Hochschild Cohomology versus De Rham Cohomology without Formality Theorems

TL;DR

This paper establishes an isomorphism between Hochschild cohomology of quantum function algebras and De Rham cohomology of symplectic manifolds without relying on the formality theorem, revealing new structural insights.

Contribution

It introduces a novel isomorphism between Hochschild and De Rham cohomology without using the formality theorem, expanding understanding of symplectic geometry and quantum algebra relations.

Findings

Isomorphism between Hochschild cohomology and De Rham cohomology established

Gerstenhaber bracket on H(M,C((h))) is shown to vanish

Discussion of equivariant properties and algebraic geometric analogues

Abstract

We exploit the Fedosov-Weinstein-Xu (FWX) resolution proposed in q-alg/9709043 to establish an isomorphism between the ring of Hochschild cohomology of the quantum algebra of functions on a symplectic manifold M and the ring H(M, C((h))) of De Rham cohomology of M with the coefficient field C((h)) without making use of any version of the formality theorem. We also show that the Gerstenhaber bracket induced on H(M,C((h))) via the isomorphism is vanishing. We discuss equivariant properties of the isomorphism and propose an analogue of this statement in an algebraic geometric setting.