassicalBVComplexOnManifoldAsDerivedCriticalLocus} together say that the BV-complex of a manifold $X$ for a volume form $\mu_S$ shifted from a background volume form $\mu$ by a function $\exp\left(\frac{1}{\hbar} S\right)$ is an $\hbar$-deformation of the [[derived critical locus]] of $S$ by a contribution of the background volume form $\mu$.

We call $(\wedge^\bullet \Gamma(T X), \iota_{d S})$ the **classical BV complex** and $(\wedge^\bullet \Gamma(T X), \iota_{d S} + \hbar \Delta_{\mu} )$ the **quantum BV complex** of the manifold $X$ equipped with the function $S$ and the voume form $\mu$.

=--

The crucial idea now is the following.

+-- {: .num_remark}
###### Remark
**(central idea of BV quantization)**

In the above discussion of BV complexes over finite-dimensional manifolds, the construction of the **classical BV complex** in remark \ref{ClassicalAndQuantumBVComplexOverManifold} as a [[derived critical locus]] directly makes sense in great generality for [[action functionals]] $S$ defined on spaces of [[field (physics)|fields]] more general than finite-dimensional [[smooth manifolds]]. (It makes sense in a general context of [[differential cohesion]], see at _[differential  cohesive infinity-topos -- critical locus](differential+cohesive+infinity-topos#CriticalLocus)_). On the other hand, the construction of the quantum BV complex as the dual to the de Rham complex by a [[volume form]] by def. \ref{TheDualBVComplexOfTheDeRhamComplexOnAManifolds} breaks down as soon as the space of fields is no longer a finite dimensional manifold, hence breaks down for all but the most degenerate [[quantum field theories]]. But by remark \ref{ClassicalAndQuantumBVComplexOverManifold} we may instead think of the quantum BV complex as a certain **deformation** of the classical BV complex, and _that_ notion continues to make sense in full generality.

And once such a deformation of a critical locus has been obtained, we may read prop. \ref{ExpectationValueOfFunctionOnManifoldByBVCohomology} the other way round and regard the [[cochain cohomology]] of the deformed complex as the _definition_ of quantum expectation values of [[observables]].

=--

See for instance ([Park, 2.1](#Park))

In order to implement this idea, we need to axiomatize those properties of classical BV complexes and their quantum deformation as above which we demand to be preserved by the generalization away from finite dimensional manifolds. This is what the following definitions do.

+-- {: .num_defn #ClassicalBVComplexAsPoisson0Algebra}
###### Definition

A **classical BV complex** is a [[cochain complex]] equipped with the structure of a [[Poisson 0-algebra]].

=--

+-- {: .num_defn #BeilinsonDrinfeldAlgebra}
###### Definition

A **quantum BV complex** or **Beilinson-Drinfeld algebra** is a $\mathbb{Z}$-[[graded algebra]] $A$ over the ring $\mathbb{R} [ [ \hbar ] ]$ of [[formal power series]] in a formal constant $\hbar$, equipped with a [[Poisson 0-algebra|Poisson bracket]] $\{-,-\}$ of degree 1 and with an operator $\Delta \colon A \to A$ of degree 1 which satisfies:

1. $\Delta^2 = 0$

1. $\Delta( a b) = (\Delta a) b  + (-1)^{\vert a\vert} a (\Delta b) + \hbar \{a,b\}$ for all homogenous elements $a, b \in A$

=--

In ([Gwilliam 2013](#Gwilliam)) this is def. 2.2.5.

+-- {: .num_remark}
###### Remark

A [[Beilinson-Drinfeld algebra]] is _not_ a [[dg-algebra]] with [[differential]] $\Delta$: the Poisson bracket $\hbar \{-,-\}$ measures the failure for the differential to satisfy the [[Leibniz rule]]. In particular the $\Delta$-[[cohomology]] is _not_ an [[associative algebra]].

In this respect the notion of BV-quantization via BD-algebras differs from other traditional notions of BV-quantization, where one demands the quantum BV-complex to be a noncommutative dg-algebra deformation of the classical BV complex. But instead the BD-algebras induced by a [[local action functional]] and varying over open subsets of [[spacetime]]/[[worldvolume]] form a [[factorization algebra]] and _that_ encodes the [[algebra of observables]]: the _[[factorization algebra of observables]]_ (see there for more).

=--

But:

+-- {: .num_defn #ClassicalLimitOfBeilinsonDrinfeldAlgebra}
###### Definition

For $A_\hbar$ a [[Beilinson-Drinfeld algebra]], its [[classical limit]] is the [[tensor product of algebras]]

$$
  A_{\hbar = 0}
  \coloneqq
  A_\hbar \otimes_{\mathbb{R}[ [ \hbar ] ]} \mathbb{R}
$$

hence the result of setting the formal parameter $\hbar$ ("[[Planck's constant]]") to 0.

=--

+-- {: .num_remark}
###### Remark

The classical limit of a [[Beilinson-Drinfeld algebra]] is canonically a classical BV-complex, def. \ref{ClassicalBVComplexAsPoisson0Algebra}.

=--

+-- {: .num_defn #BVQuantizationByBDAlgebra}
###### Definition

For $A_{\hbar = 0}$ a classical BV complex, def. \ref{ClassicalBVComplexAsPoisson0Algebra}, a **BV quantization** of it is a [[Beilinson-Drinfeld algebra]] $A_{\hbar}$, def. \ref{BeilinsonDrinfeldAlgebra} whose classical limit, def. \ref{ClassicalLimitOfBeilinsonDrinfeldAlgebra}, is the given $A_{\hbar = 0}$.

=--

In ([Gwilliam 2013](#Gwilliam)) this is def. 2.2.6.

[[!include action (physics) - table]]

#### Quantum observables by BV-cohomology
 {#PathIntegrationAndQuantumObservablesByBVCohomology}

+-- {: .num_defn }
###### Definition

Given a [[quantum BV-complex]], its [[cochain cohomology]] are the **[[expectation values]] of [[observables]]** of the [[theory (physics)|theory]].

Specifically, an observable is a closed element $f$ in the quantum BV-complex and its _expectation value_ is its image $[f]$ in [[cochain cohomology]].

=--

+-- {: .num_remark }
###### Remark

Given a [[quantum BV-complex]] by def. \ref{BVQuantizationByBDAlgebra}
its [[cochain cohomology]] is, by definition, a perturbation of that of its [[classical limit]] BV complex, def. \ref{ClassicalLimitOfBeilinsonDrinfeldAlgebra}. Accordingly, the quantum observables may be computed from the classical observables by the [[homological perturbation lemma]]. For [[free field theories]] this yields [[Wick's lemma]] and [[Feynman diagrams]] for computing observables. ([Gwilliam 2013, section 2.3](#Gwilliam)).

=--

+-- {: .num_remark }
###### Remark

For local theories (...) [[gauge fixing operator]] (...) [[Hodge theory]] (...)

=--

(...)

### Poincaré duality on Hochschild (co)homology and framed little disk algebra
 {#PoincareDualityOnHochschild}

The [above](#HomologicalIntegration) duality between [[differential forms]] and [[multivector field]] may be understood in a more general context.

Multivector fields may be understood in terms of [[Hochschild cohomology]] of $C$.
Under the identification of
[[Hochschild homology]]/[[cyclic homology]] with  the [[de Rham complex]] the product of the  [[action functional]]
$\exp(i S(-))$ with a formal [[measure]] $vol$
on $C$ is regarded as a [[cycle]] in [[cyclic homology]]. Or rather,
an [[isomorphism]] with [[Hochschild cohomology]] is picked,
and interpreted as a choice of [[volume form]] $vol$ and
$\exp(i S(-))$ is regarded as a cocycle in cyclic cohomology, hence
as a [[multivector field]] whose closure condition
$\Delta \exp(i S(-)) = 0 $
is the quantum master equation of BV-formalism.

By the identification of [[Hochschild cohomology]]
with functions on [[derived loop space]]s we know that
the operator $\Delta$ encodes the rotation of loops.
Accordingly, the resuling [[BV-algebra]] has an interpretation
as an algebra over (the homology of) the
[[framed little disk operad]].

For certain algebras $A$ there exists [[Poincaré duality]] between [[Hochschild cohomology]] and [[Hochschild homology]]

$$
  \tau : HH_i(A) \to HH^{n-i}(A)
$$

([VanDenBergh](#VanDenBergh)) and this takes the [[Connes coboundary operator]] to the [[BV operator]] ([Ginzburg](#Ginzburg)).

### Non-perturbative

On [[non-perturbative quantum field theory|non-perturbative]] enhancements of BV-BRST formalism cast in ([[differential cohesion|differential]]) [[cohesive (infinity,1)-topos|cohesive $(\infty,1)$-topos]] theory:

* {#AlfonsiYoung23} [[Luigi Alfonsi]], [[Charles A. S. Young]], *Towards non-perturbative BV-theory via derived differential cohesive geometry* [[arXiv:2307.15106](https://arxiv.org/abs/2307.15106)]

## Hamiltonian BFV -- Homotopical Poisson reduction
 {#HamiltonianBV}

The following is a rough survey of homotopical Poisson reduction, following ([Stasheff 96](#Stasheff96)).

Let $(X, \{-,-\})$ be a [[smooth manifold|smooth]] [[Poisson manifold]].

Let $A \coloneqq C^\infty(X)$ be its [[associative algebra|algebra]] of [[smooth function]]s.

Consider

* an [[ideal]] $I \subset A$

* that is closed under the Poisson bracket

  $\{I,I\} \subset I$

  (one says that we have _[[first class constraint]]_ or that the 0-locus of $I$ is _coisotropic_)

By the Poisson bracket $I$ acts on $A$. The **Poisson reduction** of $X$ by $I$ is the combined

1. passage to the 0-locus of $I$, which algebraically (dually) is passage to the quotient algebra $A/I$;

1. passage to the quotient of $X$ by the $I$-[[action]], which dually is the passage to the invariant subalgebra $A^I$.

This may be achieved in different orders:

+-- {: .num_defn }
###### Definition

The **Sniatycky-Weinstein reduction is the object

$$
  A_{SW} := (A/I)^I
  \,.
$$

The **Dirac reduction** is

$$
  A_{Dirac} := N(I)/I
$$

where $N(I) = \{f \in A | \{f, I\} \subset I\}$ is the "subalgebra of [[observable]]s".

=--

+-- {: .num_prop }
###### Proposition

These two algebras are [[isomorphic]]

$$
  A_{red} := A_{SW} \simeq A_{Dirac}
  \,.
$$

=--

+-- {: .num_example }
###### Example

Suppose a [[Lie algebra]] $\mathfrak{g}$ acts on the [[Poisson manifold]] $X$, by [[Hamiltonian vector fields]]. This is equivalently encoded in a [[moment map]] $\mu : X \to \mathfrak{g}^*$.

Let then $I$ be the [[ideal]] of functions that vanish on $\mu^{-1}(0)$. This is always coisotropic.

Then $A_{red}$ is the algebraic dual to the preimage $\mu^{-1}(0)$ quotiented by the Lie algebra action: the "constraint surface" quotiented by the symmetries.

In fact, if 0 is a [[regular value]] of $\mu$ then $X_{red} := \mu^{-1}(0)/G$ is a submanifold and

$$
  A_{red} \simeq C^\infty(X_{red})
  \,.
$$

=--

We now discuss the BRST-BV complex for the set of constraints $I$ on $(X, \{-,-\})$, which will be a resolution of $A_{red}$ in the following sense:

* instead of forming the [[quotient]] $X/G$ we form the [[action groupoid]] or [[quotient stack]] $X//G$. More precisely we do this for the infinitesimal action and consider a quotient [[Lie algebroid]];

* instead of forming the intersecton $X|_{I = 0}$ we consider its derived locus.

Let $\{T_1, \cdots, T_N\}$ be any finite set of generators of the [[ideal]] $I$. Then there exists a non-positively graded [[cochain complex]] on the graded algebra

$$
  A \otimes Sym(V)
  \,,
$$

where $V$ is a [[graded vector space]] in non-positive degree and $Sym(V)$ is its symmetric [[tensor algebra]]: the [[Koszul-Tate resolution]] of $C^\infty(X)/I$.

Then on

$$
  A \otimes Sym(V) \otimes Sym(V^*)
$$

(with $V^*$ in non-negative degree)

there is an evident graded generalization of the Poisson bracket on $A$, which is on $V$ and $V^*$ just the canonical pairing.

Write $\{c^\alpha\}$ for the [[basis]] for $V^*$, called the **[[ghost]]**. Write $\{\pi_\alpha\}$ for the dual basis on $V$, called the **ghost momenta**.

+-- {: .num_theorem }
###### Theorem
**(Henneaux, Stasheff et al.)**

([[homological perturbation theory]])

There exists an element

$$
  \Omega \in A \otimes S(V) \otimes S(V^*)
$$

the **BRST-BV charge** such that

* $\{\Omega, \Omega\} =  0$, so that $(A\otimes S(V) \otimes S(V^*), d := \{\Omega, -\})$ is a [[cochain complex]], in fact a [[dg-algebra]];

* the [[cochain cohomology]] is

  $$
    H^0(A \otimes S(V) \otimes S(V^*), d = \{\Omega, -\}) = A/I
    \;
  $$

  $$
    H^{\lt 0}(A \otimes S(V) \otimes S(V^*), d = \{\Omega, -\}) = 0
    \;
  $$

  (which says that this is in non-positive degree a resolution of the constraint locus $A/I$)

* If $I$ is a regular ideal (meaing that $V$ can be chosen to be concentrated in degree 1) or the vanishing ideal of a coisotropic submanifold, then the cohomology in positive degree

  $$
    H^{\bullet \geq 0}(A \otimes S(V) \otimes S(V^*), d = \{\Omega, 0\})
    \simeq
    H^\bullet(CE(A/I, I/I^2))
  $$

  is isomorphic to the [[infinity-Lie algebra cohomology|Lie algebroid cohomology]] of the [[Lie algebroid]] whose [[Lie-Rinehart algebra]] is $(A/I, I/I^2)$

  (which says that in positive degree the BRST-BV complex is a resolution of the [[action Lie algebroid]] of $\{I,-\}$ acting on $X$).

=--

+-- {: .num_theorem }
###### Theorem
**(Oh-Park, Cattaneo-Felder)** If $C \subset X$ is coisotropic, there is an [[L-infinity algebra]]-structure on $\wedge^\bullet  \Gamma(N C)$ such that the induced bracket on $H^0 = A_{red}$ is the given one;

=--

+-- {: .num_theorem }
###### Theorem
**(Schätz)** The BRST-BV complex with $\{-,-\}$ as its Lie bracket is [[quasi-isomorphism|quasi-isomorphic]] to the above.

=--

## Related concepts

* [[BRST complex]]

* [[local BV-BRST complex]]

* [[variational BV-BRST bicomplex]]

* [[homotopy BV algebra]]

* [[quantum L-infinity algebra]]

## References
 {#References}

### General
  {#ReferencesGeneral}

A classical standard references for the Lagrangian formalism is

* {#Henneaux90} [[Marc Henneaux]], _Lectures on the Antifield-BRST formalism for gauge theories_, Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106 ([pdf](http://www.math.uni-hamburg.de/home/schweigert/ws07/henneaux2.pdf))

Similarly the bulk of the textbook

* {#HenneauxTeitelboim} [[Marc Henneaux]], [[Claudio Teitelboim]], _[[Quantization of Gauge Systems]]_, Princeton University Press 1992. xxviii+520 pp.

considers the Hamiltonian formulation. Chapters 17 and 18 are about the Lagrangian ("antifield") formulation, with section 18.4 devoted to the relation between the two.

The [[L-infinity algebroid]]-structure of the [[local BV-BRST complex]] on the [[jet bundle]] is made manifest in

* {#Barnich10} [[Glenn Barnich]], equation(3) of _A note on gauge systems from the point of view of Lie algebroids_, in P. Kielanowski, V. Buchstaber, A. Odzijewicz,
M. Schlichenmaier, T Voronov,  (eds.) XXIX Workshop on Geometric Methods in Physics, vol. 1307 of AIP Conference Proceedings, 1307, 7 (2010) ([arXiv:1010.0899](https://arxiv.org/abs/1010.0899), [doi:/10.1063/1.3527427]( https://doi.org/10.1063/1.3527427))

Formulation as [[homotopy AQFT]]:

* {#BeniniBruinsmaSchenkel19} [[Marco Benini]], Simen Bruinsma, [[Alexander Schenkel]], _Linear Yang-Mills theory as a homotopy AQFT_ ([arXiv:1906.00999](https://arxiv.org/abs/1906.00999))

Review in the context of [[higher geometry]]:

* [[Luigi Alfonsi]], §5 in: *Higher geometry in physics*, in: *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.07308](https://arxiv.org/abs/2312.07308)]

### Lagrangian BV
 {#ReferencesForLagrangianBV}

#### For Lagrangian theories
 {#ReferencesForLagrangianBVForLagrangianTheories}

The original articles are

* {#BV81} [[Igor Batalin]], [[Grigori Vilkovisky]],  _Gauge Algebra and Quantization_, Phys. Lett. B 102 (1): 27–31, 1981 (<a href="https://doi.org/10.1016/0370-2693(81)90205-7">doi:10.1016/0370-2693(81)90205-7</a>)

* {#BV83} [[Igor Batalin]], [[Grigori Vilkovisky]],  (1983). _Quantization of Gauge Theories with Linearly Dependent Generators_, Phys. Rev. D 28 (10): 2567–2582. doi:10.1103/PhysRevD.28.2567.  Erratum-ibid. 30 (1984) 508 doi:10.1103/PhysRevD.30.508

* [[Igor Batalin]], [[Grigori Vilkovisky]], _Existence Theorem For Gauge Algebra_, J. Math. Phys. 26 (1985) 172-184.

Review:

* [[Joaquim Gomis]], J. Paris, S. Samuel, _Antibrackets, Antifields and Gauge Theory Quantization_ ([arXiv:hep-th/9412228](http://arxiv.org/abs/hep-th/9412228))

* {#Park} J. Park, _Pursuing the quantum world_ ([pdf](http://cds.cern.ch/record/638963/files/0308130.pdf))

* [[Alberto S. Cattaneo]], [[Pavel Mnev]], [[Michele Schiavina]], *BV Quantization*, [[Encyclopedia of Mathematical Physics 2nd ed]] [[arXiv:2307.07761](https://arxiv.org/abs/2307.07761)]

Geometrical aspects were pioneered in

* [[Albert Schwarz]], _[[semiclassical approximation|Semiclassical approximation]] in Batalin-Vilkovisky formalism_, Comm. Math. Phys.  __158__ (1993), no. 2, 373--396, [euclid](http://projecteuclid.org/euclid.cmp/1104254246)

* M. Alexandrov, [[M. Kontsevich]], [[Albert Schwarz]], O. Zaboronsky, _The geometry of the master equation and topological quantum field theory_, Int. J. Modern Phys. A 12(7):1405–1429, 1997, [hep-th/9502010](http://arxiv.org/abs/hep-th/9502010)

A systematic account of the classical master equation is also in

* [[David Kazhdan]], _The classical master equation in the finite-dimensional case_ ([pdf](http://math.berkeley.edu/~theojf/KazhdanNotes.pdf))

* {#FelderKazhdan} [[Giovanni Felder]], [[David Kazhdan]], _The classical master equation_ ([arXiv:1212.1631](http://arxiv.org/abs/1212.1631))

Generalization from [[Lie algebra]]-actions to actual [[algebraic group]]-actions:

* [[Marco Benini]], [[Pavel Safronov]], [[Alexander Schenkel]], *Classical BV formalism for group actions* ([arXiv:2104.14886](https://arxiv.org/abs/2104.14886))

Other discussions include

* [[Domenico Fiorenza]], _An introduction to the Batalin-Vilkovisky formalism_, Lecture given at the Recontres Mathématiques de Glanon, July 2003, [arXiv:math/0402057](http://arxiv.org/abs/math/0402057)

* [[Alberto Cattaneo]], _From topological field theory to deformation quantization and reduction_, ICM 2006. ([pdf](http://www.math.uzh.ch/fileadmin/math/preprints/icm.pdf))

*  M. Bächtold, _On the finite dimensional BV formalism_, 2005. ([pdf](http://www.math.uzh.ch/reports/04_05.pdf))

* Carlo Albert, Bea Bleile, [[Jürg Fröhlich]], _Batalin-Vilkovisky integrals in finite dimensions_, [arXiv/0812.0464](http://eprintweb.org/S/article/math-ph/0812.0464)

* Qiu and Zabzine, _Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and their Applications_, [arXiv/1105.2680](http://arxiv.org/pdf/1105.2680v2).

The [[perturbative quantum field theory|perturbative quantization]] of [[gauge theories]] ([[Yang-Mills theory]]) in [[causal perturbation theory]]/[[perturbative AQFT]] is discussed (for trivial [[principal bundles]] and restricted to [[gauge invariant observables]]) via [[BRST-complex]]/[[BV-formalism]] in

* {#Hollands07} [[Stefan Hollands]], _Renormalized Quantum Yang-Mills Fields in Curved Spacetime_, Rev. Math. Phys.20:1033-1172, 2008 ([arXiv:0705.3340](https://arxiv.org/abs/0705.3340))

* {#FredenhagenRejzner11a} [[Klaus Fredenhagen]], [[Kasia Rejzner]], _Batalin-Vilkovisky formalism in the functional approach to classical field theory_, Commun. Math. Phys. 314(1), 93–127 (2012) ([arXiv:1101.5112](https://arxiv.org/abs/1101.5112))

* {#FredenhagenRejzner11b} [[Klaus Fredenhagen]], [[Kasia Rejzner]], _Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory_, Commun. Math. Phys. 317(3), 697–725 (2012) ([arXiv:1110.5232](https://arxiv.org/abs/1110.5232))

* {#Rejzner11} [[Katarzyna Rejzner]], _Batalin-Vilkovisky formalism in locally covariant field theory_ ([arXiv:1111.5130](https://arxiv.org/abs/1111.5130))

* {#Rejzner13} [[Katarzyna Rejzner]], _Remarks on local symmetry invariance in perturbative algebraic quantum field theory_ ([arXiv:1301.7037](https://arxiv.org/abs/1301.7037))

and surveyed in

* {#Rejzner16} [[Kasia Rejzner]], section 7 of _Perturbative algebraic quantum field theory_ Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

Discussion for field theories with [[boundary conditions]] and going in the direction of [[extended TQFT|extended field theory]]/[[local quantum field theory]] is in

* {#CattaneoMnevReshetikhin12} [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], _Classical BV theories on manifolds with boundary_, [arXiv:1201.0290](http://arxiv.org/abs/1201.0290); _Classical and quantum Lagrangian field theories with boundary_, [arXiv:1207.0239](http://arxiv.org/abs/1207.0239);  _Perturbative quantum gauge theories on manifolds with boundary_, [arxiv/1507.01221](http://arxiv.org/abs/1507.01221)

A discussion of BV-BRST formalism in the general context of [[perturbative quantum field theory]] is in

* [[Kevin Costello]], _[[Renormalization and Effective Field Theory]]_

Relation to [[Feynman diagrams]] is made explicit in

* [[Owen Gwilliam]], [[Theo Johnson-Freyd]], _How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism_ (2011) ([arXiv:1202.1554](http://arxiv.org/abs/1202.1554))

See also

* [[Theo Johnson-Freyd]], _Star-quantization via lattice topological field theory_, talk at [String-Math 2013](http://scgp.stonybrook.edu/events/event-pages/string-math-2013) ([pdf](http://media.scgp.stonybrook.edu/presentations/20130618_johnson-freyd.pdf))

The interpretation of the BV quantum master equation as a description of closed differential forms acting as measures on infinite-dimensional spaces of fields is described in

* [[Edward Witten]], _A note on the antibracket formalism_, Modern Physics Letters A, __5__, n. 7, 487--494, [MR91h:81178](http://www.ams.org/mathscinet-getitem?mr=91h:81178), [doi](http://dx.doi.org/10.1142/S0217732390000561), [scan](http://ccdb4fs.kek.jp/cgi-bin/img_index?9004090)
{#Witten90}

* {#Schwarz92} [[Albert Schwarz]], _Geometry of Batalin-Vilkovisky quantization_ ([arXiv:hep-th/9205088](http://arxiv.org/abs/hep-th/9205088))

This isomorphisms between the [[de Rham complex]] and the complex of [[polyvector field]]s is reviewed for instance on p. 3 of

* Thomas Willwacher, Damien Calaque _Formality of cyclic cochains_ ([arXiv:0806.4095](http://arxiv.org/abs/0806.4095))
{#WillwacherCalaque}

and in section 2 of

* [[Alberto Cattaneo]], [[Domenico Fiorenza]], Riccardo Longoni, _On the Hochschild-Kostant-Rosenberg map for graded manifolds_ ([pdf](http://www.math.uzh.ch/fileadmin/math/preprints/05-06.pdf))
{#CattaneoFiorenzaLongoni}

A discussion in the general context of [[BV-algebras]] is in

* {#Roger} [[Claude Roger]], _Gerstenhaber and Batalin-Vilkovisky algebras_, Archivum mathematicum, Volume 45 (2009), No. 4 ([pdf](http://www.emis.de/journals/AM/09-4/roger.pdf))

The generalization of this to [[Poincaré duality]] on Hochschild (co)homollogy is in

* M. Van den Bergh, _A relation between Hochschild homology and cohomology for Gorenstein rings_ . Proc. Amer. Math. Soc. 126 (1998), 1345–1348; ([JSTOR](http://www.jstor.org/stable/118786))

  Correction: Proc. Amer. Math. Soc. 130 (2002), 2809–2810.
{#VanDenBergh}

with more on that in

* U. Krähmer, _Poincaré duality in Hochschild cohomology_
  ([pdf](http://www.maths.gla.ac.uk/~ukraehmer/brussels.pdf))

* [[Victor Ginzburg]], _Calabi-Yau Algebras_  ([arXiv](http://arxiv.org/abs/math.AG/0612139))
{#Ginzburg}

The application in [[string theory]]/[[string field theory]] is discussed in

* B. Zwiebach, _Closed string field theory: Quantum action and the B-V master equation_, Nucl. Phys. B 390, 33-152 (1993)

A mathematically oriented reformulation of some of this (in the context of [[TCFT]] ) is in

* [[Kevin Costello]], _The Gromov-Witten potential associated to a TCFT_ ([pdf](http://www.math.northwestern.edu/~costello/0509264.pdf))

Here the analog of the [[virtual fundamental class]] on the [[moduli space]] of surfaces is realized as a solution to the BV-master equation.

The perspective on the BV-complex as a [[schreiber:derived critical locus]] is indicated in

* {#CostelloGwilliam} [[nLab:Kevin Costello]], [[nLab:Owen Gwilliam]], _Factorization algebras in perturbative quantum field theory -- Derived critical locus_ ([web](http://math.northwestern.edu/~costello/factorization_public.html#[[Derived%20critical%20locus]]))

A clear discussion of the BV-complex as a means for homological [[path integral]] [[quantization]] is in

* {#Gwilliam} [[Owen Gwilliam]], _Factorization algebras and free field theories_ PhD thesis (2013) ([[GwilliamThesis.pdf:file]])

Related Chern-Simons type graded action functionals are discussed also in

* M.V. Movshev, [[Albert Schwarz]], _Generalized Chern-Simons action and maximally supersymmetric gauge theories_ ([arXiv:1304.7500](http://arxiv.org/abs/1304.7500))

Lectures, discussing also the relation to the [[graph complex]] are

* Jian Qiu, [[Maxim Zabzine]], _Introduction to graded geometry, Batalin-Vilkovisky formalism and their applications_, [arxiv/1105.2680](http://arxiv.org/abs/1105.2680); _Knot weight systems from graded symplectic geometry_, [arxiv/1110.5234](http://arxiv.org/abs/1110.5234); _Odd Chern-Simons theory, Lie algebra cohomology and characteristic classes_, [arxiv/0912.1243](http://arxiv.org/abs/0912.1243)
* Klaus Fredenhagen, Katarzyna Rejzner, _Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory_, [arxiv/1110.5232](http://arxiv.org/abs/1110.5232)

Gluing aspects are in focus of the program explained in

* Alberto S. Cattaneo, Pavel Mnev, Nicolai Reshetikhin, _Perturbative BV theories with Segal-like gluing_, [arxiv/1602.00741](http://arxiv.org/abs/1602.00741)

On extracting [[L-infinity algebras|$L_\infty$-algebras]] from [[BV-formalism]] around a solutio, encoding [[tree level]] [[Feynman diagrams]]:

* [[Maxim Grigoriev]], [[Dmitry Rudinsky]], *Notes on the $L_\infty$-approach to local gauge field theories*, Journal of Geometry and Physics **190** (2023) 104863 [[arXiv2303.08990](https://arxiv.org/abs/2303.08990), [doi:10.1016/j.geomphys.2023.104863](https://doi.org/10.1016/j.geomphys.2023.104863)]

#### For non-Lagrangian theories
 {#ReferencesForNonLagrangianEquations}

The whole formalism also applies to the locus of solutions of [[differential equation]]s that are not necessarily the [[Euler-Lagrange equation]]s of an [[action functional]]. Discussion of this more general case is in

* D.S. Kaparulin, S.L. Lyakhovich, [[A.A. Sharapov]], _Local BRST cohomology in (non-)Lagrangian field theory_ ([arXiv:1106.4252](http://arxiv.org/abs/1106.4252))

* D.S. Kaparulin, S.L. Lyakhovich, A.A. Sharapov, _Rigid Symmetries and Conservation Laws in Non-Lagrangian Field Theory _ ([arXiv:1001.0091](http://arxiv.org/abs/1001.0091))

* S.L. Lyakhovich, A.A. Sharapov, _Quantizing non-Lagrangian gauge theories: an augmentation method_ ([arXiv:hep-th/0612086](http://arxiv.org/abs/hep-th/0612086))

* S.L. Lyakhovich, A.A. Sharapov, _BRST theory without Hamiltonian and Lagrangian_ ([arXiv:hep-th/0411247](http://arxiv.org/abs/hep-th/0411247))

Section 4.5 of

* [[Gennadi Sardanashvily]], _Advanced Classical Field Theory_ (2009) ([pdf](http://www.g-sardanashvily.ru/book09.pdf))

This also makes the connection to

* P. Olver, _Applications of Lie Groups to Differential Equations_ (Springer-Verlag, Berlin) (1986)

See also via [[quantum L-infinity algebras]]:

* [[Martin Doubek]], [[Branislav Jurčo]], [[Ján Pulmann]], *Quantum L∞ Algebras and the Homological Perturbation Lemma*  Comm. Math. Phys. **367** (2019) 215–240 [[arXiv:1712.02696](https://arxiv.org/abs/1712.02696), [doi:10.1007/s00220-019-03375-x](https://doi.org/10.1007/s00220-019-03375-x)]

* [[Branislav Jurčo]], [[Tommaso Macrelli]], [[Christian Sämann]], [[Martin Wolf]], *Loop Amplitudes and Quantum Homotopy Algebras*, Journal of High Energy Physics **2020** 3 (2020) [<a href="https://doi.org/10.1007/JHEP07(2020)003">doi:10.1007/JHEP07(2020)003</a>, [arXiv:1912.06695](https://arxiv.org/abs/1912.06695)]

Review:

* Sebastian Albrecht: *Formulation of Batalin-Vilkovisky Field Theories as Homotopy Lie Algebras*, Munich (2022) [[pdf](https://www.theorie.physik.uni-muenchen.de/TMP/theses/sebastian_albrecht.pdf)]

#### For CFT/vertex algebras

A class of "free" vertex algebras are also quantized using Batalin-Vilkovisky formalism, with results on quantization of [[BCOV theory]] important for understanding [[mirror symmetry]], in

* Si Li, _Vertex algebras and quantum master equation_, [arxiv/1612.01292](https://arxiv.org/abs/1612.01292)

### Hamiltonian BFV

BRST formalism is discussed in

* [[Glenn Barnich]], Friedemann Brandt, [[Marc Henneaux]], _Local BRST cohomology in gauge theories_, Phys. Rep. __338__ (2000), no. 5, 439–569, [hep-th/0002245](http://xxx.lanl.gov/abs/hep-th/0002245), <a href="http://dx.doi.org/10.1016/S0370-1573(00)00049-1">doi</a>

The original references on Hamiltonian BFV formalism are

* [[Igor Batalin]], [[Grigori Vilkovisky]], _Relativistic S-matrix of dynamical systems with boson and fermion constraints_ , Phys. Lett. __B69__ (1977) 309-312;

* [[Igor Batalin]], [[Efim Fradkin]], _A generalized canonical formalism and quantization of reducible gauge theories_ , Phys. Lett. B122 (1983) 157-164.

Homological Poisson reduction is discussed in

* {#Stasheff96} [[Jim Stasheff]], _Homological Reduction of Constrained Poisson Algebras_, J. Differential Geom. Volume 45, Number 1 (1997), 221-240 ([arXiv:q-alg/9603021](http://arxiv.org/abs/q-alg/9603021), [Euclid](https://projecteuclid.org/euclid.jdg/1214459757))

Remarks  on the [[homotopy theory]] interpretation of BRST-BV are in

* [[Jim Stasheff]], _The (secret?) homological algebra of the Batalin-Vilkovisky approach_ ([arXiv](http://arxiv.org/abs/hep-th/9712157))

A standard textbook on the application of BRST-BV to [[gauge theory]] is

* [[Marc Henneaux]], [[Claudio Teitelboim]], _Quantization of gauge systems_, Princeton University Press 1992. xxviii+520 pp.

* [[Glenn Barnich]], Friedemann Brandt, [[Marc Henneaux]], _Local BRST cohomology in the antifield formalism. I. General theorems_, [euclid](http://projecteuclid.org/euclid.cmp/1104275094), [MR97c:81186](http://www.ams.org/mathscinet-getitem?mr=97c:81186)

 * _Basics of Poisson reduction_ ([blog](http://golem.ph.utexas.edu/category/2008/07/poisson_reduction.html))

 * Alejandro Cabrera, _Homological BV-BRST methods: from QFT to Poisson reduction_ ([pdf](http://www.math.uni-hamburg.de/home/schreiber/Charla_IMPA_BRST.pdf))

 * [[Jeremy Butterfield]], _On symplectic reduction in classical mechanis_ ([pdf](http://philsci-archive.pitt.edu/archive/00002373/01/ButterfieldNHSympRed.pdf))

* S. Lyakhovich, A. Sharapov,  _BRST theory without Hamiltonian and Lagrangian_ ([pdf](http://arxiv.org/PS_cache/hep-th/pdf/0411/0411247v2.pdf))

* [[Florian Schätz]], _BFV-complex and higher homotopy structures_ ([pdf](http://www.math.ist.utl.pt/~fschaetz/BFV-complex.pdf))

* MO: [what is the BV formalism and its uses](http://mathoverflow.net/questions/30352/what-is-the-batalin-vilkovisky-formalism-and-what-are-its-uses-in-mathematics/32443#32443)

### Multisymplectic BRST
 {#ReferencesMultisymplectic}

In the context of [[multisymplectic geometry]]

* Sean Hrabak, _Ambient Diffeomorphism Symmetries of Embedded Submanifolds, Multisymplectic BRST and Pseudoholomorphic Embeddings_ ([arXiv:math-ph/9904026](http://arxiv.org/abs/math-ph/9904026))

* Sean Hrabak, _On a Multisymplectic Formulation of the Classical BRST symmetry for First Order Field Theories Part I: Algebraic Structures_ ([arXiv:math-ph/9901012](http://arxiv.org/abs/math-ph/9901012))

* Sean Hrabak, _On a Multisymplectic Formulation of the Classical BRST Symmetry for First Order Field Theories Part II: Geometric Structures_ ([arXiv:math-ph/9901013](http://arxiv.org/abs/math-ph/9901013))

based on

* I. Kanatchikov, _On field theoretic generalizations of a Poisson algebra_, Rept.Math.Phys. 40 (1997) 225 ([arXiv:hep-th/9710069](http://arxiv.org/abs/hep-th/9710069))

[[!redirects Batalin-Vilkovisky formalism]]
[[!redirects Batalin-Vilkovisky theory]]

[[!redirects BV-formalism]]
[[!redirects BV formalism]]

[[!redirects BV-theory]]
[[!redirects BV theory]]
[[!redirects BV-BRST formalism]]
[[!redirects BV-BRST quantization]]
[[!redirects BV-BRST quantisation]]
[[!redirects BV/BRST formalism]]
[[!redirects BV/BRST quantization]]
[[!redirects BV/BRST quantisation]]

[[!redirects BRST complex]]
[[!redirects BRST-complex]]

[[!redirects BV-BRST complex]]
[[!redirects BRST-BV complex]]
[[!redirects BV-BRST complexes]]
[[!redirects BRST-BV complexes]]

[[!redirects BRST-BV formalism]]

[[!redirects antifield formalism]]

[[!redirects BV-differential]]
[[!redirects BV-differentials]]

[[!redirects BV differential]]
[[!redirects BVdifferentials]]

[[!redirects BV-BRST-differential]]
[[!redirects BV-BRST-differentials]]

[[!redirects BV-BRST differential]]
[[!redirects BV-BRST differentials]]

[[!redirects BV-complex]]
[[!redirects BV-complexes]]
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[[!redirects BV complexes]]

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[[!redirects classical BV-complexes]]
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[[!redirects classical BV complexes]]

[[!redirects BV-quantization]]
[[!redirects BV quantization]]

[[!redirects quantum BV-complex]]
[[!redirects quantum BV-complexes]]
[[!redirects quantum BV complex]]
[[!redirects quantum BV complexes]]

[[!redirects BV-BRST cohomology]]
[[!redirects BV-BRST cohomologies]]

[[!redirects BV-BRST theory]]

[[!redirects BV-BRST Lagrangian density]]
[[!redirects BV-BRST Lagrangian densities]]

[[!redirects BFV-formalism]]

[[!redirects BV-BFV formalism]]


BV-BRST variational bicomplex

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[variational calculus]], the _[[variational bicomplex]]_ of a [[fiber bundle]] serves to give a local description of the [[variational derivative]] $\delta$. In applications one is typically interested in 1) specifying a [[differential equation]] which carves out its [[solution]] locus ([[diffiety]]) inside the [[jet bundle]] and 2) forming the [[quotient]] by [[infinitesimal symmetries]] of the [[PDE]]. Moreover, it is usually useful to perform both these operations in [[homotopy theory]]/[[higher differential geometry]]. Using tools from [[homological algebra]], under suitable conditions the resulting combined [[homotopy intersection]] in and [[homotopy quotient]] is modeled by a combination of the [[variational bicomplex]] with a _[[local BV-BRST complex]]_.

## Definition

+-- {: .num_defn #BVVariationalBicomplex}
###### Definition
**([[variational BV-bicomplex]])**

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] ([this def.](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime))
equipped with a [[gauge parameter bundle]]   $\mathcal{G}$ ([this def.\](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#GaugeParametrizedInfinitesimalGaugeTransformation)) which is closed ([this def..](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#GaugeParametrizedInfinitesimalGaugeTransformation#GaugeParametersClosed)).
Consider the [[Lie algebroid]] $E/(\mathcal{G} \times_\Sigma T \Sigma)$ from [this example](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#GaugeParametrizedInfinitesimalGaugeTransformation#GaugeParametersClosed#LocalOffShellBRSTComplex),
whose [[Chevalley-Eilenberg algebra]] is the [[local BRST complex]] of the theory.

Then its [[Weil algebra]] $W(E/(\mathcal{G} \times_\Sigma T \Sigma))$
has as differential the [[variational derivative]] ([this def.](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#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 $\mathcal{E}^\infty_{BV}$ ([this def.](geometry+of+physics+--+A+first+idea+of+quantum+field+theory#DerivedProlongedShell)) has differential

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

Since $s$ is the [[BV-BRST differential]] this defines the "BV-BRST [[variational bicomplex]]".

=--

[[!redirects variational BV-bicomplex]]
[[!redirects variational BRST-bicomplex]]
[[!redirects variational BV-BRST-bicomplex]]

[[!redirects variational BV bicomplex]]
[[!redirects variational BRST bicomplex]]
[[!redirects variational BV-BRST bicomplex]]


BV-Lagrangian density

#Contents#
* table of contents
{:toc}

## Idea

In [[BV-BRST formalism]] the plain [[Lagrangian density]] $\mathbf{L}$ of a given [[Lagrangian field theory]] is accompanied by an auxiliary Lagrangian density $\mathbf{L}_{BRST}$ depending on [[auxiliary fields]] such as [[ghost fields]] and [[antifields]], such that the [[local antibracket]] with the sum $-\mathbf{L} + \mathbf{L}_{BRST}$ yields the [[BV-BRST differential]]

$$
  s
  \;=\;
  \left\{
     - \mathbf{L}
     + \mathbf{L}_{BRST}
     \,,\,
     -
  \right\}
$$

which models the [[derived critical locus]] $\delta_{EL}\mathbf{L} \simeq 0$ inside the [[homotopy quotient]] modeled by the [[BRST complex]] of the  theory.

## Related concepts

* [[gauge fixing Lagrangian density]]

[[!redirects BV-Lagrangian densities]]


BV-Laplacian

#Contents#
* table of contents
{:toc}

## Idea

The perturbation of the [[differential]] in the passage from a [[classical BV-complex]] to a [[quantum BV-complex]] is called a _BV-Laplacian_. As discussed at _[[BV-complex]]_ the BV Laplacian is a [[homological algebra|homological]] incarnation of a [[measure]].

## Examples

In the archetypical example of the BV-complex of a [[smooth manifold]] of finite [[dimension]] $n$ and equipped with a [[volume form]] $vol$, the BV-Laplacian is the operation on [[multivector fields]] which is the image of the [[de Rham differential]] under the [[isomorphism]] between [[multivector fields]] and [[differential forms]] induced by contraction with the [[volume form]].

Specifically over a [[Cartesian space]] $\mathbb{R}^n$ with its canonical volume form and with $\{x^i, \xi_i\}$ the canonical [[basis]] elements of the algebra of multivectorfields, the BV-Laplacian has the expression

$$
  \Delta = \sum_{i = 1}^n \frac{\partial}{ \partial x^i} \frac{\partial}{\partial \xi_i}
$$

(it [[differentiation|differentiates]] the function and "removes a vector field" for each derivative, being the dual operation to "wedging with $d x^i$").

## Related concepts

[[BV-BRST formalism]]

* [[antifield]]

* [[ghost]]

[[!include action (physics) - table]]

## References

See at _[[BV-BRST formalism]]_ for general references.

Discussion of the BV-operation in [[renormalization]] and [[effective field theory]] is in section 10.1 of

* [[Kevin Costello]], _Renormalisation and the Batalin-Vilkovisky formalism_ ([arXiv:0706.1533](http://arxiv.org/abs/0706.1533))

[[!redirects BV Laplacian]]

[[!redirects BV operator]]
[[!redirects BV-operator]]


BV-operator

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of _BV-operator_ has two different but (somewhat subtly) related meanings:

1. In [[perturbative quantum field theory]] (discussed [below](#InPerturbativeQuantumFieldTheory)) the _BV-operator_ or _BV-Laplacian_ ([Batalin-Vilkovisky 81](#BatalinVilkovisky81)) is essentially the difference between the action of the [[BV differential]] on the [[algebra of observables]] before and after [[quantization]] of the [[free field theory]] around which the [[perturbative quantum field theory|perturbative quantization]] is considered; it is a quantum correction $i \hbar \Delta_{BV}$ (of order of [[Planck's constant]] $\hbar$) to the [[BV differential]] $s_{BV} = \{-S',-.\}$. Where the condition $\left(\{-S',-\}\right)^2 = 0$ for the [[BV-differential]] to be a [[differential]] is called the "[[master equation]]" in [[BV-BRST theory]], the quantum corrected version $\left( \{-S',-\} + i \hbar \Delta\right)^2 = 0$ is called the _[[quantum master equation]]_.

1. In [[higher algebra]], under the identification of a [[BV-algebra]] with the [[chain homology]] of a [[little k-cubes operad|E2-algebra]], the _BV-operator_ corresponds to the operation of rotating a little disk around.

For the relation between the two see _[[relation between BV and BD]]_.

## In perturbative quantum field theory
 {#InPerturbativeQuantumFieldTheory}

In [[perturbative quantum field theory]] the BV-operator $i \hbar \Delta$ may be understood intuitively as reflecting the contribution of the [[Gaussian measure]] in the [[path integral]] of the [[free field theory]] around which the perturbative quantization takes place. This intuition may be made precise for finite-dimensional toy path integrals. This we discuss in:

* _[In finite-dimensional toy path integrals](#ForFiniteDimensionalToyPathIntegrals)_

In the rigorous construction of [[relativistic field theory|relativistic]] [[perturbative quantum field theory]] via [[causal perturbation theory]]/[[perturbative AQFT]] there is a rigorous incarnation of the BV-operator ([Fredenhagen-Rejzner 11b, section 2](#FredenhagenRejzner11b), [Rejzner 11, section 5.1.2](#Rejzner11)): The would-be path integral is reflected in the [[perturbative S-matrix]], hence in the [[time ordered products]], and the BV-operator on [[regular polynomial observables]] is the difference between the classical [[BV-differential]] and its [[conjugation]] into the [[time-ordered products]] (def. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator} below). This we discuss in

* _[In causal perturbation theory](#InCausalPerturbationTheory)_

### For finite-dimensional toy path integrals
 {#ForFiniteDimensionalToyPathIntegrals}

If $X$ is a [[finite number|finite]] [[dimension|dimensional]] [[closed manifold|closed]] [[orientation|oriented]] [[smooth manifold]], then [[integration of differential forms]] of top degree over $X$ may be identified with sending such differential forms to their [[image]] in the [[de Rham cohomology]] of $X$.

This may be slightly reformulated: Fixing a [[volume form]] $\mu$ on $X$ it induces by contraction with [[vector fields]] degreewise a [[linear isomorphism]]

$$
  \mu \;\colon\;
  \Gamma_X(\wedge^n T^\ast X)
   \overset{\simeq}{\longrightarrow}
  \Gamma_X(\wedge^{dim(X)-n} T X)
$$

between the spaces of [[differential n-forms]] on $X$ and the space of [[multivector fields]] on $X$ of degree $n-dim(X)$. Under this isomorphism the [[de Rham differential]] induces a differential on [[multivector fields]]:

$$
  \Delta_{BV}
  \;\coloneqq\;
  \mu \circ d_{dR} \circ \mu^{-1}
$$

This $\Delta_{BV}$ is the _BV-operator_ in this simple situation.

The above statement about [[integration]] now translates into saying that for $f$ any [[smooth function]] on $X$, then its [[integration of differential forms]] $\int f \mu$ may be identified with the [[image]] of $f$ in the [[chain homology]] of the BV-operator $\Delta_{BV}$.

If one thinks of $X$ as a space of [[field configurations]] and of  $f = \exp(i \hbar S)$ as an exponentiated [[action functional]], the one may think of this integral $\int \exp(i \hbar S) \mu$ as the finite-dimensional toy version of a [[path integral]].

While this is in general not defined in the actual non-finite dimensional situations in [[field theory]], the above re-formulation in terms of the [[chain homology]] of a BV-operator does make sense whenever an appropriate kind of differential is given. One may then try to axiomatize which chain complexes qualify as BV-complex and try to interpret their chain homology as computing perturbative [[path integrals]].

For more on this perspective see at _[[BV-BRST formalism]]_ the section _[Quantum BV as homological (path-)integration](BV-BRST+formalism#HomologicalIntegration)_

### In causal perturbation theory
  {#InCausalPerturbationTheory}

For more context for the following see at _[[A first idea of quantum field theory]]_ the chapter _[Free quantum fields](A+first+idea+of+quantum+field+theory#FreeQuantumFields)_.

#### Background

Recall the following context

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] admitting a [[gauge fixing]], and let $\mathbf{L}' - \mathbf{L}'_{BRST}$ be its BV-BRST [[Lagrangian density]] after [[gauge fixing]] ([this def.](A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity)), so that the gauge-fixed [[local BRST complex|local BV-BRST differential]] is given by the [[local antibracket]] as

$$
  s'
    \;=\;
  \left\{ -\mathbf{L}' + \mathbf{L}'_{BRST}, - \right\}
$$

The corresponding global BV-BRST differential on [[regular polynomial observables]] is ([this def.](A+first+idea+of+quantum+field+theory#ComplexBVBRSTGlobal))

$$
  \label{GaugeFixedGlobalBVBRSTDifferentialRecalled}
  \left\{ -S' + S'_{BRST} \;,\; -\right\}
  \;\coloneqq\;
  \left\{ -\tau_\Sigma\mathbf{L}'(x) + \tau_\Sigma\mathbf{L}'_{BRST}(x), - \right\}
  \;\colon\;
  PolyObs(E)_{reg} \longrightarrow PolyObs(E)_{reg}
  \,.
$$

By definition of [[gauge fixing]] ([this def.](A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity)), the [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] for $\mathbf{L}'$ are [[Green hyperbolic differential equations|Green hyperbolic]] and hence have a [[causal propagator]] $\Deta = \Delta_+ - \Delta_-$ and admit a compatible _[[Wightman propagator]]_ $\Delta_H = \tfrac{i}{2}(\Delta_+ - \Delta_-) + H$ and the corresponding [[Feynman propagator]] $\Delta_F \coloneqq \tfrac{i}{2}(\Delta_+ + \Delta_-) + H$.  The [[star products]] with respect to these ([this def.](star+product#PropagatorStarProduct)) on [[regular polynomial observables]]

$$
  \star_H, \star_F
  \;\colon\;
  PolyObs(E)_{reg}[ [\hbar] ]
    \longrightarrow
  PolyObs(E)_{reg}[ [\hbar] ]
$$

are, respectively, the [[Wick algebra]] product ([[operator product]], see [this def.](Wick+algebra#NormalOrderedProductNotation))

$$
  A_1 A_2
  \;\coloneqq\;
  A_1 \star_H A_2
  \;\coloneqq\;
  ((-) \cdot (-))
  \circ
  \exp\left(
    \hbar
    \underset{\Sigma \times \Sigma}{\int}
      \Delta_{H}(x,y)^{a b}
      \frac{\delta}{\delta \mathbf{\Phi}^a(x)}
        \otimes
      \frac{\delta}{\delta \mathbf{\Phi}^b(y)}
      \, dvol_\Sigma(x) \, dvol_\Sigma(y)
  \right)
  (A_1 \otimes A_2)
$$

and the [[time-ordered product]] (see again [this def.](Wick+algebra#NormalOrderedProductNotation))

$$
  T(A_1 A_2)
  \;\coloneqq\;
  A_1 \star_F A_2
  \;\coloneqq\;
  ((-) \cdot (-))
  \circ
  \exp\left(
    \hbar
    \underset{\Sigma \times \Sigma}{\int}
      \Delta_{F}(x,y)^{a b}
      \frac{\delta}{\delta \mathbf{\Phi}^a(x)}
        \otimes
      \frac{\delta}{\delta \mathbf{\Phi}^b(y)}
      \, dvol_\Sigma(x)\, dvol_\Sigma(y)
  \right)
  (A_1 \otimes A_2)
  \,,
$$

Since the [[Feynman propagator]] is symmetric ([this prop.](A+first+idea+of+quantum+field+theory#SymmetricFeynmanPropagator)), the latter [[time ordered product]] on [[regular polynomial observables]] is [[isomorphism|isomorphic]] (via [this prop.](star+product#SymmetricContribution)) to the pointwise product, via

$$
  \label{RecallIsomorphsimTimeOrdering}
  \mathcal{T}A
  \;\coloneqq\;
  \exp\left(
    \tfrac{1}{2}
    \hbar
    \underset{\Sigma}{\int}
     \Delta_F(x,y)^{a b}
     \frac{\delta^2}{\delta \mathbf{\Phi}^a(x) \delta \mathbf{\Phi}^b(y)}
  \right)
  A
$$

as

$$
  A_1 \star_{F} A_2
  \;=\;
  \mathcal{T}( \mathcal{T}^{-1}A_1 \cdot \mathcal{T}^{-1}A_2 )
$$

([this prop.](time-ordered+product#IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise)).

#### The BV-Operator
 {#BVOperatorInCausalPerturbationTheory}

+-- {: .num_defn #AntibracketTimeOrdered}
###### Definition
**([[time-ordered product|time-ordered]] [[antibracket]])

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] ([this def.](A+first+idea+of+quantum+field+theory#FreeFieldTheory)) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ ([this def.](A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity)) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$.

Then the _time-ordered global [[antibracket]]_ on [[regular polynomial observables]]

$$
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
   \otimes
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \overset{\{-,-\}_{\mathcal{T}}}{\longrightarrow}
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$

is the [[conjugation]] of the global [[antibracket]] ([this def. ](A+first+idea+of+quantum+field+theory#ComplexBVBRSTGlobal)) by the time-ordering operator $\mathcal{T}$ (from [this prop.](time-ordered+product#IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise)):

$$
  \{-,-\}_{\mathcal{T}}
  \;\coloneqq\;
  \mathcal{T}\left(\left\{ \mathcal{T}^{-1}(-), \mathcal{T}^{-1}(-)\right\}\right)
$$

hence

$$
  \array{
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
      \otimes
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
      &\overset{\{-,-\}}{\longrightarrow}&
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \\
    {}^{\mathllap{\mathcal{T}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mathcal{T}}}_\simeq
    \\
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
      \otimes
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
      &\overset{ \{-,-\}_{\mathcal{T}} }{\longrightarrow}&
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
  }
$$

=--

([Fredenhagen-Rejzner 11, (27)](#FredenhagenRejzner11), [Rejzner 11, (5.14)](#Rejzner11))

+-- {: .num_prop #GaugeFixedActionFunctionalTimeOrderedAntibracket}
###### Proposition
**([[time-ordered product|time-ordered]] [[antibracket]] with [[gauge fixing|gauge fixed]] [[action functional]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] ([this def.](A+first+idea+of+quantum+field+theory#FreeFieldTheory)) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ ([this def.](A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity)) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$.

Then the [[time-ordered product|time-ordered]] [[antibracket]] (def. \ref{AntibracketTimeOrdered}) with the gauge fixed BV-[[action functional]] $-S'$ (def. \ref{ComplexBVBRSTGlobal}) equals the [[conjugation]] of the global [[BV-differential]] with the [[isomorphism]] $\mathcal{T}$ from the pointwise to the [[time-ordered product]] of [[observables]] (from [this prop.](time-ordered+product#IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise))

$$
  \{-S',-\}_{\mathcal{T}}
  \;=\;
  \mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-1}
  \,,
$$

hence

$$
  \array{
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
     &\overset{ \{-S',-\} }{\longrightarrow}&
    PoyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
    \\
    {}^{\mathllap{\mathcal{T}}}\downarrow
      &&
    \downarrow^{\mathrlap{\mathcal{T}}}
    \\
    PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
     &\overset{ \{-S',-\}_{\mathcal{T}} }{\longrightarrow}&
    PoyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
  }
$$

=--

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

By the assumption that $(E,\mathbf{L})$ is a [[free field theory]] its [[Euler-Lagrange equations]] are linear in the fields, and hence $S'$ is quadratic in the fields. This means that

$$
  \mathcal{T}^{-1}S' = S' + const
  \,,
$$

where the second term on the right is independent of the fields, and hence that

$$
  \{\mathcal{T}^{-1}(-S'),-\}
  =
  \{-S', - \}
  \,.
$$

This implies the claim:

$$
  \begin{aligned}
    \{-S',-\}_{\mathcal{T}}
    & \coloneqq
    \mathcal{T}\left(\{ \mathcal{T}^{-1}(-S'), \mathcal{T}^{-1}(-) \}\right)
    \\
   & =
   \mathcal{T}\left(\{ -S', \mathcal{T}^{-1}(-) \}\right)
   \\
   & =
   \mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-1}
   \,.
  \end{aligned}
$$

=--

+-- {: .num_defn #ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}
###### Definition
**([[BV-operator]] for [[gauge fixing|gauge fixed]] [[free field theory|free]] [[Lagrangian field theory]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] ([this def.](A+first+idea+of+quantum+field+theory#FreeFieldTheory)) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ ([this def.](A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity)) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$ and with corresponding gauge-fixed global [[BV-BRST differential]] on graded [[regular polynomial observables]]

$$
  \{-S' + S'_{BRST}, -\}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
  \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$

([this def.](A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity)).

Then the corresponding _[[BV-operator]]_

$$
  \Delta_{BV}
  \;\colon\;
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
  \longrightarrow
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$

on [[regular polynomial observables]] is, up to a factor of $i \hbar$, the difference between the free component $\{-S',-\}$ of the gauge fixed global BV differential and its time-ordered version (def. \ref{AntibracketTimeOrdered})

$$
  \Delta_{BV}
    \;\coloneqq\;
  \tfrac{1}{i \hbar}
  \left(
    \left\{
      -S',-
    \right\}_{\mathcal{T}}
    -
    \left\{ -S',(-) \right\}
  \right)
  \,,
$$

hence

$$
  \label{BVOperatorDefiningRelation}
  \{-S',-\}_{\mathcal{T}}
  \;=\;
  \{-S',-\} + i \hbar \Delta_{BV}
  \,.
$$

=--

+-- {: .num_prop #ComponentsBVOperator}
###### Proposition
**([[BV-operator]] in components)**

If the [[field bundles]] of all [[field (physics)|fields]], [[ghost fields]] and [[auxiliary fields]] are [[trivial vector bundles]], with field/ghost-field/auxiliary-field coordinates collectively denoted $(\phi^A)$ then
the [[BV-operator]] $\Delta_{BV}$ from prop. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator} is given explicitly by

$$
  \Delta_{BV}
  \;=\;
  \underset{a}{\sum} (-1)^{deg(\Phi^A)}
  \underset{\Sigma}{\int}
    \frac{\delta}{\delta \Phi^A(x)}
    \frac{\delta}{\delta \Phi^{\ddagger}_A(y)}
  dvol_\Sigma
$$

Since this formula exhibits a graded [[Laplace operator]], the BV-operator is also called the _BV-Laplace operator_ or _BV-Laplacian_, for short.

=--

([Fredenhagen-Rejzner 11, (29)](#FredenhagenRejzner11), [Rejzner 11, (5.20)](#Rejzner11))

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

By prop. \ref{GaugeFixedActionFunctionalTimeOrderedAntibracket} we have equivalently

$$
  i \hbar \Delta_{BV}
  \;=\;
  \mathcal{T} \circ \{-S',-\} \circ \mathcal{T}^{-1}
  \,-\,
  \{-S',-\}
$$

and by [this example](A+first+idea+of+quantum+field+theory#BVDifferentialGlobal) the second term on the right is

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

With this we compute as follows:

$$
  \label{AAA}
  \begin{aligned}
    \{-S',-\}_{\mathcal{T}}
    & =
    \mathcal{T} \circ \left\{ -S,-\right\} \circ \mathcal{T}^{-1}
    \\
    & =
    \exp\left(
     \left[
       \hbar \tfrac{1}{2} \Delta_F
       \left(
         \frac{\delta}{\delta \mathbf{\Phi}},
         \frac{\delta}{\delta \mathbf{\Phi}}
        \right)
        \,,\,
        -
      \right]
    \right)
    \left(
      \{-S',-\}
    \right)
    \\
    & =
    \{-S',-\}
    +
    \left[
      \hbar \tfrac{1}{2}
      \Delta_F
      \left(
        \frac{\delta}{\delta \mathbf{\Phi}},
        \frac{\delta}{\delta \mathbf{\Phi}}
      \right)
      \,,
      \{-S',-\}
    \right]
    + \underset{ = 0 }{\underbrace{\hbar^2(...)}}
    \\
    & =
    \phantom{+}
    \left\{ -S' , -\right\}
    \\
    &
    \phantom{=}
    +
    \left[
      \tfrac{1}{2}\hbar
      \underset{\Sigma \times \Sigma}{\int}
      \Delta_F^{A B}(x,y)
      \frac{\delta^2}{\delta \mathbf{\Phi}^A(x) \delta \mathbf{\Phi}^B(y)}
      \, dvol_\Sigma(x)
      \, dvol_\Sigma(y)
      \;,\;
      \underset{a}{\sum}
      (-1)^{deg(\phi^A)}
      \underset{\Sigma}{\int}
        (P\mathbf{\Phi})_A(x)
        \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
      \,
      dvol_\Sigma(x)
    \right]
   \\
   & =
   \left\{ -S', -\right\}
   \\
   & \phantom{=}
   +
   \underset{A}{\sum}
    (-1)^{deg(\phi^A)}
   \underset{\Sigma \times \Sigma}{\int}
     \underset{ =  i \delta(x-y) }{\underbrace{P_x \Delta_F(x,y)}}
     \frac{\delta}{\delta \mathbf{\Phi}^A(x)}
     \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(y)}
    \, dvol_\Sigma(x)
    \, dvol_\Sigma(y)
    \\
   & = \left\{ -S', -\right\}
   + i \hbar
   \underset{A}{\sum}
   (-1)^{deg(\phi^A)}
   \underset{\Sigma}{\int}
    \frac{\delta}{\delta \mathbf{\Phi}^A(x)}
    \frac{\delta}{\delta \mathbf{\Phi}^\ddagger_A(x)}
    \,
    dvol_\Sigma(x)
  \end{aligned}
$$

Here we used

1. under the first brace that by assumption of a [[free field theory]], $\{-S',-\}$ is linear in the fields, so that the first [[commutator]] with the [[Feynman propagator]] is independent of the fields, and hence all the higher commutators vanish;

1.  under the second brace that the [[Feynman propagator]] is $+i$ times the [[Green function]] for the [[Green hyperbolic differential equation|Green hyperbolic]] [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] ([this cor.](A+first+idea+of+quantum+field+theory#GreenFunctionFeynmanPropagator)).

=--

#### Relation to time-ordered antibracket
 {#InCausalPerturbationTheoryRelationToTimeOrderedAntibracket}

+-- {: .num_prop #AntibracketBVOperatorRelation}
###### Proposition
**(global [[antibracket]] exhibits failure of [[BV-operator]] to be a [[derivation]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] ([this def.](A+first+idea+of+quantum+field+theory#FreeFieldTheory)) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ ([this def.](A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity)) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$

The [[BV-operator]] $\Delta_{BV}$ (def. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}) and the global [[antibracket]] $\{-,-\}$ ([this def.](A+first+idea+of+quantum+field+theory#ComplexBVBRSTGlobal)) satisfy for all [[polynomial observables]] ([this def.](A+first+idea+of+quantum+field+theory#PolynomialObservables)) $A_1, A_2 \in PolyObs(E_{\text{BV-BRST}})[ [\hbar] ]$ the relation

$$
  \label{GlobalAntibracketInteractingWithBVOperator}
  \{A_1, A_2\}
  \;=\;
  (-1)^{deg(A_2)}
  \,
  \Delta_{BV}(A_1 \cdot A_2)
  -
  (-1)^{deg(A_2)}
  \,
  \Delta_{BV}(A_1) \cdot A_2
  -
  A_1 \cdot \Delta_{BV}(A_2)
$$

for $(-) \cdot (-)$ the pointwise product of observables (def. \ref{Observable}).

Moreover, it commutes on [[regular polynomial observables]] with the [[time-ordered product|time-ordering operator]] $\mathcal{T}$ ([this prop.](time-ordered+product#IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise))

$$
  \Delta_{BV} \circ \mathcal{T}
  =
  \mathcal{T} \circ \Delta_{BV}
  \phantom{AAA}
  \text{on}
  \,\,
  PolyObs(E_{\text{BV-BRST}})_{reg}[ [\hbar] ]
$$

and hence satisfies the analogue of relation (eq:GlobalAntibracketInteractingWithBVOperator) also for the time-ordered antibracket $\{-,-\}_{\mathcal{T}}$ (def. \ref{AntibracketTimeOrdered}) and the [[time-ordered product]] $\star_F$ on regular polynomial observables

$$
  \{A_1, A_2\}_{\mathcal{T}}
  \;=\;
  (-1)^{deg(A_2)}
  \,
  \Delta_{BV}(A_1 \star_F A_2)
  -
  (-1)^{deg(A_2)}
  \Delta_{BV}(A_1) \star_F A_2
  -
  A_1 \star_F \Delta_{BV}(A_2)
  \,.
$$

=--

(e.g. [Henneaux-Teitelboim 92, (15.105d)](antibracket#HenneauxTeitelboim92))

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

With prop. \ref{ComponentsBVOperator} the first statement is a graded version of the analogous relation for an ordinary [[Laplace operator]] $\Delta \coloneqq g^{a b} \partial_a \partial_b$ acting on [[smooth functions]] on [[Cartesian space]], which on [[smooth functions]] $f,g$ satisfies

$$
  \Delta(f \cdot g)
  \;=\;
  (\nabla f, \nabla g) - \Delta(f) g - f \Delta(g)
  \,,
$$

by the [[product law]] for [[differentiation]], where now $\nabla f \coloneqq (g^{a b} \partial_b f)$ is the [[gradient]] and $(v,w) \coloneqq g_{a b} v^a w b$ the [[inner product]]. Here one just needs to carefully record the relative signs that appear.

That the BV-operator commutes with the time-ordering operator is clear from the fact that both of these are given by [[partial derivative|partial]] [[functional derivatives]] with _[[constant function|constant]]_ [[coefficients]]. This immediately implies the last statement from the first.

=--

+-- {: .num_example #TimeOrderedExponentialBVOperator}
###### Example
**([[BV-operator]] on [[time-ordered product|time-ordered]] [[exponentials]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] ([this def.](A+first+idea+of+quantum+field+theory#FreeFieldTheory)) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ ([this def.](A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity)) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$.

Let moreover $V \in PolyObs(E_{\text{BV-BRST}})_{reg, deg = 0}[ [\hbar] ]$  be a [[regular polynomial observable]] (def. \ref{PolynomialObservables}) of degree zero. Then the application of the [[BV-operator]] $\Delta_{BV}$ (def. \ref{ForGaugeFixedFreeLagrangianFieldTheoryBVOperator}) to the [[time-ordered product|time-ordered]] [[exponential]] $\exp_{\mathcal{T}}(V)$ ([this example](time-ordered+product#RegularObservablesExponentialTimeOrdered)) is the [[time-ordered product]] of the time-ordered exponential with the sum of $\Delta_{BV}(V)$ and the global [[time-ordered product|time-ordered]] [[antibracket]] $\tfrac{1}{2}\{V,V\}_{\mathcal{T}}$ (def. \ref{AntibracketTimeOrdered}) of $V$ with itself:

$$
  \Delta_{BV}
  \left(
    \exp_{\mathcal{T}}(V)
  \right)
  \;=\;
  \left(
    \Delta_{BV}(V)
    +
    \tfrac{1}{2}\{V,V\}_{\mathcal{T}}
  \right)
   \star_F
  \exp_{\mathcal{T}}(V)
$$

=--

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

By prop. \ref{AntibracketBVOperatorRelation} $\Delta_{BV}$ acts as a [[derivation]] on the [[time-ordered product]] up to a correction given by the antibracket of the two factors. This yields the result by the usual combinatorics of [[exponentials]].

$$
  \begin{aligned}
    \Delta_{BV}
    \left(
      1 + V + \tfrac{1}{2}V \star_F V + \cdots
    \right)
    & =
    \Delta_{BV}(V)
      +
    \tfrac{1}{2}\left( \Delta_{BV}(V) \star_F V + V \star_F \Delta_{BV}(V) \right)
      +
    \tfrac{1}{2}\{V,V\}_{\mathcal{T}}
      +
    \cdots
    \\
    & =
    \Delta_{BV}(V) +  \Delta_{BV}(V) \star_F V + \tfrac{1}{2}\{V,V\}_{\mathcal{T}}
    + \cdots
  \end{aligned}
$$

=--

#### Schwinger-Dyson equation
  {#SchwingerDysonEquation}

A special case of the general occurence of the [[BV-operator]] is the following important property of [[on-shell]] [[time-ordered products]]:

+-- {: .num_prop #DysonSchwinger}
###### Proposition
**([[Schwinger-Dyson equation]])**

Let $(E,\mathbf{L})$ be a [[free field theory|free]] [[Lagrangian field theory]] ([this def.](A+first+idea+of+quantum+field+theory#FreeFieldTheory)) with [[gauge fixing|gauge fixed]] BV-BRST [[Lagrangian density]] $-\mathbf{L}' + \mathbf{L}'_{BRST}$ ([this def.](A+first+idea+of+quantum+field+theory#GaugeFixingLagrangianDensity)) on a graded BV-BRST [[field bundle]] $E_{\text{BV-BRST}} \coloneqq T^\ast[-1]_{\Sigma,inf}(E \times_\Sigma \mathcal{G}[1] \times_{\Sigma} A \times_\Sigma A[-1])$.

Let

$$
  \label{SchwingerDysonTestObservable}
  A
  \;\coloneqq\;
  \underset{\Sigma}{\int}
    A^a(x) \cdot \mathbf{\Phi}^\ddagger_a(x)
  \, dvol_\Sigma(x)
  \;\in\;
  PolyObs_{reg}(E_{\text{BV-BRST}})
$$

be an [[off-shell]] [[regular polynomial observable]] which is [[linear map|linear]] in the [[antifield]] [[field observables]] $\mathbf{\Phi}^\ddagger$. Then

$$
  \label{EquationSchwingerDyson}
  \mathcal{T}^{\pm 1}
  \left(
    \underset{\Sigma}{\int}
      \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)}
      \cdot
      A^a(x)
      \, dvol_\Sigma(x)
  \right)
  \;=\;
  \pm i \hbar
  \,
  \mathcal{T}^{\pm}
  \left(
    \underset{\Sigma}{\int}
    \frac{\delta A^a(x)}{\delta \mathbf{\Phi}^a(x)}
    \,
    dvol_\Sigma(x)
  \right)
  \phantom{A}
  \in
  \underset{
    \text{on-shell}
  }{
  \underbrace{
    PolyObs_{reg}(E_{\text{BV-BRST}}, \mathbf{L'})
  }}
  \,.
$$

This is called the _[[Schwinger-Dyson equation]]_.

=--

The proof below is due to ([Rejzner 16, remark 7.7](#Rejzner16)), following the informal traditional argument ([Henneaux-Teitelboim 92, (15.108b)](#HenneauxTeitelboim92)).

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

Applying the inverse time-ordering map $\mathcal{T}^{-1}$ ([this prop.](time-ordered+product#IsomorphismOnRegularPolynomialObservablesTimeOrderedandPointwise)) to equation (eq:BVOperatorDefiningRelation) applied to $A$ yields

$$
  \label{BVDerivationOfSchwingerDyson}
  \underset{
    \mathcal{T}^{-1}
    \underset{\Sigma}{\int}
      \frac{\delta S'}{\delta \mathbf{\Phi}^a(x)}
      \cdot
      A^a(x)
    dvol_\Sigma(x)
  }{
  \underbrace{
    \mathcal{T}^{-1}\left\{ -S', A\right\}
  }
  }
  \;=\;
  -
  \underset{
    i \hbar
    \mathcal{T}^{-1}
    \underset{\Sigma}{\int}
      \frac{\delta A^a(x)}{\delta \mathbf{\Phi}^a(x)}
    \, dvol_\Sigma
  }{
  \underbrace{
    i \hbar \mathcal{T}^{-1}\Delta_{BV}(A)
  }
  }
  +
  \underset{
    \{-S',\mathcal{T}^{-1}(A)\}
  }{
  \underbrace{
    \mathcal{T}^{-1}\left\{ -S',A\right\}_{\mathcal{T}}
  }
  }
$$

where we have identified the terms under the braces by 1) the component expression for the BV-differential $\{-S',-\}$ from [this prop](A+first+idea+of+quantum+field+theory#BVDifferentialGlobal), 2) prop. \ref{ComponentsBVOperator} and 3) prop. \ref{GaugeFixedActionFunctionalTimeOrderedAntibracket}.

The last term is manifestly in the [[image]] of the [[BV-differential]] $\{-S',-\}$ and hence vanishes when passing to [[on-shell]] observables along the [[isomorphism]] ([this equation](A+first+idea+of+quantum+field+theory#eq;OnShellPolynomialObservablesAsBVCohomology))

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

(by [this example.](A+first+idea+of+quantum+field+theory#BVDifferentialGlobal)).

The same argument with the replacement $\mathcal{T} \leftrightarrow \mathcal{T}^{-1}$ throughout yields the other version of the equation (with time-ordering instead of reverse time ordering and the sign of the $\hbar$-term reversed).

=--

+-- {: .num_remark }
###### Remark
**("Schwinger-Dyson operator")**

The proof of the [[Schwinger-Dyson equation]] in prop. \ref{DysonSchwinger} shows that, up to [[time-ordered product|time-ordering]], the [[Schwinger-Dyson equation]] is the on-shell vanishing of the "quantized" [[BV-differential]] (eq:BVOperatorDefiningRelation)

$$
  \{-S',-\}_{\mathcal{T}}
  \;=\;
  \{-S', -\}
  \,+\,
  i \hbar \, \Delta_{BV}
  \,,
$$

where the [[BV-operator]] is the quantum correction of order $\hbar$. Therefore this is also called the _Schwinger-Dyson operator_ ([Henneaux-Teitelboim 92, (15.111)](#HenneauxTeitelboim92)).

=--

+-- {: .num_remark #SchwingerDysonDistributional}
###### Remark
**([[distribution|distributional]] [[Schwinger-Dyson equation]])**

Often the [[Schwinger-Dyson equation]] (prop. \ref{DysonSchwinger}) is displayed before spacetime-smearing of [[field observables]] in terms of [[operator products]] of [[operator-valued distributions]], taking the observable $A$ in (eq:SchwingerDysonTestObservable) to be

$$
  A^a(x)
    \;\coloneqq\;
  \delta(x-x_0) \delta^a_{a_0}
  \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)
  \,.
$$

This choice makes (eq:EquationSchwingerDyson) become the [[distribution|distributional]] [[Schwinger-Dyson equation]]

$$
  \begin{aligned}
  & T
  \left(
    \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)}
      \cdot
    \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)
  \right)
  \\
  &
  \underset{\text{on-shell}}{=}
  - i \hbar
  \underset{k}{\sum}
  T
  \left(
    \mathbf{\Phi}^{a_1}(x_1)
      \cdots
    \mathbf{\Phi}^{a_{k-1}}(x_{k-1})
      \cdot
    \delta(x_0 - x_k) \delta^{a_0}_{a_k}
      \cdot
    \mathbf{\Phi}^{a_{k+1}}(x_{k+1})
      \cdots
    \mathbf{\Phi}^{a_n}(x_m)
  \right)
  \end{aligned}
$$

(e.q. [Dermisek 09](Schwinger-Dyson+equation#Dermisek09)).

In particular this means that if $(x_0,a_0) \neq (x_k, a_k)$ for all $k \in \{1,\cdots ,n\}$ then

$$
  T
  \left(
    \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)}
      \cdot
    \mathbf{\Phi}^{a_1}(x_1) \cdots \mathbf{\Phi}^{a_n}(x_n)
  \right)
  \;=\;
  0
  \phantom{AAA}
  \text{on-shell}
$$

Since by the [[principle of extremal action]] ([this prop.](A+first+idea+of+quantum+field+theory#PrincipleOfExtremalAction)) the equation

$$
  \frac{\delta S}{\delta \mathbf{\Phi}^{a_0}(x_0)}
  \;=\;
  0
$$

is  the [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] (for the [[classical field theory]]) "at $x_0$", this may be interpreted as saying that the classical equations of motion for fields at $x_0$ still hold for [[time-ordered product|time-ordered]] [[quantum theory|quantum]] [[expectation values]], as long as all other observables are evaluated away from $x_0$; while if observables do coincide at $x_0$ then there is a correction measured by the [[BV-operator]].

=--

+-- {: .num_remark #FreeFieldQuantumShell}
###### Remark
**(the "quantum shell")**

Beware that, superficially, it might seem that in equation (eq:BVDerivationOfSchwingerDyson) not only the term $\{-S',\mathcal{T}^{-1}(A)\}$ on the right vanishes [[on-shell]], but also the term $\mathcal{T}^{-1}\left\{ -S', A\right\}$ on the left, since the latter is the image under the linear map $\mathcal{T}^{-1}$ of an observable that vanishes on-shell.

To sort this out, notice that the isomorphism (eq:OnShellObservablesAsQuotientOfOffShellObservablesRecalled) tells us that the observables that vanish when passing from off-shell to on-shell observables are precisely those in the ideal generated by the image of $\{-S',(-)\}$.
But while $\mathcal{T}^{-1}$ is an isomorphism on (regular off-shell observables), it need not (and in general does not) preserve this ideal! Hence $\mathcal{T}^{-1}(\{-S',A\})$ need not (and in general is not) an element of that ideal, and this is why it remains when passing to the algebra of on-shell observables and thus makes its crucial appearance in the [[Schwinger-Dyson equation]].

=--

#### Quantum master equation

Passing from [[free field theory]] to [[perturbative quantum field theory|perturbative]] [[quantization]] of [[interacting field theory]], the above BV-operator of the underying free field theory appears as a quantum correction to the "[[classical master equation]]" which expresses the nilpotency of the [[BV-differential]]. This "master equation" corrected by the BV-operator is called the _[[quantum master equation]]_. See [this prop.](quantum+master+equation#QuantumMasterEquation).

## References

The concept originates with

* {#BatalinVilkovisky81} [[Igor Batalin]], [[Grigori Vilkovisky]],  _Gauge Algebra and Quantization_, Phys. Lett. B 102 (1): 27–31, 1981 (<a href="https://doi.org/10.1016/0370-2693(81)90205-7">doi:10.1016/0370-2693(81)90205-7</a>)

Traditional review includes

* {#HenneauxTeitelboim92} [[Marc Henneaux]], [[Claudio Teitelboim]], section 15.5.3  of _[[Quantization of Gauge Systems]]_, Princeton University Press, 1992

The understanding of the BV-operator in the rigorous formulation of [[relativistic field theory|relativistic]] [[perturbative quantum field theory]] via [[causal perturbation theory]]/[[perturbative AQFT]] is due to

* {#FredenhagenRejzner11b} [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], section 2 of _Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory_, Commun. Math. Phys. (2013) 317: 697 ([arXiv:1110.5232](https://arxiv.org/abs/1110.5232), [doi:10.1007/s00220-012-1601-1](https://doi.org/10.1007/s00220-012-1601-1))

* {#Rejzner11} [[Katarzyna Rejzner]], section 5.1.2 of _Batalin-Vilkovisky formalism in locally covariant field theory_ ([arXiv:1111.5130](https://arxiv.org/abs/1111.5130))

surveyed in

* {#Rejzner16} [[Katarzyna Rejzner]], section 7.3 of _[[Perturbative Algebraic Quantum Field Theory]]_, Mathematical Physics Studies, Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

See at _[[BV-formalism]]_ for more references.

[[!redirects BV-operators]]

[[!redirects BV-Laplacian]]
[[!redirects BV-Laplacians]]

[[!redirects BV-Laplace operator]]
[[!redirects BV-Laplace operators]]

[[!redirects Schwinger-Dyson operator]]
[[!redirects Schwinger-Dyson operators]]


BV-theory > history

< [[BV-theory]]
[[!redirects BV-theory -- history]]


Byeongho Lee

Postdoc

Center for Applications of Mathematical Principles

National Institute for Mathematical Sciences

Daejeon, South Korea


Byungdo Park

* [webpage](https://byungdo.github.io/)

## Selected writings

On [[twisted differential K-theory]]:

* [[Byungdo Park]], _Geometric models of twisted differential K-theory I_,  J. Homotopy Relat. Struct. 13, 143–167 (2018) ([arXiv:1602.02292](https://arxiv.org/abs/1602.02292), [doi:10.1007/s40062-017-0177-z](https://doi.org/10.1007/s40062-017-0177-z))

On [[equivariant ordinary differential cohomology]]:

* {#ParkRedden19} [[Byungdo Park]], [[Corbett Redden]], _A classification of equivariant gerbe connections_, Communications in Contemporary MathematicsVol. 21, No. 02, 1850001 (2019) ([doi:10.1142/S0219199718500013](https://doi.org/10.1142/S0219199718500013), [arXiv:1709.06003](https://arxiv.org/abs/1709.06003))

On [[ordinary differential cohomology]] with emphasis on relation to [[bundle gerbes]] [[connection on a bundle gerbe|with connection]]:

* [[Byungdo Park]], *Differential cohomology and gerbes: An introduction to higher differential geometry*, lecture notes (2023) [[pdf](https://byungdo.github.io/seminars/IMSRS.pdf), [[Park-DifferentialCohomology.pdf:file]]]

category: people


Bénabou cosmos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[enriched category theory]], a *Bénabou cosmos* is a [[monoidal category]] $\mathcal{V}$ with good properties that make it a [[base of enrichment]] well-suited for $\mathcal{V}$-[[enriched category theory]]. (There are variants of this idea, see at *[[cosmos]]* for more).

The archetypical Bénabou cosmos is the category [[Set]] of all sets, equipped with its [[Cartesian product]]-monoidal structure; and $Set$-[[enriched category theory]] is the ordinary theory of [[locally small categories]], whose [[hom-sets]] are [[sets]].

The point of the notion of Bénabou cosmoi is to allow more general [[hom-objects]] than just [[sets]], while imposing enough conditions on the category $\mathcal{V}$ which these form in order that most standard definitions and proofs of plain [[category theory]] (e.g. properties of [[enriched functor categories]] and specifically of [[categories of presheaves]]) generalize to $\mathcal{V}$-enriched category theory (e.g. concerning [[enriched presheaves]]).

## Definition

\begin{definition}\label{BenabouCosmosAccordingToStreet}
A [[Bénabou cosmos]]  is ([Street 74, p. 1](#Street74)) a

* [[complete category|complete]] and [[cocomplete category|cocomplete]] (hence [[bicomplete category|bicomplete]])

* [[closed monoidal category|closed]]

* [[symmetric monoidal category|symmetric]]

* [[monoidal category|monoidal]]

[[category]] (cf *[[symmetric monoidal closed category]]*).
\end{definition}

\begin{remark}
There is a generalization of Def. \ref{BenabouCosmosAccordingToStreet} to the context of enriched [[indexed categories]] ([Shulman 2013](#Shulman13)).  While Def. \ref{BenabouCosmosAccordingToStreet} is not "[[elementary topos|elementary]]" (as it involves infinitary (non-[[finite limit|finite]]) [[limits]] and [[colimits]]), the indexed version *is* elementary, as the infinitary structure is folded into the indexing base category.  The notion of [[Bénabou cosmoi]] is recovered as particular indexed cosmoi over [[Set]].
\end{remark}

## Examples
 {#Examples}

\begin{example}\label{TopoiAreCosmoi}
**([[topoi]] are [[cosmoi]])**
\linebreak
  Every [[Grothendieck topos]] is a Bénabou cosmos, where the [[symmetric monoidal category|symmetric monoidal]] structure is [[cartesian monoidal category|cartesian]]. Examples in this class include:

* The archetypical such example is the category [[Set]] of [[set]]. $Set$-[[enriched categories]] are ordinary [[locally small categories]].

* The category [[sSet]] of [[simplicial sets]] is the [[presheaf topos]] over the [[simplex category]], whose cartesian monoidal structure is discussed at *[[products of simplicial sets]]*. $sSet$-[[enriched categories]] are often called *[[simplicial categories]]* (though see there for ambiguity in this terminology). They serve as foundation for much of [[homotopy theory]], see at *[[simplicial homotopy theory]]* and *[[simplicial model categories]]* for more.

\end{example}

## Related concepts

* [[cosmos]], [[base of enrichment]]

* [[infinity-cosmos|$\infty$-cosmos]]

## References

Apparently there is no explicit written account by [[Jean Bénabou]] on the notion, but one finds it recounted in [Street 74, p. 1](#Street74):

> to J. Benabou the word means "bicomplete symmetric monoidal category", such categories $\mathcal{V}$ being rich enough so that the theory of categories enriched in  $\mathcal{V}$ develops to a large extent just as the theory of ordinary categories.

* {#Street74} [[Ross Street]],  *Elementary cosmoi I.* in *Category Seminar*, Lecture Notes in Mathematics **420**, Springer (1974) [[doi:10.1007/BFb0063103](https://doi.org/10.1007/BFb0063103)]

See also:

* Wikipedia, *Cosmos_(théorie_des_catégories)*

The [[indexed category|indexed]] generalization:

* {#Shulman13} [[Mike Shulman]], *Enriched indexed categories*, Theory and Applications of Categories, **28** 21 (2013) 616-695 ([tac:28-21](http://www.tac.mta.ca/tac/volumes/28/21/28-21abs.html))

[[!redirects Bénabou cosmoi]]

[[!redirects Benabou cosmos]]
[[!redirects Benabou cosmoi]]

[[!redirects cosmos for enriched category theory]]
[[!redirects cosmoi for enriched category theory]]