Kontsevich's Swiss Cheese Conjecture

TL;DR

This paper proves Kontsevich's conjecture that the Hochschild cohomology of an $E_{d-1}$ algebra forms the universal $E_d$ algebra acting on it, using the swiss cheese operad framework.

Contribution

It establishes that the swiss cheese operad is generated up to homotopy by its degree 0 and 1 pieces, confirming a key conjecture in operad theory.

Findings

Hochschild cohomology of $E_{d-1}$ algebra is the universal $E_d$ algebra.

The swiss cheese operad is generated by its degree 0 and 1 pieces.

Proof of Kontsevich's conjecture on $E_d$ algebra actions.

Abstract

We prove a conjecture of Kontsevich which states that if $A$ is an $E_{d-1}$ algebra then the Hochschild cohomology object of $A$ is the universal $E_d$ algebra acting on $A$. The notion of an $E_d$ algebra acting on an $E_{d-1}$ algebra was defined by Kontsevich using the swiss cheese operad of Voronov. The degree 0 and 1 pieces of the swiss cheese operad can be used to build a cofibrant model for $A$ as an $E_{d-1}-A$ module. The theorem amounts to the fact that the swiss cheese operad is generated up to homotopy by its degree 0 and 1 pieces.