Hochschild cohomology of quantized symplectic orbifolds and the Chen-Ruan cohomology

TL;DR

This paper proves a conjecture linking Hochschild cohomology of invariant quantum algebras to Chen-Ruan cohomology of symplectic orbifolds, establishing a deep connection between algebraic and geometric invariants.

Contribution

It proves the additive version of Ginzburg and Kaledin's conjecture relating Hochschild cohomology to Chen-Ruan cohomology for symplectic orbifolds.

Findings

Hochschild cohomology of invariant quantum algebras matches Chen-Ruan cohomology

The conjecture holds for orbifolds modeled on quotients of smooth affine symplectic varieties

Provides a bridge between algebraic invariants and geometric stringy cohomology

Abstract

We prove the additive version of the conjecture proposed by Ginzburg and Kaledin. This conjecture states that if X/G is an orbifold modeled on a quotient of a smooth affine symplectic variety X (over C) by a finite group G\subset Aut(X) and A is a G-stable quantum algebra of functions on X then the graded vector space HH(A^G) of the Hochschild cohomology of the algebra A^G of invariants is isomorphic to the graded vector space H_{CR}(X/G)((h)) of the Chen-Ruan (stringy) cohomology of the orbifold X/G.