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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
in the construction of reduced phase spaces of Lagrangian field theories, in particular including gauge theories; (“Lagrangian BV”)
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:
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 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
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-)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.
The path integral in quantum field theory is supposed to be the integral over a configuration space $X$ of fields $\phi$ using a measure $\mu_S$ which is thought of in the form
for $\mu$ some other measure and $S : X \to \mathbb{R}$ 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
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 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.
If one thinks of $X$ as an ordinary $(d \lt \infty)$-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
which is usefully thought of as reversing degrees. Under this isomorphism the deRham differential maps to a divergence operator, the BV-operator, conventionally denoted
which interacts naturally with the canonical bracket on multivector fields: the Schouten bracket. (See at polyvector field for more details.)
For $X$ an oriented smooth manifold of dimension $n \in \mathbb{N}$ and for $\mu \in \Omega^n(X)$a volume form, write
for the cochain complex induced on multivector fields by dualizing the de Rham differential with $\mu$.
The Schouten bracket on $BV(X,\mu)$ makes this cochain complex a Poisson 0-algebra.
For more see at relation between BV and BD.
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 $L_\infty$-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 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 $L_\infty$-algebroid are the so-called “anti-fields”, while those dual to the higher degree elements are the so-called “anti-ghosts”.
The following proposition about integration of differential $n$-forms is the archetype for interpreting cohomology in BV-complexes in terms of integration.
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 oriented smooth manifold of finite number dimension $n$ and let $\mu_S \in \Omega^n(X)$ be any volume form. Let again
be the corresponding dual cochain complex of the de Rham complex by def. above.
For $f \in C^\infty(X)$ a smooth function, its expectation value with respect to $\mu_S$ is
Write $[-]_{BV}$ for the cochain cohomology classes in the BV complex $BV(X, \mu_S)$.
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. times the cohomology class of the unit function 1:
See (Gwilliam 13, lemma 2.2.2).
Let $X$ be a closed manifold as above and write $BV(X, \mu)$ for the BV-complex def. , induced by a given volume form $\mu \in \Omega^n(X)$.
If $S \in C^\infty(X)$ then the BV-complex induced via def. by the volume form
(for any constant $\hbar$ to be read as Planck's constant) has BV-differential related to that of $\mu$ itself by
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.
The complex
is the derived critical locus of the function $S$.
By the discussion at derived critical locus.
Prop. and prop. 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.
(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 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. 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 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. 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.
A classical BV complex is a cochain complex equipped with the structure of a Poisson 0-algebra.
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 bracket $\{-,-\}$ of degree 1 and with an operator $\Delta \colon A \to A$ of degree 1 which satisfies:
$\Delta^2 = 0$
$\Delta( a b) = (\Delta a) b + (-1)^{\vert a\vert} a (\Delta b) + \hbar \{a,b\}$ for all homogenous elements $a, b \in A$
In (Gwilliam 2013) this is def. 2.2.5.
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:
For $A_\hbar$ a Beilinson-Drinfeld algebra, its classical limit is the tensor product of algebras
hence the result of setting the formal parameter $\hbar$ (“Planck's constant”) to 0.
The classical limit of a Beilinson-Drinfeld algebra is canonically a classical BV-complex, def. .
For $A_{\hbar = 0}$ a classical BV complex, def. , a BV quantization of it is a Beilinson-Drinfeld algebra $A_{\hbar}$, def. whose classical limit, def. , is the given $A_{\hbar = 0}$.
In (Gwilliam 2013) this is def. 2.2.6.
action functional | kinetic action | interaction | path integral measure |
---|---|---|---|
$\exp(-S(\phi)) \cdot \mu =$ | $\exp(-(\phi, Q \phi)) \cdot$ | $\exp(I(\phi)) \cdot$ | $\mu$ |
BV differential | elliptic complex + | antibracket with interaction + | BV-Laplacian |
$d_q =$ | $Q$ + | $\{I,-\}$ + | $\hbar \Delta$ |
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.
Given a quantum BV-complex by def. its cochain cohomology is, by definition, a perturbation of that of its classical limit BV complex, def. . 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).
For local theories (…) gauge fixing operator (…) Hodge theory (…)
(…)
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 $\exp(i S(-))$ 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 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
(VanDenBergh) and this takes the Connes coboundary operator? to the BV operator (Ginzburg).
The following is a rough survey of homotopical Poisson reduction, following (Stasheff 96).
Let $(X, \{-,-\})$ be a smooth Poisson manifold.
Let $A := C^\infty(X)$ 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
passage to the 0-locus of $I$, which algebraically (dually) is passage to the quotient algebra $A/I$;
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:
The Sniatycky-Weinstein reduction is the object
The Dirac reduction is
where $N(I) = \{f \in A | \{f, I\} \subset I\}$ is the “subalgebra of observables”.
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
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
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
(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.
(Henneaux, Stasheff et al.)
(homological perturbation theory)
There exists an element
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
(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
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 $X$).
(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;
(Schätz) The BRST-BV complex with $\{-,-\}$ as its Lie bracket is quasi-isomorphic to the above.
A classical standard references for the Lagrangian formalism is
Similarly the bulk of the textbook
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
M. Schlichenmaier, T Voronov, (eds.) XXIX Workshop on Geometric Methods in Physics, vol. 1307 of AIP Conference Proceedings, 1307, 7 (2010) (arXiv:1010.0899, doi:/10.1063/1.3527427)
Formulation as homotopy AQFT:
The original articles are
Igor Batalin, Grigori Vilkovisky, Gauge Algebra and Quantization, Phys. Lett. B 102 (1): 27–31, 1981 (doi:10.1016/0370-2693(81)90205-7)
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.
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)
Generalization from Lie algebra-actions to actual algebraic group-actions:
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
Alberto 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.
The 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
Stefan Hollands, Renormalized Quantum Yang-Mills Fields in Curved Spacetime, Rev. Math. Phys.20:1033-1172, 2008 (arXiv:0705.3340)
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)
Klaus Fredenhagen, Kasia Rejzner, Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory, Commun. Math. Phys. 317(3), 697–725 (2012) (arXiv:1110.5232)
Katarzyna Rejzner, Batalin-Vilkovisky formalism in locally covariant field theory (arXiv:1111.5130)
Katarzyna Rejzner, Remarks on local symmetry invariance in perturbative algebraic quantum field theory (arXiv:1301.7037)
and surveyed in
Discussion for field theories with boundary conditions and going in the direction of extended field theory/local quantum field theory is in
A discussion of BV-BRST formalism in the general context of perturbative quantum field theory is in
Relation to Feynman diagrams is made explicit in
See also
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
This isomorphisms between the de Rham complex and the complex of polyvector fields is reviewed for instance on p. 3 of
and in section 2 of
A discussion in the general context of BV-algebras is in
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
A mathematically oriented reformulation of some of this (in the context of TCFT ) is in
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
A clear discussion of the BV-complex as a means for homological path integral quantization is in
Related Chern-Simons type graded action functionals are discussed also in
Lectures, discussing also the relation to the graph complex are
Gluing aspects are in focus of the program explained in
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
This also makes the connection to
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
BRST formalism is discussed in
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
Remarks on the homotopy theory interpretation of BRST-BV are in
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)
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
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