nLab
BV-BRST formalism

under construction

Idea

The $n$POV

What in physics is called BV-BRST formalism or just BV-formalism is from the nPOV the following:

the configuration space $\mathrm{Conf}$ of a physical system – notably that of a gauge theory – is in general not a naive space, such as a manifold, but instead a space in the general sense of higher geometry: it is in fact an object in the (∞,1)-topos of ∞-stacks/(∞,1)-sheaves on the (∞,1)-site

$$C:=({\mathrm{dgAlg}}_k^{-}{)}^{\mathrm{op}}$$C :=(dgAlg_k^-)^{op}

of formal duals to cochain differential graded algebras in non-postive degree:

$$\mathrm{Conf}\in H={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C).$$Conf \in \mathbf{H} = Sh_{(\infty,1)}(C) \,.

The way to understand how such ∞-stacks are spaces is described at motivation for sheaves, cohomology and higher stacks:

we think here of an object $\mathrm{Spec}A\in ({\mathrm{dgAlg}}_k^{-}{)}^{\mathrm{op}}$ as a test space with derived algebra of functions $A\in \mathrm{dgAlg}$. An object in ${\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(C)$ is an ∞-groupoid with geometric structure given by such test objects: it can be probed by such test objects.

Every such ∞-groupoid $X$ modeled on objects in $C$ has also a global function algebra $𝒪(X)\in \mathrm{dgAlg}$. This is in general an unbounded differential graded algebra: it may be nontrivial in positive and negative degrees. For $X=\mathrm{Conf}$ the configuration space of a gauge theory, the dg-algebra

$$𝒪(\mathrm{Conf})\in \mathrm{dgAlg}$$\mathcal{O}(Conf) \in dgAlg

is called the BV-BRST-complex of the physical system. Roughly, in positive degrees this dg-algebra remembers the k-morphisms of the $\mathrm{\infty }$-groupoid $\mathrm{Conf}$, while in negative degrees it remembers the negative degrees of the derived function algebra of the space of objects of the $\mathrm{\infty }$-groupoid.

In the physics literature

• the elements in degree 0 of $𝒪(\mathrm{Conf})$ are called the fields (really they are the functions on the naive space of fields);

• the elements in positive degree are called the ghosts – this are really the duals to the k-morphisms of the L-∞-algebroid $\mathrm{Conf}$;

• the elements in negative degree are called anti-fields and anti-ghosts. (This is just formal terminology made up in physics for lack of a better term. It has in particular nothing to do with the notion of anti-particles !)

The operation $X\mapsto 𝒪(X)$ extends to an (∞,1)-functor $𝒪:H\to {\mathrm{dgAlg}}^{\mathrm{op}}$, which is part of an (∞,1)-adjunction

$$(𝒪⊣\mathrm{Spec}):H\stackrel{\stackrel{\mathrm{Spec}}{\leftarrow }}{\stackrel{𝒪}{\to }}{\mathrm{dgAlg}}^{\mathrm{op}}.$$(\mathcal{O} \dashv Spec) \;\; : \;\; \mathbf{H} \stackrel{\overset{Spec}{\leftarrow}}{\overset{\mathcal{O}}{\to}} dgAlg^{op} \,.

In detail, $𝒪$ acts as follows: every ∞-stack $X$ may be written as a (colimit) over representable $\mathrm{Spec}{A}_i\in {\mathrm{dgAlg}}_i$

$$X\simeq \underset{{\to }^i}{\lim }Y(\mathrm{Spec}{A}_i),$$X \simeq \lim_{\to^i} Y(Spec A_i) \,,

where $Y:({\mathrm{dgAlg}}^{-}{)}^{\mathrm{op}}\to H$ is the Yoneda embedding.

The functor $𝒪$ takes any such colimit-description, and simply reinterprets the colimit in ${\mathrm{dgAlg}}^{\mathrm{op}}$, i.e. the limit in $\mathrm{dgAlg}$:

$$𝒪(X)=\underset{{\leftarrow }^i}{\lim }{A}_i.$$\mathcal{O}(X) = \lim_{\leftarrow^i} A_i \,.

This is proposition 3.1 in

• David Ben-Zvi, David Nadler, Loop Spaces and Connections (arXiv:math/1002.3636)

The pedestrian POV

BV theory is the answer to the second of two different questions:

• Hamiltonian BFV: taking quotients of constraint surfaces in Poisson manifolds by group actions and more generally by the foliation determined by first class constraints;

• Lagrangian BV: integrating forms over NQ-supermanifolds.

Hamiltonian BFV

The F is for Fradkin. In this context, the BFV-complex is a homological resolution of the problem of taking quotients of symplectic manifolds by group actions.

Question: Can you explain more about this? What do you mean by a “homological resolution of the problem”? Is there a nice example? I went over the blog entry but it seemed to talk about symplectic/Poisson reduction in its own right and didn’t yet make the link with the BV formalism. (Bruce)

(Jim) Thanks, Bruce. My initial edit here is just to set the record straight - BV $\ne$ BFV. Will expand further and try to answer your query.

See homological resolution.

Comment: Kevin Costello is preparing a book on Renormalization of quantum field theories, available on his webpage. He has a section entitled The BV construction as symplectic reduction . Could you somehow link your explanation to that in some way?

Urs says: I have (only) a vague hunch that Lagrangian and Hamiltonian BV are related in some way by “holography” of sorts, in a way that explains why the master equation in Lagrangian BV – $\Delta \exp (S)=0$ – looks like a Schrödinger equation if one re-interprets the space of histories with the space of states of a system of one dimension higher. Some very useful observations in this regard are in S.L. Lyakhovich, A.A. Sharapov, Quantization of Donaldson-Uhlenbeck-Yau theory arXiv, which I talk about at the end of this.

(Jim) I would hope they were related by a homological version of the usual Hamiltonian - Lagrangian correspondence.

Zoran: It sound like, there might be a role of microlocalization in understanding this correspondence.

Poisson/symplectic reduction

• Basics of Poisson reduction (blog)

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

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

Homological interpretation of BV-Poisson reduction

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

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

The latter is NOT in a Poisson context, any more than Lagrangians are only for symplectic manifolds.

Lagrangian BV

Idea

• Lagrangian BV-formalism is a means to describe integration over supermanifolds for NQ-supermanifolds given by $L_{\mathrm{\infty }}$-algebroids.

  Maybe nowadays?, but not originally. It was meant for Lagrangians with symmetries.

  (Recall that in the physics literture the function algebra of these NQ-supermanifolds is addressed as the BRST complex.)

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

$${\mu }_S(\varphi )=\exp (\frac{i}{\hslash }S(\varphi ))\mu (\varphi )\varphi \in X,$$\mu_S(\phi) = \exp(\frac{i}{\hbar} S(\phi)) \mu(\phi) \;\;\;\; \phi \in X \,,

for $\mu$ some other measure and $S:X\to ℝ$ the action functional.

If one thinks of $X$ as an ordinary $(d<\mathrm{\infty })$-dimensional smooth manifold, $d{\mu }_S$ will be given by a volume form, ${\mu }_S\in {\Omega }^d(X)$. By contraction of multivector fields with forms, every choice of volume form on $X$ induces an isomorphism between differential forms and multivectors

$$\mu :{\Omega }^{•}(X)\stackrel{\simeq }{\to }{\wedge }^{-•}\Gamma (TX),$$\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 This idea can be found recalled for instance on p.3 of Willwacher, Calaque Formality of cyclic cochains.

The point to notice now is

• if we think of

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

  • the exponentiated action functional $\exp (\frac{i}{\hslash }S(-))$ 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 $L_{\mathrm{\infty }}$-algebroids 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”.

Examples

See examples for Lagrangian BV.

Relation to groupoid cardinality

There ought to be a close relation between the integration over $L_{\mathrm{\infty }}$-algebroids using BV-formalism and the notion of groupoid cardinality for finite groupoids, which was recently generalized to a notion of volume of a Lie groupoid.

References and related entries

• homotopy BV algebra

A comprehensive recent review is

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

Other introductions include

• D. 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)

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

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

• E. Witten, A note on the antibracket formalism, Modern Physics Letters A, 5 7, 487–494 (scan)

• MO: what is the BV formalism and its uses