Orbifold cup products and ring structures on Hochschild cohomologies

TL;DR

This paper investigates the Hochschild cohomology ring structures of orbifold convolution algebras and their deformation quantizations, revealing connections with twisted polyvectorfields, Chen--Ruan product, and equivariant cohomology models.

Contribution

It introduces a new ring structure on Hochschild cohomology for orbifolds and relates it to deformation quantization and equivariant orbifold cohomology.

Findings

Ring structure given by wedge product on twisted polyvectorfields

Deformation quantization induces a product on orbifold cohomology

Provides a de Rham model for equivariant orbifold cohomology

Abstract

In this paper we study the Hochschild cohomology ring of convolution algebras associated to orbifolds, as well as their deformation quantizations. In the first case the ring structure is given in terms of a wedge product on twisted polyvectorfields on the inertia orbifold. After deformation quantization, the ring structure defines a product on the cohomology of the inertia orbifold. We study the relation between this product and an $S^1$-equivariant version of the Chen--Ruan product. In particular, we give a de Rham model for this equivariant orbifold cohomology.