Rational BV-algebra in String Topology

Let $M$ be a 1-connected closed manifold and $LM$ be the space of free loops on $M$. In \cite{C-S} M. Chas and D. Sullivan defined a structure of BV-algebra on the singular homology of $LM$, $H_{\ast }(LM;\text{bk})$. When the field of coefficients is of characteristic zero, we prove that there exists a BV-algebra structure on ${\text{hH}}^{\ast }(C^{\ast }(M);C^{\ast }(M))$ which carries the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between ${\text{hH}}^{\ast }(C^{\ast }(M);C^{\ast }(M))$ and the shifted $H_{\ast +m}(LM;\text{bk})$. We also prove that the Chas-Sullivan product and the BV-operator behave well with the Hodge decomposition of $H_{\ast }(LM)$.