Formality theorem for Hochschild cochains via transfer

TL;DR

This paper constructs an extended operad G^+ that unifies structures governing homotopy Gerstenhaber algebras and open-closed homotopy algebras, demonstrating its role in formality and non-formality results for Hochschild cochains.

Contribution

It introduces the G^+ operad extending existing operads, and applies it to transfer formality structures in Hochschild cochains of A-infinity algebras.

Findings

G^+ extends Tamarkin's G-structure to pairs (C(A), A)

A formality quasi-isomorphism is obtained via transfer of G^+

G^+ is a sub operad of the first sheet E^1(SC) of a spectral sequence

Abstract

We construct a 2-colored operad G^+ which, on the one hand, extends the operad G governing homotopy Gerstenhaber algebras and, on the other hand, extends the 2-colored operad governing open-closed homotopy algebras (OCHA). We show that Tamarkin's G-structure on the Hochschild cochain complex C(A) of an A-infinity algebra A extends naturally to a G^+ structure on the pair (C(A), A). We show that a formality quasi-isomorphism for the Hochschild cochains of the polynomial algebra can be obtained via transfer of this G^+ structure to the cohomology of the pair (C(A), A). We show that G^+ is a sub DG operad of the first sheet E^1(SC) of the homology spectral sequence for the Fulton-MacPherson version SC of Voronov's Swiss Cheese operad. Finally, we prove that the DG operads G^+ and E^1(SC) are non-formal.