A Proof of Tsygan's Formality Conjecture for an Arbitrary Smooth Manifold

TL;DR

This paper proves Tsygan's formality conjecture for Hochschild chains on any smooth manifold using Fedosov resolutions and existing formality quasi-isomorphisms, enabling advances in algebraic and geometric quantization.

Contribution

It introduces Fedosov resolutions for Hochschild complexes on smooth manifolds and proves Tsygan's conjecture using Kontsevich and Shoikhet's formality quasi-isomorphisms.

Findings

Proves Tsygan's formality conjecture for arbitrary smooth manifolds.

Constructs functorial formality quasi-isomorphism for Hochschild chains.

Applications to equivariant quantization and Hochschild homology.

Abstract

Proofs of Tsygan's formality conjectures for chains would unlock important algebraic tools which might lead to new generalizations of the Atiyah-Patodi-Singer index theorem and the Riemann-Roch-Hirzebruch theorem. Despite this pivotal role in the traditional investigations and the efforts of various people the most general version of Tsygan's formality conjecture has not yet been proven. In my thesis I propose Fedosov resolutions for the Hochschild cohomological and homological complexes of the algebra of functions on an arbitrary smooth manifold. Using these resolutions together with Kontsevich's formality quasi-isomorphism for Hochschild cochains of R[[y_1, >..., y_d]] and Shoikhet's formality quasi-isomorphism for Hochschild chains of R[[y_1,..., y_d]] I prove Tsygan's formality conjecture for Hochschild chains of the algebra of functions on an arbitrary smooth manifold. The construction of the formality quasi-isomorphism for Hochschild chains is manifestly functorial for isomorphisms of the pairs (M,\nabla), where M is the manifold and \nabla is an affine connection on the tangent bundle. In my thesis I apply these results to equivariant quantization, computation of Hochschild homology of quantum algebras and description of traces in deformation quantization.