Continuous Hochschild Cohomology and Formality

TL;DR

This paper develops a homological framework for deformation theory of complete locally convex dg-algebras, proving formality theorems for various algebraic structures on manifolds and complex varieties.

Contribution

It introduces a new homological setting for deformation theory using Positselski's contraderived categories and establishes formality theorems for several important classes of algebras.

Findings

Formality theorems for Fréchet algebras of smooth functions, de Rham, and Dolbeault algebras.

Hochschild cohomology matches known deformation complexes in special cases.

Computed continuous Hochschild cohomology for matrix factorisation categories.

Abstract

We define the appropriate homological setting to study deformation theory of complete locally convex (curved) dg-algebras based on Positselski's contraderived categories. We define the corresponding Hochschild complex controlling deformations and prove formality theorems for the Fr\'echet algebras of smooth functions on a manifold, the de Rham algebra and for the Dolbeault algebra of a complex manifold. In the latter case, the Hochschild cohomology is equivalent to Kontsevich's extended deformation complex, the Hochschild cohomology of the derived category in case $X$ is a smooth projective variety and to Gualtieri's deformation complex of $X$ viewed as generalized complex manifold. We also compute the continuous Hochschild cohomology for various categories of matrix factorisations.