Van den Bergh isomorphisms in string topology

Abstract

Let $M$ be a path-connected closed oriented $d$-dimensional smooth manifold and let $\mathbb{k}$ be a principal ideal domain. By Chas and Sullivan, the shifted free loop space homology of $M$, $H_{\ast +d}(\mathrm{L}M)$ is a Batalin–Vilkovisky algebra. Let $G$ be a topological group such that $M$ is a classifying space of $G$. Denote by $S_{\ast }(G)$ the (normalized) singular chains on $G$. Suppose that $G$ is discrete or path-connected. We show that there is a Van Den Bergh type isomorphism

Therefore, the Gerstenhaber algebra ${\mathrm{HH}}^{\ast }(S_{\ast }(G),S_{\ast }(G))$ is a Batalin–Vilkovisky algebra and we have a linear isomorphism

This linear isomorphism is expected to be an isomorphism of Batalin–Vilkovisky algebras. We also give a new characterization of Batalin–Vilkovisky algebra in terms of the derived bracket.