Van Den Bergh isomorphisms in String Topology

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 $M$, $H_{\ast +d}(LM)$ 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 $$H{H}^{-p}(S_{\ast }(G),S_{\ast }(G))\cong H{H}_{p+d}(S_{\ast }(G),S_{\ast }(G))\mathrm{.}$$ Therefore, the Gerstenhaber algebra $H{H}^{\ast }(S_{\ast }(G),S_{\ast }(G))$ is a Batalin-Vilkovisky algebra and we have a linear isomorphism $$H{H}^{\ast }(S_{\ast }(G),S_{\ast }(G))\cong H_{\ast +d}(LM)\mathrm{.}$$ This linear isomorphism is expected to be an isomorphism of Batalin-Vilkovisky algebras. We also give a new characterization of Batalin-Vilkovisky algebra in term of derived bracket.