The Homotopy Braces Formality Morphism

TL;DR

This paper extends Kontsevich's formality morphism to homotopy versions for braces and Gerstenhaber structures, demonstrating their homotopy equivalence to Tamarkin's morphism with the Alekseev-Torossian associator, unifying various graphical formality constructions.

Contribution

It introduces homotopy braces and Gerstenhaber morphisms, proving their homotopy equivalence to Tamarkin's morphism, thus resolving longstanding problems in deformation quantization.

Findings

Homotopy braces morphism extended from Kontsevich's formality.

Homotopy Gerstenhaber morphism constructed and related to Tamarkin's morphism.

Unification of multiple graphical formality constructions.

Abstract

We extend M. Kontsevich's formality morphism to a homotopy braces morphism and to a homotopy Gerstenhaber morphism. We show that this morphism is homotopic to D. Tamarkin's formality morphism, obtained using formality of the little disks operad, if in the latter construction one uses the Alekseev-Torossian associator. Similar statements can also be shown in the "chains" case, i.e., on Hochschild homology instead of cohomology. This settles two well known and long standing problems in deformation quantization and unifies the several known graphical constructions of formality morphisms and homotopies by Kontsevich, Shoikhet, Calaque, Rossi, Alm, Cattaneo, Felder and the author.