Continuous Hochschild Cohomology and Formality
Patrick Antweiler
TL;DR
The paper develops a framework to study deformations of complete locally convex (curved) dg-algebras using continuous Hochschild cohomology within contraderived categories, addressing the locality issues of polydifferential Hochschild theory. It introduces contraderived categories $D^{ctr}(A)$ for complete locally convex algebras and constructs a canonical dg-model via graded-projective modules, establishing a robust setting for $HH_{cont}^n(A,A)$. Formally, the authors prove that, for manifolds, the continuous Hochschild complexes of $C^ olinebreak{ ext{∞}}(M)$, the de Rham algebra $ abla ext{Ω}(M)$, and the Dolbeault algebra $ ablaar{ ext{A}}(X)$ are quasi-isomorphic to the corresponding geometric deformation complexes (HKR-type identifications and $L_ extinfty$-extensions), linking to Kontsevich's ambient formality and Gualtieri's generalized complex deformations. They also compute $HH_{cont}$ for matrix factorisations and outline the behavior for curved algebras via curved bar constructions, suggesting a unified analytic deformation theory with potential applications to enhanced derived categories and analytic quasi-coherent sheaves.
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 Fréchet 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.