A Batalin-Vilkovisky algebra morphism from double loop spaces to free loops

Let $M$ be a compact oriented $d$-dimensional smooth manifold and $X$ a topological space. Chas and Sullivan \cite{Chas-Sullivan:stringtop} have defined a structure of Batalin-Vilkovisky algebra on ${\mathbb{H}}_{\ast }(LM):=H_{\ast +d}(LM)$. Getzler \cite{Getzler:BVAlg} has defined a structure of Batalin-Vilkovisky algebra on the homology of the pointed double loop space of $X$, $H_{\ast }({\mathrm{\Omega }}^{2}X)$. Let $G$ be a topological monoid with a homotopy inverse. Suppose that $G$ acts on $M$. We define a structure of Batalin-Vilkovisky algebra on $H_{\ast }({\mathrm{\Omega }}^{2}BG)\otimes {\mathbb{H}}_{\ast }(M)$ extending the Batalin-Vilkovisky algebra of Getzler on $H_{\ast }({\mathrm{\Omega }}^{2}BG)$. We prove that the morphism of graded algebras $$H_{\ast }({\mathrm{\Omega }}^{2}BG)\otimes {\mathbb{H}}_{\ast }(M)\to {\mathbb{H}}_{\ast }(LM)$$ defined by Felix and Thomas \cite{Felix-Thomas:monsefls}, is in fact a morphism of Batalin-Vilkovisky algebras. In particular, if $G=M$ is a connected compact Lie group, we compute the Batalin-Vilkovisky algebra ${\mathbb{H}}_{\ast }(LG;\mathbb{Q})$.