Compatibility with cap-products in Tsygan's formality and homological Duflo isomorphism

TL;DR

This paper proves that the isomorphism from Tsygan's formality for Hochschild chains respects cap-products, extending known compatibility results from cohomology to homology in a very general algebraic setting.

Contribution

It establishes the compatibility of the Tsygan formality isomorphism with cap-products in tangent homology, generalizing previous cohomological results and clarifying the role of homotopy structures.

Findings

Proves compatibility of Tsygan's formality with cap-products in tangent homology.

Clarifies the role of Kontsevich eye and I-cube in the proof.

Extends the compatibility results to a broad class of Maurer-Cartan elements.

Abstract

In this paper we prove, with details and in full generality, that the isomorphism induced on tangent homology by the Shoikhet-Tsygan formality $L_\infty$-quasi-isomorphism for Hochschild chains is compatible with cap-products. This is a homological analog of the compatibility with cup-products of the isomorphism induced on tangent cohomology by Kontsevich formality $L_\infty$-quasi-isomorphism for Hochschild cochains. As in the cohomological situation our proof relies on a homotopy argument involving a variant of {\bf Kontsevich eye}. In particular we clarify the r\^ole played by the {\bf I-cube} introduced in \cite{CR1}. Since we treat here the case of a most possibly general Maurer-Cartan element, not forced to be a bidifferential operator, then we take this opportunity to recall the natural algebraic structures on the pair of Hochschild cochain and chain complexes of an $A_\infty$-algebra. In particular we prove that they naturally inherit the structure of an $A_\infty$-algebra with an $A_\infty$-(bi)module.