Equivalence of formalities of the little discs operad

Abstract

There is a remarkable Drinfeld associator given by Kontsevich’s integrals over configuration spaces of points in the plane. Using this Drinfeld associator we show that Kontsevich’s and Tamarkin’s formalities of the little discs operad are homotopic. The basic technical tool for this result is an $L_{\infty }$-algebra of graphs whose cohomology is the Drinfeld-Kohno Lie algebra of infinitesimal pure braids.