Deformations of algebras over operads and Deligne's conjecture

TL;DR

This paper develops a deformation theory for operads and their algebras, applying it to prove Deligne's conjecture by constructing a geometric operad acting on the Hochschild complex and exploring related group actions.

Contribution

It introduces a new geometric operad acting on Hochschild complexes and advances the deformation theory of operads, providing a novel proof of Deligne's conjecture.

Findings

Constructed a geometric operad acting on Hochschild complexes.

Proved Deligne's conjecture using the developed deformation theory.

Showed the Grothendieck-Teichmüller group acts on the moduli space of 2-algebra structures.

Abstract

In present paper we develop the deformation theory of operads and algebras over operads. Free resolutions (constructed via Boardman-Vogt approach) are used in order to describe formal moduli spaces of deformations. We apply the general theory to the proof of Deligne's conjecture. The latter says that the Hochschild complex of an associative algebra carries a canonical structure of a dg-algebra over the chain operad of the little discs operad. In the course of the proof we construct an operad of geometric nature which acts on the Hochschild complex. It seems to be different from the brace operad (the latter was used in the previous approaches to the Deligne's conjecture). It follows from our results that the Grothendieck-Teichm\"uller group acts (homotopically) on the moduli space of structures of 2-algebras on the Hochschild complex. In the Appendix we develop a theory of piecewise algebraic chains and forms. It is suitable for real semialgebraic manifolds with corners (like Fulton-Macpherson compactifications of the configuration spaces of points).