Batalin-Vilkovisky algebra structures on Hochschild Cohomology

TL;DR

This paper proves that the Hochschild cohomology of the singular cochains on a compact simply-connected manifold has a Batalin-Vilkovisky algebra structure, linking it to string topology and loop space homology.

Contribution

It establishes the existence of a BV algebra structure on Hochschild cohomology of singular cochains, confirming a conjecture and connecting to string topology.

Findings

Hochschild cohomology extends to a BV algebra

Negative cyclic cohomology has a Lie bracket

Connections to Chas-Sullivan string bracket

Abstract

Let $M$ be any compact simply-connected $d$-dimensional smooth manifold and let $\mathbb{F}$ be any field. We show that the Gerstenhaber algebra structure on the Hochschild cohomology on the singular cochains of $M$, $HH^*(S^*(M);S^*(M))$, extends to a Batalin-Vilkovisky algebra. Such Batalin-Vilkovisky algebra was conjecturated to exist and is expected to be isomorphic to the Batalin-Vilkovisky algebra on the free loop space homology on $M$, $H_{*+d}(LM)$ introduced by Chas and Sullivan. We also show that the negative cyclic cohomology $HC^*_-(S^*(M))$ has a Lie bracket. Such Lie bracket is expected to coincide with the Chas-Sullivan string bracket on the equivariant homology $H_*^{S^1}(LM)$.