Batalin-Vilkovisky algebra structures on Hochschild Cohomology

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 $M$, $H{H}^{\ast }(S^{\ast }(M);S^{\ast }(M))$, 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 $M$, $H_{\ast +d}(LM)$ introduced by Chas and Sullivan. We also show that the negative cyclic cohomology $H{C}_{-}^{\ast }(S^{\ast }(M))$ has a Lie bracket. Such Lie bracket is expected to coincide with the Chas-Sullivan string bracket on the equivariant homology $H_{\ast }^{S^1}(LM)$.

minor corrections, essentially typos