Triviality of the higher formality theorem

Abstract

It is noted that the higher version of M. Kontsevich’s Formality Theorem is much easier than the original one. Namely, we prove that the higher Hochschild-Kostant-Rosenberg map taking values in the n-Hochschild complex already respects the natural E$_{n+1}$ operad action whenever n $\ge,$ 2. To this end we introduce a higher version of the braces operad, which—analogously to the usual braces operad—acts naturally on the higher Hochschild complex, and which is a model of the E$_{n+1}$operad.