Types of quantum field thories
Given a local action functional
on some configuration space , BRST-BV formalism provides a construction of a symplectic reduced phase space suitable for quantization (deformation quantization, geometric quantization) in the context of derived dg-geometry.
Notice that if is a local action functional (is the integral over a Lagrangian on the jet bundle of some bundle over spacetime ) then the covariant phase space (the critical locus) of is canonically equipped with presymplectic structure. The quotient of 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 which is genuinely symplectic, and whose deformation quantization or geometric quantization is the desired quantization of .
But 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 in the context of derived dg-geometry:
One distinguishes two somewhat different constructions
Lagrangian BV formalism (or “field-antifield formalism”) constructs the phase space starting from an action functional by restricting homologically to the locus where and then weakly dividing out gauge group actions;
Hamiltonian BFV formalism implements a homological version of symplectic reduction.
In either case BRST-BV complex 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:
The following is a rough survey of homotopical Poisson reduction, following (Stasheff 96).
that is closed under the Poisson bracket
(one says that we have first class constraint or that the 0-locus of is coisotropic)
By the Poisson bracket acts on . The Poisson reduction of by is the combined
passage to the 0-locus of , which algebraically (dually) is passage to the quotient algebra ;
passage to the quotient of by the -action, which dually is the passage to the invariant subalgebra .
This may be achieved in different orders:
The Sniatycky-Weinstein reduction is the object
The Dirac reduction is
where is the “subalgebra of observables”.
These two algebras are isomorphic
Let then be the ideal of functions that vanish on . This is always coisotropic.
Then is the algebraic dual to the preimage quotiented by the Lie algebra action: the “constraint surface” quotiented by the symmetries.
In fact, if 0 is a regular value? of then is a submanifold and
We now discuss the BRST-BV complex for the set of constraints on , which will be a resolution of in the following sense:
instead of forming the intersecton we consider its derived locus.
(with in non-negative degree)
there is an evident graded generalization of the Poisson bracket on , which is on and just the canonical pairing.
(Henneaux, Stasheff et al.)
There exists an element
the BRST-BV charge such that
the cochain cohomology is
(which says that this is in non-positive degree a resolution of the constraint locus )
If is a regular ideal (meaing that can be chosen to be concentrated in degree 1) or the vanishing ideal of a coisotropic submanifold, then the cohomology in positive degree
(which says that in positive degree the BRST-BV complex is a resolution of the action Lie algebroid of acting on ).
(Oh-Park, Cattaneo-Felder) If is coisotropic, there is an L-infinity algebra-structure on such that the induced bracket on is the given one;
(Schätz) The BRST-BV complex with as its Lie bracket is quasi-isomorphic to the above.
Given a non-degenerate action functional (i.e., one that does not possess gauge symmetries), the derived manifold of Lagrangian BV is constructed by extending to an element of the algebra of multivector fields (“antifields”) of , such that
When the action 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 .
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 be a field of characteristic 0. Write for the category of graded-commutative dg-algebras over (not assumed to be finitely generated and not assumed to be bounded). For the present discussion we regared the opposite category as our category of spaces and write
to indicate that a space is defined as having an algebra of functions .
is the polynomial algebra over on a single generator.
The starting point of standard Lagrangian BV is
a morphism in
called the action functional .
Dually is a morphism
By the defining free property of and since is assumed to be concentrated in degree 0, this morphism is fixed by its image and hence we may identify as an element in
and -module homomorphism
to be thought of a giving by evaluating a 1-form on the vector field corresponding to the derivation.
Conversely, the fixed Kähler differential
defines a -linear function
We define the following notions
is called the Euler-Lagrange ideal of . The space whose function algebra is the quotient
is the unresolved covariant phase space of .
Consider then the dg-algebra
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
that is a quasi-isomorphism. Under suitable conditions on and , this is a resolution of by a complex of projective objects in the category of -modules, hence a cofibrant resolution of the unresolved covariant phase space with function algebra in a typical model structure on dg-algebras. Under non-suitable conditions 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 .
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.
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 ).
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.
If one thinks of as an ordinary -dimensional smooth manifold, then will be given by a volume form, . By contraction of multivector fields with differential forms, every choice of volume form on induces an isomorphism between differential forms and polyvector fields
if we think of
the measure as some closed reference differential form on ;
the exponentiated action functional as a multivector field on ;
the expression as the contraction of this multivector field with
then the BV quantum master equaton says nothing but that is a closed differential form.
If we furthermore take into account that in the presence of gauge symmetries the space is not a plain manifold but the -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 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 -algebroid are the so-called “anti-fields”, while those dual to the higher degree elements are the so-called “anti-ghosts”.
On the open ball of dimension , the integration of differential forms of compact support 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 .
This “integration without integration” is discussed in more detail at Lie integration.
Write for the cochain cohomology classes in the BV complex .
See (Gwilliam 13, lemma 2.2.2).
If then the BV-complex induced via def. 2 by the volume form
(for any constant to be read as Planck's constant) has BV-differential related to that of itself by
is the derived critical locus of the function .
By the discussion at derived critical locus.
Prop. 3 and prop. 4 together say that the BV-complex of a manifold for a volume form shifted from a background volume form by a function is an -deformation of the derived critical locus of by a contrinution of the background volume form .
We call the classical BV complex and the quantum BV complex of the manifold equipped with the function and the voume form .
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 2 as a derived critical locus directly makes sense in great generality for action functionals 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.
A quantum BV complex or Beilinson-Drinfeld algebra is a -graded algebra over the ring of formal power series in a formal constant , equipped with a Poisson bracket of degree 1 and with an operator of degree 1 which satisfies:
for all homogenous elements
In (Gwilliam 2013) this is def. 2.2.5.
A Beilinson-Drinfeld algebra is not a dg-algebra with differential : the Poisson bracket measures the failure for the differential to satisfy the Leibniz rule. In particular the -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).
hence the result of setting the formal parameter (“Planck's constant”) to 0.
In (Gwilliam 2013) this is def. 2.2.6.
|action functional||kinetic action||interaction||path integral measure|
|BV differential||elliptic complex +||antibracket with interaction +||BV-Laplacian|
Specifically, an observable is a closed element in the quantum BV-complex and its expectation value is its image in cochain cohomology.
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).
Multivector fields may be understood in terms of Hochschild cohomology of . Under the identification of Hochschild homology/cyclic homology with the de Rham complex the product of the action functional with a formal measure on 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 and is regarded as a cocycle in cyclic cohomology, hence as a multivector field whose closure condition 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 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.
A classical standard references is
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
The extension of this approach to higher dimensions is being worked out in terms of factorization algebra in
The general classical BV formalism (chapter 17 of Henneaux-Teitelboim) is formalized in the same language in
and in the book
A systematic/axiomatic account from the point of view of higher geometry is given in
The original articles are
Reviews are in
J. Park, Pursuing the quantum world (pdf)
Geometrical aspects were pioneered in
A systematic account of the classical master equation is also in
Other discussions include
M. Bächtold, On the finite dimensional BV formalism, 2005. (pdf)
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
Relation to Feynman diagrams is made explicit in
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
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)
A mathematically oriented reformulation of some of this (in the context of TCFT ) is in
The perspective on the BV-complex as a derived critical locus is indicated in
Related Chern-Simons type graded action functionals are discussed also in
Lectures, discussing also the relation to the graph complex are
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, 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
BRST formalism is discussed in
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
Remarks on the homotopy theory interpretation of BRST-BV are in
A standard textbook on the application of BRST-BV to gauge theory is
Basics of Poisson reduction (blog)
Alejandro Cabrera, Homological BV-BRST methods: from QFT to Poisson reduction (pdf)
S. Lyakhovich, A. Sharapov, BRST theory without Hamiltonian and Lagrangian (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)