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

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 rôle 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.

The first and second Section on $B_{\infty }$-algebras and modules have been completely re-written, with new results; partial revision of Section 3; the proofs in Section 4 and 5 have been re-formulated in a more general context; we added Section 8 on globalisation

References in corpus (6)

• Hochschild cohomology and Atiyah classes
• Nonhomogeneous quadratic duality and curvature
• Formality of the homotopy calculus algebra of Hochschild (co)chains
• Caldararu's conjecture and Tsygan's formality
• A Counterexample to the Quantizability of Modules
• Shoikhet's Conjecture and Duflo Isomorphism on (Co)Invariants

Cited by in corpus (9)

• The Homotopy Braces Formality Morphism
• Caldararu's conjecture and Tsygan's formality
• Bimodules and branes in deformation quantization
• Gravity algebra structure on the negative cyclic homology of Calabi-Yau algebras
• Categorical Saito theory, I: A comparison result
• Poisson cohomology, Koszul duality, and Batalin-Vilkovisky algebras
• Shoikhet's Conjecture and Duflo Isomorphism on (Co)Invariants
• The character map in deformation quantization
• Geometry of Localized Effective Theories, Exact Semi-classical Approximation and the Algebraic Index