Topic list and references for a seminar on BV-BRST formalism in quantum field theory and derived critical loci, spring 2011.
For background see the previous Seminar on derived differential geometry and the next Seminar on quantum field theory.
A rich source of examples of higher geometry/derived geometry comes from quantum field theory: given an action functional $S : Conf \to \mathbb{A}^1$ its corresponding covariant phase space is
the quotient $Conf/G_{gauge}$ of configuration space $Conf$ by the gauge symmetry group $G_{gauge}$;
the restriction $Conf^{d S = 0}/G_{gauge}$ to the critical locus of $S$.
To make this combined colimit/limit space exist in a reasonable way, one needs to form genuine (β,1)-colimits/(β,1)-limits and consider the resulting object in derived geometry.
In the context of dg-geometry these homotopical reductions have been studied for a long time in the context of BV-BRST formalism . The plan is that we proceed in the following steps
understand objects in dg-geometry as L-β algebras and L-β algebroids;
study the traditional theory of BV-BRST formalism;
re-interpret that formalism in derived geoemtry as being a model for derived critical loci;
find this way a more abstract understanding for instance of the BV-algebra structure that controls this formalism.
The notion of L-β algebra (or strongly homotopy Lie algebra / sh Lie algebra ) is the homotopical refinement of the notion of Lie algebra. Since a Lie algebra is the algebra over an operad of the Lie operad, an L-β algebra is an algebra over a resolution thereof: an L-β operad.
By Koszul duality for operads, $L_\infty$-algebras $\mathfrak{g}$ may dually equivalently be encoded in semifree dg-algebras: the higher analogs of the Chevalley-Eilenberg algebra $CE(\mathfrak{g})$ of a Lie algebra. Many dg-algebraic structures appearing in the literature β such as the BRST-complex discussed below β are therefore secret incarnations of L-β algebras, or more generally of L-β algebroids if there is a nontrivial base space.
The classical correspondence between Lie algebras and (simply connected) Lie groups has analogs in β-Lie theory. Notably there is an operation of $\infty$-Lie integration that sends an L-β algebra to the corresponding smooth β-groupoid that integrates it.
It turns out that at its heart, and applied to the dual CE algebras, this operation is the classical Sullivan construction in rational homotopy theory that assigns a simplicial set and hence a (rational topological space) topological space to a dg-algebra. This Sullivan construction may be thought of as producing the discrete β-groupoid underlying the smooth β-groupoid that integrates the $L_\infty$-algebra dual to the given dg-algebra (Hinich, Getzler09). A refinement of this construction that produces internal β-groupoids in Banach spaces was given in (Henriques)
This way, in turn, many dg-algebraic constructions in the literature are secretly infinitesimal models for β-groupoids. For instance the BRST complex discussed below is a model for the Lie algebroid that corresponds to the action groupoid of a Lie group acting on a smooth manifold.
The original references on L-β algebras are
Tom Lada and Jim Stasheff, Introduction to sh Lie algebras for physicists, Int. J. Theo. Phys. 32 (1993), 1087β1103. (arXiv)
Tom Lada, Martin Markl, Strongly homotopy Lie algebras (arXiv:hep-th/9406095)
The discussion of Lie integration of an L-β algebra to a discrete β-groupoid goes back (somewhat implicitly) to the Sullivan construction in rational homotopy theory, which has been made explicit as a construction in Lie theory in
The Lie integration to a genuinely smooth β-groupoid, or at least to internal β-groupoid in Banach spaces was described in
A simple version of higher derived geometry is dg-geometry , where spaces are modeled as formal duals to commutative dg-algebras.
We have seen above that
a dg-algebra in non-negative degree models a weak quotient: it may be thought of as the Chevalley-Eilenberg algebra of an L-β algebra or L-β algebroid,
a dg-algebra in non-positive degree models a derived intersection: it may be regared as an analog of Koszul-Tate resolution of its degree-0 cohomology.
These two dual aspects are notably unified in derived Poisson geometry:
for $(X, \{-,-\})$ a Poisson manifold and $I \subset C^\infty(X)$ an ideal that is closed under the Poisson bracket, the Poisson reduction of $X$ induced by $I$ is the operation consisting of passage
to the quotient of $X$ by the $\{I,-\}$-action of $I$;
to the intersection of $X$ with the 0-locus of $I$.
The result of performing both of these steps in derived dg-geometry is (somewhat informally) known as the BRST-BV complex of $X$ and $I$.
One speaks of
A standard reference on homotopical Poisson reduction and BRST-BV formalism is
Lagrangian BV-formalism is a way to restate the following simple finite-dimensional situation in terms of more general Hochschild cohomology/cyclic cohomology.
If we have an action functional $\exp(i S) : C \to S^1$ on a finite-dimensional compact smooth manifold $C$ with volume form $\omega$ then $\exp(i S(-)) \wedge vol$ is a closed differential form
whose integral
is the path integral that defines the corresponding quantum field theory.
It was amplified in (Witten) that in the presence of the volume form we may identify differential forms with multivector fields and rephrase the closure condition on the action by what is called the master equation
where $\Delta$ is the image of the de Rham differential under the isomorphism between forms and multivector fields (see multivector field) for details. Together with the canonical Schouten bracket on multivector field this makes these a BV-algebra, with $\Delta$ the BV-operator.
We may understand this more generally by realizing that differential forms are encoded by Hochschild homology and multivector fields by Hochschild cohomology. There is generally a BV-algebra structure on Hochschild cohomology. And for sufficiently nice spaces there is PoincarΓ© duality
takes the de Rham differention to the BV-operator.
Moreover, we may understand the Hochschild homology of $C$ as the cohomology of the derived loop space $\mathcal{L}C$ of $C$ and under this identification the BV-operator corresponds to the rotation of loops.
Accordingly, the resuling BV-algebra has an interpretation as an algebra over (the homology of) the framed little disk operad. This we discuss below.
A modern geometric way for describing variational calculus and Lagrangian BV-formalism has been initiated in section 2
in terms of D-geometry: the geometry over infinitesimal path β-groupoids?/de Rham stacks $\mathbf{\Pi}_{inf}(X)$: the jet bundle of a bundle $E \to X$ is its direct image along the constant infinitesimal path inclusion $X \to \mathbf{\Pi}_{inf}(X)$.
A useful exposition of this D-geometric way to BV-formalism with an eye on actual applications in physics is in
The formal interpretation of the Lagrangian BV-formalism is apparently due to
Geometrical aspects and BV-integration are discussed by Schwarz:
Albert Schwarz, Geometry of Batalin-Vilkovisky quantization, Commun. Math. Phys. 155, 249 (1993), euclid
Albert Schwarz, Semiclassical approximation in Batalin-Vilkovisky formalism, Comm. Math. Phys. 158 (1993), no. 2, 373β396, euclid
A survey discussion of the BV-algebra arising in BV-quantization is for instance in
PoincarΓ© duality on Hochschild (co)homology
M. Van den Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings . Proc. Amer. Math. Soc. 126 (1998), 1345β1348; (JSTOR)
Correction: Proc. Amer. Math. Soc. 130 (2002), 2809β2810.
with more on that in
and
The BV-BRST formalism was thought up long before derived geometry was fully conceived. It is in retrospect that one recognized this old construction of an example of this new geometry.
In
we give one precise version of the folklore that the BV-BRST complex indeed represents the derived critical locus of an action functional, by showing that it indeed is a homotopy fiber of $d S : Conf \to T^* Conf$.
The rich structure of Lagrangian BRST-BV formalismes is to a large extent due to the graded Poisson algebra structure that these carry. Graded and higher Poisson algebras are sometimes known as Poisson n-algebras. A Poisson 2-algebra is a Gerstenhaber algebra.
In addition the BV-Laplacian makes this Gerstenhaber algebra a BV-algebra and a homotopy BV-algebra for higher $n$.
Higher Poisson algebras turn out to have a natural origin: they are the algebras over an operad over a little disk operad. Moreover, their refinement to a homotopy BV-algebra is given by a refinement to an alfgebra over the homology of a framed little disk operad.
These little disk operads control the quantum field theory encoded by BRST-BV complexes in physics. See factorization algebra.
The relation of BV-algebras to framed little disk operad is due to
The generalizations to the framed little n-disk operads is discussed in