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 have defined a structure of Batalin-Vilkovisky algebra on ℍ*(LM) := H*+d(LM). Getzler (1994) has defined a structure of Batalin-Vilkovisky algebra on the homology of the pointed double loop space of X, H*(Ω2 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*(Ω2 BG) ⊗ ℍ*(M) extending the Batalin-Vilkovisky algebra of Getzler on H*(Ω2BG). We prove that the morphism of graded algebras H*(Ω2BG) ⊗ ℍ* (M) → ℍ* (LM) defined by Felix and Thomas (2004), 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 ℍ*(LG;ℚ)