Moduli theory associated to Hochschild pairs

TL;DR

This paper provides a moduli-theoretic interpretation of the algebraic structures on Hochschild pairs in stable infinity-categories, using operads and deformation theory to deepen understanding of Hochschild invariants.

Contribution

It introduces a moduli-theoretic framework for Hochschild pairs, connecting algebraic structures to deformation theory via the Kontsevich-Soibelman operad.

Findings

Algebraic structures on Hochschild pairs are encoded by the Kontsevich-Soibelman operad.

Cyclic and equivariant deformations are central to understanding Hochschild invariants.

Provides a new perspective linking deformation theory and Hochschild (co)homology.

Abstract

We consider an $A$-linear stable infinity-category $\mathcal{C}$ and the pair $(\mathcal{HH}^\bullet(\mathcal{C}/A),\mathcal{HH}_\bullet(\mathcal{C}/A))$ of the Hochschild cohomology spectrum (Hochschild cochain complex) and the Hochschild homology spectrum (Hochschild chain complex). The purpose of this paper is to provide a moduli-theoretic interpretation of the algebraic structure on the Hochschild pair of $\mathcal{C}$. The algebraic structure on the Hochschild pair is encoded by means of a two-colored topological operad called Kontsevich-Soibelman operad. The notions of cyclic deformations and equivariant deformations (of the Hochschild chain complex) associated to deformations of $\mathcal{C}$ play a central role.