Symmetric multiplicative formality of the Kontsevich operad

TL;DR

This paper proves that the Kontsevich operad is formal over the reals as a multiplicative symmetric operad for dimensions greater than two, extending known formality results.

Contribution

It establishes the formality of the Kontsevich operad as a multiplicative symmetric operad over the reals for d > 2, a previously unresolved case.

Findings

Kontsevich operad is formal over reals as a symmetric operad for d > 2

Extends known formality results to the symmetric multiplicative case

Supports the broader understanding of operad formality in deformation quantization

Abstract

In his famous paper entitled "Operads and motives in deformation quantization", Maxim Kontsevich constructed (in order to prove the formality of the little d-disks operad) a topological operad, which is called in the literature the Kontsevich operad, and which is denoted Kd in this paper. This operad has a nice structure: it is a multiplicative symmetric operad, that is, it comes with a morphism from the symmetric associative operad. There are many results in the literature concerning the formality of Kd. It is well known (by Kontsevich) that Kd is formal over reals as a symmetric operad. It is also well known that Kd is formal as a multiplicative nonsymmetric operad. In this paper, we prove that the Kontsevich operad is formal over reals as a multiplicative symmetric operad, when d > 2.