Equivalence of formalities of the little discs operad

TL;DR

This paper demonstrates that two different formality constructions of the little disks operad, one by Kontsevich and another by Tamarkin, are homotopic under a specific choice of associator derived from the Alekseev-Torossian connection.

Contribution

It establishes the homotopy equivalence between Kontsevich's and Tamarkin's formality of the little disks operad for a particular associator.

Findings

Homotopy equivalence between Kontsevich's and Tamarkin's formality.

Use of Alekseev-Torossian connection to define the associator.

Specific choice of Drinfeld associator links the two formality constructions.

Abstract

We show that Kontsevich's formality of the little disk operad, obtained using graphs, is homotopic to Tamarkin's formality, for a special choice of a Drinfeld associator. The associator is given by parallel transport of the Alekseev-Torossian connection.