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 CC;

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

Classical BV as homological resolution of reduced phase space

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

A first idea of quantum field theory, chapter 11. Reduced phase space

Quantum BV as homological (path-)integration

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 idea of path integal quantization

The path integral in quantum field theory is supposed to be the integral over a configuration space XX of fields ϕ\phi using a measure μ S\mu_S which is thought of in the form

μ S(ϕ)≔exp(iℏS(ϕ))⋅μ(ϕ)ϕ∈X, \mu_S(\phi) \coloneqq \exp\left(\frac{i}{\hbar} S\left(\phi\right)\right) \cdot \mu(\phi) \;\;\;\; \phi \in X \,,

for μ\mu some other measure and S:X→ℝS : X \to \mathbb{R} the action functional of the theory.

For ff 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=∫ ϕ∈Fieldsf(ϕ)⋅μ(ϕ)∫ ϕ∈Fieldsμ(ϕ) \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 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 XX as an ordinary (d<∞)(d \lt \infty)-dimensional smooth manifold, then μ S\mu_S will be given by a volume form, μ S∈Ω d(X)\mu_S \in \Omega^d(X). By contraction of multivector fields with differential forms, every choice of volume form on XX induces an isomorphism between differential forms and polyvector fields

μ:Ω •(X)⟶≃∧ d−•Γ(TX), \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

Δ≔μ∘d dR∘μ −1 \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.)

For more see at relation between BV and BD.

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

Observe that

if we think of
- the measure μ\mu as some closed reference differential form on XX;
- the exponentiated action functional exp(iℏS(−))exp\left(\frac{i}{\hbar}S\left(-\right)\right) as a multivector field on XX;
- the expression exp(iℏS(−))μexp(\frac{i}{\hbar}S(-)) \mu as the contraction of this multivector field with μ\mu
then the BV quantum master equaton Δexp(iℏS)=0\Delta \exp(\frac{i}{\hbar}S) = 0 says nothing but that exp(iℏS(−))μexp(\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 XX is not a plain manifold but the L ∞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(iℏS)μ\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 ∞L_\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 nn-forms is the archetype for interpreting cohomology in BV-complexes in terms of integration.

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 XX be a closed oriented smooth manifold of finite number dimension nn and let μ S∈Ω n(X)\mu_S \in \Omega^n(X) be any volume form. Let again

BV(X,μ S)≔(∧ •Γ(TX),Δ μ 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. above.

Definition

For f∈C ∞(X)f \in C^\infty(X) a smooth function, its expectation value with respect to μ S\mu_S is

⟨f⟩ μ S≔∫ Xf⋅μ S∫ Xμ S. \langle f\rangle_{\mu_S} \coloneqq \frac{ \int_X f \cdot \mu_S }{\int_X \mu_S } \,.

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

Proposition

For f∈BV(X,μ S) 0≃C ∞(X)f \in BV(X,\mu_S)_0 \simeq C^\infty(X) the cohomology class of ff in the BV complex is the expectation value of ff, def. times the cohomology class of the unit function 1:

[f] BV=⟨f⟩ μ S[1] BV. [f]_{BV} = \langle f\rangle_{\mu_S} [1]_{BV} \,.

See (Gwilliam 13, lemma 2.2.2).

BV quantization

Let XX be a closed manifold as above and write BV(X,μ)BV(X, \mu) for the BV-complex def. , induced by a given volume form μ∈Ω n(X)\mu \in \Omega^n(X).

Proposition

If S∈C ∞(X)S \in C^\infty(X) then the BV-complex induced via def. by the volume form

μ S≔exp(1ℏS)⋅μ \mu_S \coloneqq \exp\left(\frac{1}{\hbar} S\right) \cdot \mu

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

Δ μ S=Δ μ+1ℏι dS, \Delta_{\mu_S} = \Delta_\mu + \frac{1}{\hbar}\iota_{d S} \,,

where ι dS:∧ •Γ(TX)→∧ •−1Γ(TX)\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 SS by differentiation, extended as a graded derivation to multivector fields.

Proposition

The complex

BV cl(X,S)≔(∧ •Γ(TX),ι dS) BV_{cl}(X, S) \coloneqq (\wedge^\bullet \Gamma(T X), \iota_{d S})

is the derived critical locus of the function SS.

By the discussion at derived critical locus.

Prop. and prop. together say that the BV-complex of a manifold XX for a volume form μ S\mu_S shifted from a background volume form μ\mu by a function exp(1ℏS)\exp\left(\frac{1}{\hbar} S\right) is an ℏ\hbar-deformation of the derived critical locus of SS by a contribution of the background volume form μ\mu.

We call (∧ •Γ(TX),ι dS)(\wedge^\bullet \Gamma(T X), \iota_{d S}) the classical BV complex and (∧ •Γ(TX),ι dS+ℏΔ μ)(\wedge^\bullet \Gamma(T X), \iota_{d S} + \hbar \Delta_{\mu} ) the quantum BV complex of the manifold XX equipped with the function SS and the voume form μ\mu.

The crucial idea now is the following.

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

Δ 2=0\Delta^2 = 0
Δ(ab)=(Δa)b+(−1) |a|a(Δb)+ℏ{a,b}\Delta( a b) = (\Delta a) b + (-1)^{\vert a\vert} a (\Delta b) + \hbar \{a,b\} for all homogenous elements a,b∈Aa, b \in A

In (Gwilliam 2013) this is def. 2.2.5.

But:

Definition

For A ℏ=0A_{\hbar = 0} a classical BV complex, def. , a BV quantization of it is a Beilinson-Drinfeld algebra A ℏA_{\hbar}, def. whose classical limit, def. , is the given A ℏ=0A_{\hbar = 0}.

In (Gwilliam 2013) this is def. 2.2.6.

Quantum observables by BV-cohomology

(…)

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 CC. Under the identification of Hochschild homology/cyclic homology with the de Rham complex the product of the action functional exp(iS(−))\exp(i S(-)) with a formal measure volvol on CC 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 volvol and exp(iS(−))\exp(i S(-)) is regarded as a cocycle in cyclic cohomology, hence as a multivector field whose closure condition Δexp(iS(−))=0\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 Δ\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 AA there exists Poincaré duality between Hochschild cohomology and Hochschild homology

τ:HH i(A)→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).

Non-perturbative

On non-perturbative enhancements of BV-BRST formalism cast in (differential) cohesive ( ∞ , 1 ) (\infty,1) -topos theory:

Luigi Alfonsi, Charles A. S. Young, Towards non-perturbative BV-theory via derived differential cohesive geometry [arXiv:2307.15106]

Hamiltonian BFV – Homotopical Poisson reduction

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

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

Let A≔C ∞(X)A \coloneqq C^\infty(X) be its algebra of smooth functions.

Consider

an ideal I⊂AI \subset A
that is closed under the Poisson bracket

{I,I}⊂I\{I,I\} \subset I

(one says that we have first class constraint or that the 0-locus of II is coisotropic)

By the Poisson bracket II acts on AA. The Poisson reduction of XX by II is the combined

passage to the 0-locus of II, which algebraically (dually) is passage to the quotient algebra A/IA/I;
passage to the quotient of XX by the II-action, which dually is the passage to the invariant subalgebra A IA^I.

This may be achieved in different orders:

Definition

The Sniatycky-Weinstein reduction is the object

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

The Dirac reduction is

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

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

Proposition

These two algebras are isomorphic

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

Example

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

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

Then A redA_{red} is the algebraic dual to the preimage μ −1(0)\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 μ\mu then X red:=μ −1(0)/GX_{red} := \mu^{-1}(0)/G is a submanifold and

A red≃C ∞(X red). A_{red} \simeq C^\infty(X_{red}) \,.

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

instead of forming the quotient X/GX/G we form the action groupoid or quotient stack X//GX//G. More precisely we do this for the infinitesimal action and consider a quotient Lie algebroid;
instead of forming the intersecton X| I=0X|_{I = 0} we consider its derived locus.

Let {T 1,⋯,T N}\{T_1, \cdots, T_N\} be any finite set of generators of the ideal II. Then there exists a non-positively graded cochain complex on the graded algebra

A⊗Sym(V), A \otimes Sym(V) \,,

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

Then on

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

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

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

Write {c α}\{c^\alpha\} for the basis for V *V^*, called the ghost. Write {π α}\{\pi_\alpha\} for the dual basis on VV, called the ghost momenta.

Theorem

(Henneaux, Stasheff et al.)

(homological perturbation theory)

There exists an element

Ω∈A⊗S(V)⊗S(V *) \Omega \in A \otimes S(V) \otimes S(V^*)

the BRST-BV charge such that

{Ω,Ω}=0\{\Omega, \Omega\} = 0, so that (A⊗S(V)⊗S(V *),d:={Ω,−})(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⊗S(V)⊗S(V *),d={Ω,−})=A/I H^0(A \otimes S(V) \otimes S(V^*), d = \{\Omega, -\}) = A/I \;
H <0(A⊗S(V)⊗S(V *),d={Ω,−})=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/IA/I)
If II is a regular ideal (meaing that VV can be chosen to be concentrated in degree 1) or the vanishing ideal of a coisotropic submanifold, then the cohomology in positive degree

H •≥0(A⊗S(V)⊗S(V *),d={Ω,0})≃H •(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)(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,−}\{I,-\} acting on XX).

Theorem

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

Theorem

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

References

General

A classical standard references for the Lagrangian formalism is

Marc Henneaux, Lectures on the Antifield-BRST formalism for gauge theories, Nuclear Physics B (Proceedings Supplement) 18A (1990) 47-106 (pdf)

Similarly the bulk of the textbook

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

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, doi:/10.1063/1.3527427)

Formulation as homotopy AQFT:

Marco Benini, Simen Bruinsma, Alexander Schenkel, Linear Yang-Mills theory as a homotopy AQFT (arXiv: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]

Lagrangian BV

For Lagrangian theories

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.

Review:

Joaquim Gomis, J. Paris, S. Samuel, Antibrackets, Antifields and Gauge Theory Quantization (arXiv:hep-th/9412228)
J. Park, Pursuing the quantum world (pdf)
Pavel Mnev. Lectures on Batalin-Vilkovisky formalism and its applications in topological quantum field theory (2017). (arXiv:1707.08096).
Alberto S. Cattaneo, Pavel Mnev, Michele Schiavina, BV Quantization, Encyclopedia of Mathematical Physics 2nd ed [arXiv:2307.07761]

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:

Marco Benini, Pavel Safronov, Alexander Schenkel, Classical BV formalism for group actions (arXiv: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
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

Kasia Rejzner, section 7 of Perturbative algebraic quantum field theory Springer 2016 (web)

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

Alberto Cattaneo, Pavel Mnev, Nicolai Reshetikhin, Classical BV theories on manifolds with boundary, arXiv:1201.0290; Classical and quantum Lagrangian field theories with boundary, arXiv:1207.0239; Perturbative quantum gauge theories on manifolds with boundary, arxiv/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)

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)

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, doi: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) [doi:10.1007/JHEP07(2020)003, arXiv:1912.06695]

Review:

Sebastian Albrecht: Formulation of Batalin-Vilkovisky Field Theories as Homotopy Lie Algebras, Munich (2022) [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

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

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

Jim Stasheff, Homological Reduction of Constrained Poisson Algebras, J. Differential Geom. Volume 45, Number 1 (1997), 221-240 (arXiv:q-alg/9603021, Euclid)

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)

Last revised on June 6, 2024 at 00:18:20. See the history of this page for a list of all contributions to it.