the restriction to the critical locus of .
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-∞ 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-∞ algebra is an algebra over a resolution thereof: an L-∞ operad.
By Koszul duality for operads, -algebras may dually equivalently be encoded in semifree dg-algebras: the higher analogs of the Chevalley-Eilenberg algebra of a Lie algebra. Many dg-algebraic structures appearing in the literature – such as the BRST-complex discussed below – are therefore secret incarnations of L-∞ algebras, or more generally of L-∞ algebroids if there is a nontrivial base space.
The classical correspondence between Lie algebras and (simply connected) Lie groups has analogs in ∞-Lie theory. Notably there is an operation of -Lie integration that sends an L-∞ algebra to the corresponding smooth ∞-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 ∞-groupoid underlying the smooth ∞-groupoid that integrates the -algebra dual to the given dg-algebra (Hinich, Getzler09). A refinement of this construction that produces internal ∞-groupoids in Banach spaces was given in (Henriques)
This way, in turn, many dg-algebraic constructions in the literature are secretly infinitesimal models for ∞-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-∞ algebras are
The discussion of Lie integration of an L-∞ algebra to a discrete ∞-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
We have seen above that
These two dual aspects are notably unified in derived Poisson geometry:
to the quotient of by the -action of ;
to the intersection of with the 0-locus of .
One speaks of
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 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 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 Poincaré duality
takes the de Rham differention to the BV-operator.
Moreover, we may understand the Hochschild homology of as the cohomology of the derived loop space of and under this identification the BV-operator corresponds to the rotation of loops.
A modern geometric way for describing variational calculus and Lagrangian BV-formalism has been initiated in section 2
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:
A survey discussion of the BV-algebra arising in BV-quantization is for instance 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
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.
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.
The generalizations to the framed little n-disk operads is discussed in