nLab
BV-BRST formalism

Idea

Given a local action functional

$$\exp (iS):C\to U(1)$$\exp(i S) : C \to U(1)

on some configuration space $C$, BRST-BV formalism provides a construction of a symplectic reduced phase space $P:=(C_{\{dS=0\}}{)}_{\mathrm{red}}$ suitable for quantization (deformation quantization, geometric quantization) in the context of derived dg-geometry.

Notice that if $S$ is a local action functional (is the integral $S(\varphi )={\int }_{X}L(\varphi ,\dot{\varphi },\cdots )$ over a Lagrangian $L$ on the jet space of some bundle over spacetime $X$) then the covariant phase space $C_{\{dS=0\}}$ (the critical locus) of $S$ is canonically equipped with presymplectic structure. The quotient of $C$ by the action of the flow of those vector fields on which the presymplectic form is degenerate – the gauge transformations of the action functional – is the reduced phase space $C_{\{dS=0\}}{}_{\mathrm{red}}$ which is genuinely symplectic, and whose deformation quantization or geometric quantization is the desired quantization of $S$.

But $C_{\{dS=0\}}{}_{\mathrm{red}}$ may either not even exist as a suitable geometric space, and even if it does exist it is in generally intractable in practice. The BRST-BV construction guarantees the existence of a tractable presentation of $(C_{\{dS=0\}}{)}_{\mathrm{red}}$ in the context of derived dg-geometry:

it is constructed as the formal dual of a graded-commutative dg-algebra called the BRST-BV complex $C^{\mathrm{\infty }}(P_{\mathrm{BV}})$ , equipped with the structure of a differential-graded Poisson algebra

$$\{-,-\}:C^{\mathrm{\infty }}(P_{\mathrm{BV}})\otimes C^{\mathrm{\infty }}(P_{\mathrm{BV}})\to C^{\mathrm{\infty }}(P_{\mathrm{BV}}).$$\{-,-\} : C^\infty(P_{BV}) \otimes C^\infty(P_{BV}) \to C^\infty(P_{BV}) \,.

One distinguishes two somewhat different constructions

• Lagrangian BV formalism (or “field-antifield formalism”) constructs the phase space starting from an action functional $S$ by restricting homologically to the locus where $dS=0$ and then weakly dividing out gauge group actions;

• Hamiltonian BFV formalism implements a homological version of symplectic reduction.

In either case BRST-BV complex $C^{\mathrm{\infty }}(P^{\mathrm{BV}})$ is a model in dg-geometry of a joint homotopical quotient and intersection, hence of an (∞,1)-colimit and (∞,1)-limit, of a space in higher geometry/derived geometry, in the presence of or induced by Poisson structure: it is the formal dual to a restriction, up to homotopy, to the Euler-Lagrange equations and to a quotient, up to homotopy, by the (higher) symmetries.

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 constraintts (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).

Hamiltonian BFV

Homotopical Poisson reduction

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

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

Let $A:=C^{\mathrm{\infty }}(X)$ be its algebra of smooth functions.

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 $A/I$;

2. 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:

Definition

The Sniatycky-Weinstein reduction is the object

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

The Dirac reduction is

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

where $N(I)=\{f\in A\mid \{f,I\}\subset I\}$ is the “subalgebra of observables”.

Proposition

These two algebras are isomorphic

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

Example

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

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

Then $A_{\mathrm{red}}$ 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 $X_{\mathrm{red}}:={\mu }^{-1}(0)/G$ is a submanifold and

$$A_{\mathrm{red}}\simeq C^{\mathrm{\infty }}(X_{\mathrm{red}}).$$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 $A_{\mathrm{red}}$ 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 $X{\mid }_{I=0}$ we consider its derived locus.

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

$$A\otimes \mathrm{Sym}(V),$$A \otimes Sym(V) \,,

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

Then on

$$A\otimes \mathrm{Sym}(V)\otimes \mathrm{Sym}(V^{*})$$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.

Theorem

(Henneaux, Stasheff et al.)

(homological perturbation theory)

There exists an element

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

  $$H^{•\ge 0}(A\otimes S(V)\otimes S(V^{*}),d=\{\Omega ,0\})\simeq H^{•}(\mathrm{CE}(A/I,I/I^2))$$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 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 $X$).

Theorem

(Oh-Park, Cattaneo-Felder) If $C\subset X$ is coisotropic, there is an L-infinity algebra-structure on ${\wedge }^{•}\Gamma (NC)$ such that the induced bracket on $H^0=A_{\mathrm{red}}$ is the given one;

Theorem

(Schätz) The BRST-BV complex with $\{-,-\}$ as its Lie bracket is quasi-isomorphic to the above.

Lagrangian BV

Given a non-degenerate action functional $S:C\to ℝ$ (i.e., one that does not possess gauge symmetries), the derived manifold of Lagrangian BV is constructed by extending $S$ to an element $S^{\mathrm{BV}}\in {𝒳}^{•}(C)$ of the algebra of multivector fields (“antifields”) of $C$, such that

$$(S^{\mathrm{BV}},S^{\mathrm{BV}})=0$$(S^{BV},S^{BV}) = 0

(called the classical master equation) with respect to the Schouten bracket $(-,-):{𝒳}^{•}(X)\otimes {𝒳}^{•}(C)\to {𝒳}^{•}(C)$ (the “anti-bracket”) and then considering the formal dual of the dg-algebra $({𝒳}^{•}(C),d=(S^{\mathrm{BV}},-))$.

When the action $S$ is degenerate, the BV complex has to be extended further.

The central theorem says that formal integration in this dg-manifold over Lagrangian submanifolds with respect to the Schouten bracket regarded as an odd Poisson bracket is independent of the choice of Lagrangian submanifold precisely due to the equation $(S,S)=0$.

The standard construction

We discuss the standard constructions and theorems in Lagrangian BV formalism. The discussion here is supposed to be a direct formalization of the informal discussion in the standard physics literature (e.g. HenneauxTeitelboim) but more pedestrian and more lightweight than for instance the more powerful formalization of (BeilinsonDrinfeld).

Let $k$ be a field of characteristic 0. Write ${\mathrm{dgcAlg}}_k$ for the category of graded-commutative dg-algebras over $k$ (not assumed to be finitely generated and not assumed to be bounded). For the present discussion we regared the opposite category $\mathrm{Space}:={\mathrm{dgAlg}}_k^{\mathrm{op}}$ as our category of spaces and write

$$𝒪:\mathrm{Space}\stackrel{=}{\to }{\mathrm{dgAlg}}_k^{\mathrm{op}}$$\mathcal{O} : Space \stackrel{=}{\to} dgAlg_k^{op}

to indicate that a space $X\in C$ is defined as having an algebra of functions $𝒪\in {\mathrm{dgAlg}}_k$.

See dg-geometry for a more comprehensive discussion of the ambient higher geometry.

We write

$${𝔸}^1\in \mathrm{Space}$$\mathbb{A}^1 \in Space

for the canonical line object in $\mathrm{Space}$, the affine line. This is the space defined by the fact that its dg-algebra of functions

$$𝒪({𝔸}^1)=k[x]$$\mathcal{O}(\mathbb{A}^1) = k[x]

is the polynomial algebra over $k$ on a single generator.

The starting point of standard Lagrangian BV is

1. a space $C\in \mathrm{Space}$ such that $𝒪(C)\in {\mathrm{CAlg}}_k\hookrightarrow {\mathrm{dgAlg}}_k$ is an ordinary commutative algebra over $k$, called the configuration space;

2. a morphism in $\mathrm{Space}$

   $$S:C\to {𝔸}^1,$$S : C \to \mathbb{A}^1 \,,

   called the action functional .

Dually $S$ is a morphism

$$𝒪(C)\leftarrow 𝒪({𝔸}^1)=k[x]:S^{*}.$$\mathcal{O}(C) \leftarrow \mathcal{O}(\mathbb{A}^1) = k[x] : S^* \,.

By the defining free property of ${𝔸}^1$ and since $𝒪(C)$ is assumed to be concentrated in degree 0, this morphism is fixed by its image $S^{*}(x)$ and hence we may identify $S$ as an element in $𝒪(C)$

$$S\in 𝒪(C).$$S \in \mathcal{O}(C) \,.

Write ${\Omega }^1(C)$ for the $𝒪(C)$-module of Kähler differentials on $C$. By its defining property there is a bijection between derivations

$$v:𝒪(C)\to 𝒪(C)$$v : \mathcal{O}(C) \to \mathcal{O}(C)

and $𝒪(C)$-module homomorphism

$${\iota }_v:{\Omega }^1(C)\to 𝒪(C)$$\iota_v : \Omega^1(C) \to \mathcal{O}(C)

to be thought of a giving by evaluating a 1-form on the vector field corresponding to the derivation.

Conversely, the fixed Kähler differential

$$dS\in {\Omega }^1(C)$$d S \in \Omega^1(C)

defines a $k$-linear function

$${\iota }_{dS}:\mathrm{Der}(𝒪(C))\to 𝒪(C)$$\iota_{d S} : Der(\mathcal{O}(C)) \to \mathcal{O}(C)

by $v\mapsto {\iota }_v(dS)$.

We define the following notions

• the kernel

  $$N_S\hookrightarrow \mathrm{Der}(𝒪(S))\stackrel{{\iota }_{dS}}{\to }𝒪(C)$$N_S \hookrightarrow Der(\mathcal{O}(S)) \stackrel{\iota_{d S}}{\to} \mathcal{O}(C)

  of ${\iota }_{dS}$ is called the module of Noether identities of the action functional $S$.

• the image

  $${\iota }_{dS}:\mathrm{Der}(𝒪(S))\to I_S\hookrightarrow 𝒪(C)$$\iota_{d S} : Der(\mathcal{O}(S)) \to I_S \hookrightarrow \mathcal{O}(C)

  is called the Euler-Lagrange ideal of $S$. The space whose function algebra is the quotient

  $$𝒪(C_{\{dS=0\}})=𝒪(C)/I_S$$\mathcal{O}(C_{\{d S = 0\}}) = \mathcal{O}(C)/I_S

  is the unresolved covariant phase space of $S$.

Consider then the dg-algebra

$${\mathrm{Sym}}_{𝒪(C)}(N_S\to \mathrm{Der}(𝒪(C))\stackrel{{\iota }_{dS}}{\to }𝒪(C))$$Sym_{\mathcal{O}(C)}( N_S \to Der(\mathcal{O}(C)) \stackrel{\iota_{d S}}{\to} \mathcal{O}(C) )

free on the cochain complex of $𝒪(C)$-modules

$$\begin{array}{ccccc}{N}_S& \hookrightarrow & \mathrm{Der}(𝒪(C))& \to & 𝒪(C)\\ -2& & -1& & 0\end{array}$$\array{ N_S &\hookrightarrow& Der(\mathcal{O}(C)) &\to& \mathcal{O}(C) \\ -2 && -1 && 0 }

with degrees as indicated. One says that the generators in degree 0 are the fields , the generators degree -1 the antifields and the generators in degree -2 the antighosts .

This comes with a canonical morphism

$$\begin{array}{c}{\mathrm{Sym}}_{𝒪(C)}(N_S\to \mathrm{Der}(𝒪(C))\stackrel{{\iota }_{dS}}{\to }𝒪(C))\\ \downarrow \\ 𝒪(C)/I_S\end{array}$$\array{ Sym_{\mathcal{O}(C)}( N_S \to Der(\mathcal{O}(C)) \stackrel{\iota_{d S}}{\to} \mathcal{O}(C) ) \\ \downarrow \\ \mathcal{O}(C)/I_S }

that is a quasi-isomorphism. Under suitable conditions on $𝒪(C)$ and $S$, this is a resolution of $𝒪(C){/}_S$ by a complex of projective objects in the category of $𝒪(C)$-modules, hence a cofibrant resolution of the unresolved covariant phase space with function algebra $𝒪(C)/I_S$ in a typical model structure on dg-algebras. Under non-suitable conditions $N_S$ itself needs to be further resolved in order to achieve this.

The main point of the Lagrangian BV construction is that this resolution naturally carries a useful BV-algebra structure. The Poisson 2-algebra-structure is induced by the Schouten bracket on the polyvector fields $\mathrm{Der}(𝒪)$.

(…)

As a derived critical locus

In (CostelloGwilliam) it is observed that the BV-complex ought to play the role of the critical locus of the action functional as seen in derived geometry. A precsie formulation and derivation of this statement is at derived critical locus. See at derived critical locus for more pointers.

The BV-complex and homological (path-)integration

We discuss the BV differential as a homological implementation of integration which makes the quantum BV-complex a homological implementation of path integral-quantization (in perturbation theory). See also at cohomological integration.

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-)integration over general configuration spaces of 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) (Schwarz 92). A particularly clear-sighted account of the general relation is in Gwilliam 2013 ).

Further below we discuss the generalization of these relation in terms of Poincaré duality on Hochschild (co)homology.

1. The idea of path integral quantization

2. Multivector fields dual to differential forms

3. The quantum master equation: the path integral measure is a closed form

4. Integration over manifolds by BV cohomology

5. BV-quantization

6. Path integration and quantum observables by BV-cohomology

The idea of path integal quantization

The path integral in quantum field theory is supposed to be the integral over a configuration space $X$ of field $\varphi$ using a measure ${\mu }_S$ which is thought of in the form

$${\mu }_S(\varphi )≔\exp \left(\frac{i}{\hslash }S\left(\varphi \right)\right)\cdot \mu (\varphi )\varphi \in X,$$\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 $S:X\to ℝ$ the action functional of the theory.

For $f$ a smooth function on the space of fields its value as an observable of the system is supposed to be what would be the expectation value

$$⟨f{⟩}_S=\frac{{\int }_{\varphi \in \mathrm{Fields}}f(\varphi )\cdot \mu (\varphi )}{{\int }_{\varphi \in \mathrm{Fields}}\mu (\varphi )}$$\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 most simplest situation. But there is a 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.

Multivector fields dual to differential forms

If one thinks of $X$ as an ordinary $(d<\mathrm{\infty })$-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

$$\mu :{\Omega }^{•}(X)\stackrel{\simeq }{\to }{\wedge }^{-•}\Gamma (TX),$$\mu : \Omega^\bullet(X) \stackrel{\simeq}{\to} \wedge^{-\bullet} \Gamma(T X) \,,

which is usefully thought of as reversing degrees. Under this isomorphism the deRham differential maps to a divergence operator conventionally denoted

$$\mu :d\mapsto \Delta$$\mu : d \mapsto \Delta

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

Definition

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

$$\mathrm{BV}(X,\mu )≔({\wedge }^{•}\Gamma (TX),{\Delta }_{\mu })$$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 $\mu$.

Remark

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

The quantum master equation: the path integral measure is a closed form

Observe that

• if we think of

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

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

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

• then the BV quantum master equaton $\Delta \exp (\frac{i}{\hslash }S)=0$ says nothing but that $\exp (\frac{i}{\hslash }S(-))\mu$ 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 $L_{\mathrm{\infty }}$-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 $\exp (\frac{i}{\hslash }S)\mu$ is an integrable form in the sense of 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 $L_{\mathrm{\infty }}$-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

The following proposition about integration of differential $n$-forms is the archetype for interpreting cohomology in BV-complexes in terms of integration. See also at cohomological interpretation?.

Example

On the open ball of dimension $n$, the integration of differential forms of compact support $\int :{\Omega }_{\mathrm{cp}}^n\to ℝ$ 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 $C^{\mathrm{\infty }}(B^n)\to \cdots \to {\Omega }^{n-1}(B^n)\to {\Omega }^n(B^n)$.

This “integration without integration” is discussed in more detail at Lie integration.

Let $X$ be a closed oriented smooth manifold of dimension $n$ and let ${\mu }_S\in {\Omega }^n(X)$ be any volume form. Let again

$$\mathrm{BV}(X,{\mu }_S)≔({\wedge }^{•}\Gamma (TX),{\Delta }_{{\mu }_S})$$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. 2 above.

Definition

For $f\in C^{\mathrm{\infty }}(X)$ a smooth function, its expectation value with respect to ${\mu }_S$ is

$$⟨f{⟩}_{{\mu }_S}≔\frac{{\int }_{X}f\cdot {\mu }_S}{{\int }_X{\mu }_S}.$$\langle f\rangle_{\mu_S} \coloneqq \frac{ \int_X f \cdot \mu_S }{\int_X \mu_S } \,.

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

Proposition

For $f\in \mathrm{BV}(X,{\mu }_S{)}_0\simeq C^{\mathrm{\infty }}(X)$ the cohomology class of $f$ in the BV complex is the expectation value of $f$, def. 3 times the cohomology class of the unit function 1:

$$[f{]}_{\mathrm{BV}}=⟨f{⟩}_{{\mu }_S}[1{]}_{\mathrm{BV}}.$$[f]_{BV} = \langle f\rangle_{\mu_S} [1]_{BV} \,.

See (Gwilliam 13, lemma 2.2.2).

BV quantization

Let $X$ be a closed manifold as above and write $\mathrm{BV}(X,\mu )$ for the BV-complex def. 2, induced by a given volume form $\mu \in {\Omega }^n(X)$.

Proposition

If $S\in C^{\mathrm{\infty }}(X)$ then the BV-complex induced via def. 2 by the volume form

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

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

$${\Delta }_{{\mu }_S}={\Delta }_{\mu }+\frac{1}{\hslash }{\iota }_{dS},$$\Delta_{\mu_S} = \Delta_\mu + \frac{1}{\hbar}\iota_{d S} \,,

where ${\iota }_{dS}:{\wedge }^{•}\Gamma (TX)\to {\wedge }^{•-1}\Gamma (TX)$ is the operation of acting with a vector field on $S$ by differentiation, extended as a graded derivation to multivector fields.

Proposition

The complex

$${\mathrm{BV}}_{\mathrm{cl}}(X,S)≔({\wedge }^{•}\Gamma (TX),{\iota }_{dS})$$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.

Remark

Prop. 3 and prop. 4 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 $\exp \left(\frac{1}{\hslash }S\right)$ is an $\hslash$-deformation of the derived critical locus of $S$ by a contrinution of the background volume form $\mu$.

We call $({\wedge }^{•}\Gamma (TX),{\iota }_{dS})$ the classical BV complex and $({\wedge }^{•}\Gamma (TX),{\iota }_{dS}+\hslash {\Delta }_{\mu })$ 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.

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 2 as a derived critical locus directly makes sense in great generality for action functionals $S$ defined on spaces of 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). 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. 2 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 2 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. 2 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)

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.

Definition

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

Definition

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

1. ${\Delta }^2=0$

2. $\Delta (ab)=(\Delta a)b+(-1{)}^{\mid a\mid }a(\Delta b)+\hslash \{a,b\}$ for all homogenous elements $a,b\in A$

In (Gwilliam 2013) this is def. 2.2.5.

Remark

A Beilinson-Drinfeld algebra is not a dg-algebra with differential $\Delta$: the Poisson bracket $\hslash \{-,-\}$ 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:

Definition

For $A_{\hslash }$ a Beilinson-Drinfeld algebra, its classical limit is the tensor product of algebras

$$A_{\hslash =0}≔A_{\hslash }{\otimes }_{ℝ[[\hslash ]]}ℝ$$A_{\hbar = 0} \coloneqq A_\hbar \otimes_{\mathbb{R}[ [ \hbar ] ]} \mathbb{R}

hence the result of setting the formal parameter $\hslash$ (“Planck's constant”) to 0.

Remark

The classical limit of a Beilinson-Drinfeld algebra is canonically a classical BV-complex, def. 4.

Definition

For $A_{\hslash =0}$ a classical BV complex, def. 4, a BV quantization of it is a Beilinson-Drinfeld algebra $A_{\hslash }$, def. 5 whose classical limit, def. 6, is the given $A_{\hslash =0}$.

In (Gwilliam 2013) this is def. 2.2.6.

action functional	kinetic action	interaction	path integral measure
$\exp (-S(\varphi ))\cdot \mu =$	$\exp (-(\varphi ,Q\varphi ))\cdot$	$\exp (I(\varphi ))\cdot$	$\mu$
BV differential	elliptic complex +	antibracket with interaction +	BV-Laplacian
$d_q=$	$Q$ +	$\{I,-\}$ +	$\hslash \Delta$

Quantum observables by BV-cohomology

Definition

Given a quantum BV-complex, its cochain cohomology are the expectation values of observables of the 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.

Remark

Given a quantum BV-complex by def. 7 its cochain cohomology is, by definition, a perturbation of that of its classical limit BV complex, def. 6. 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).

Remark

For local theories (…) gauge fixing operator (…) Hodge theory (…)

(…)

Poincaré duality on Hochschild (co)homology and framed little disk algebra

The above 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 $\exp (iS(-))$ with a formal measure $\mathrm{vol}$ 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 $\mathrm{vol}$ and $\exp (iS(-))$ is regarded as a cocycle in cyclic cohomology, hence as a multivector field whose closure condition $\Delta \exp (iS(-))=0$ is the quantum master equation of BV-formalism.

By the identification of Hochschild cohomology
with functions on derived loop spaces 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

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

(VanDenBergh) and this takes the Connes coboundary operator? to the BV operator (Ginzburg).

Related concepts

• BRST complex

• homotopy BV algebra

References

General

A classical standard references is

• Marc Henneaux, Claudio Teitelboim, Quantization of Gauge Systems, Princeton University Press 1992. xxviii+520 pp.

The bulk of the book 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.

This is written in the traditional informal style of the physics literature. A general formalization of Lagrangian quantum BV (chapter 18 of Henneaux-Teitelboim) in the Chiral algebra setting for perturbative quantum field theory on algebraic curves is in

• Alexander Beilinson, Vladimir Drinfeld, Chiral Algebras .

The extension of this approach to higher dimensions is being worked out in terms of factorization algebra in

• Kevin Costello, Owen Gwilliam Factorization algebras in perturbative quantum field theory (pdf)

and in

• Dennis Gaitsgory, John Francis Chiral Koszul duality (arXiv:1103.5803)

The general classical BV formalism (chapter 17 of Henneaux-Teitelboim) is formalized in the same language in

• Frédéric Paugam, Homotopical Poisson Reduction of gauge theories (pdf)

and in the book

• Frédéric Paugam, Towards the mathematics of quantum field theory (pdf)

A systematic/axiomatic account from the point of view of higher geometry is given in

• Urs Schreiber, derived critical locus

Lagrangian BV

For Lagrangian theories

The original articles are

• I. Batalin, G. Vilkovisky, Gauge Algebra and Quantization . Phys. Lett. B 102 (1): 27–31. doi:10.1016/0370-2693(81)90205-7 (1981)

• I. Batalin,G. 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

• I.A. Batalin and G.A. Vilkovisky, Existence Theorem For Gauge Algebra , J. Math. Phys. 26 (1985) 172-184.

Reviews are in

• Joaquim Gomis, J. Paris, S. Samuel, Antibrackets, Antifields and Gauge Theory Quantization (arXiv:hep-th/9412228)

• J. Park, Pursuing the quantum world (pdf)

Geometrical aspects were pioneered in

• Albert Schwarz, Semiclassical approximation in Batalin-Vilkovisky formalism, Comm. Math. Phys. 158 (1993), no. 2, 373–396, euclid

• 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

A systematic account of the classical master equation is also in

• David Kazhdan, The classical master equation in the finite-dimensional case (pdf)

• Giovanni Felder, David Kazhdan, The classical master equation (arXiv:1212.1631)

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

• A. Cattaneo, From topological field theory to deformation quantization and reduction, ICM 2006. (pdf)

• M. Bächtold, On the finite dimensional BV formalism, 2005. (pdf)

• Carlo Albert, Bea Bleile, Jürg Fröhlich, Batalin-Vilkovisky integrals in finite dimensions, arXiv/0812.0464

• Qiu and Zabzine, Introduction to Graded Geometry, Batalin-Vilkovisky Formalism and their Applications, arXiv/1105.2680.

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 nite-dimensional integrals directly from the BV formalism (2011) (arXiv:1202.1554)

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, doi, scan

• Albert Schwarz, Geometry of Batalin-Vilkovisky quantization (arXiv:hep-th/9205088)

This isomorphisms between the de Rham complex and the complex of polyvector fields is reviewed for instance on p. 3 of

• Thomas Willwacher, Damien Calaque Formality of cyclic cochains (arXiv:0806.4095)

and in section 2 of

• Alberto Cattaneo, Domenico Fiorenza, Riccardo Longoni, On the Hochschild-Kostant-Rosenberg map for graded manifolds (pdf)

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

• C. Roger, Gerstenhaber and Batalin-Vilkovisky algebras (ps)

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)

  Correction: Proc. Amer. Math. Soc. 130 (2002), 2809–2810.

with more on that in

• U. Krähmer, Poincaré duality in Hochschild cohomology (pdf)

• Victor Ginzburg, Calabi-Yau Algebras (arXiv)

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)

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 derived critical locus is indicated in

• Kevin Costello, Owen Gwilliam, Factorization algebras in perturbative quantum field theory – Derived critical locus (web)

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

• Owen Gwilliam, Factorization algebras and free field theories PhD thesis (2013) (pdf)

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)

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; Knot weight systems from graded symplectic geometry, arxiv/1110.5234; Odd Chern-Simons theory, Lie algebra cohomology and characteristic classes, arxiv/0912.1243
• Klaus Fredenhagen, Katarzyna Rejzner, Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory, arxiv/1110.5232

For non-Lagrangian theories

The whole formalism also applies to the locus of solutions of differential equations that are not necessarily the Euler-Lagrange equations 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)

• D.S. Kaparulin, S.L. Lyakhovich, A.A. Sharapov, Rigid Symmetries and Conservation Laws in Non-Lagrangian Field Theory (arXiv:1001.0091)

• S.L. Lyakhovich, A.A. Sharapov, Quantizing non-Lagrangian gauge theories: an augmentation method (arXiv:hep-th/0612086)

• S.L. Lyakhovich, A.A. Sharapov, BRST theory without Hamiltonian and Lagrangian (arXiv:hep-th/0411247)

Section 4.5 of

• Gennadi Sardanashvily, Advanced Classical Field Theory (2009) (pdf)

This also makes the connection to

• P. Olver, Applications of Lie Groups to Differential Equations (Springer-Verlag, Berlin) (1986)

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, doi

The original references on Hamiltonian BFV formalism are

• I.A. Batalin, G.A. Vilkovisky, Relativistic S-matrix of dynamical systems with boson and fermion constraints , Phys. Lett. B69 (1977) 309-312;

• I.A. Batalin, E.S. Fradkin, A generalized canonical formalism and quantization of reducible gauge theories , Phys. Lett. B122 (1983) 157-164.

Homological Poisson reduction is discussed in

• Jim Stasheff, Homological Reduction of Constrained Poisson Algebras (arXiv)

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

• Jim Stasheff, The (secret?) homological algebra of the Batalin-Vilkovisky approach (arXiv)

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, MR97c:81186

• Basics of Poisson reduction (blog)

• Alejandro Cabrera, Homological BV-BRST methods: from QFT to Poisson reduction (pdf)

• Jeremy Butterfield, On symplectic reduction in classical mechanis (pdf)

• S. Lyakhovich, A. Sharapov, BRST theory without Hamiltonian and Lagrangian (pdf)

• Florian Schätz, BFV-complex and higher homotopy structures (pdf)

• MO: what is the BV formalism and its uses

Multisymplectic BRST

In the context of multisymplectic geometry

• Sean Hrabak, Ambient Diffeomorphism Symmetries of Embedded Submanifolds, Multisymplectic BRST and Pseudoholomorphic Embeddings (arXiv: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)

• Sean Hrabak, On a Multisymplectic Formulation of the Classical BRST Symmetry for First Order Field Theories Part II: Geometric Structures (arXiv: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)