Proof of Swiss Cheese Version of Deligne's Conjecture

TL;DR

This paper proves a Swiss Cheese version of Deligne's conjecture, showing that a specific operad acting on Hochschild cochains is quasi-isomorphic to the Swiss Cheese operad, bridging algebraic and topological structures.

Contribution

It establishes a quasi-isomorphism between the operad acting on Hochschild cochains and the Swiss Cheese operad, extending Deligne's conjecture to a new topological context.

Findings

Operad acting on Hochschild cochains is quasi-isomorphic to the Swiss Cheese operad.

Provides a topological interpretation of algebraic structures in Hochschild cohomology.

Extends Deligne's conjecture to the Swiss Cheese setting.

Abstract

For an associative algebra A we consider the pair "the Hochschild cochain complex C*(A,A) and the algebra A". There is a natural 2-colored operad which acts on this pair. We show that this operad is quasi-isomorphic to the singular chain operad of Voronov's Swiss Cheese operad. This statement is the Swiss Cheese version of the Deligne conjecture formulated by M. Kontsevich in arXiv:math/9904055.