Effective Batalin--Vilkovisky theories, equivariant configuration spaces and cyclic chains

TL;DR

This paper extends Kontsevich's formality theorem to configuration spaces in the disk with rotational symmetry, constructing L-infinity morphisms linking cyclic chains and multivector fields, with applications to traces on star-product algebras.

Contribution

It introduces an equivariant approach to formality using disk configuration spaces, incorporating divergence operators and Batalin--Vilkovisky quantization for new algebraic structures.

Findings

Constructed L-infinity morphisms from cyclic chains to multivector fields.

Provided a method to define traces on star-product algebras.

Applied BV quantization to Poisson sigma models on the disk.

Abstract

Kontsevich's formality theorem states that the differential graded Lie algebra of multidifferential operators on a manifold M is L-infinity-quasi-isomorphic to its cohomology. The construction of the L-infinity map is given in terms of integrals of differential forms on configuration spaces of points in the upper half-plane. Here we consider configuration spaces of points in the disk and work equivariantly with respect to the rotation group. This leads to considering the differential graded Lie algebra of multivector fields endowed with a divergence operator. In the case of R^d with standard volume form, we obtain an L-infinity morphism of modules over this differential graded Lie algebra from cyclic chains of the algebra of functions to multivector fields. As a first application we give a construction of traces on algebras of functions with star-products associated with unimodular Poisson structures. The construction is based on the Batalin--Vilkovisky quantization of the Poisson sigma model on the disk and in particular on the treatment of its zero modes.