  Chas and Sullivan showed that the homology of the free loop space
  LM of an oriented closed smooth finite dimensional manifold M admits the
structure of a Batalin-Vilkovisky (BV) algebra equipped with an associative
product called the loop product and a Lie bracket called the loop bracket. We
show that the cap product is compatible with the above two products in the loop
homology. Namely, the cap product with cohomology classes coming from M via the
circle action acts as derivations on loop products as well as on loop brackets.
We show that Poisson identities and Jacobi identities hold for the cap product
action, extending the BV structure in the loop homology to the one including
the cohomology of M. Finally, we describe the cap product in terms of the BV
algebra structure in the loop homology.
