## Rational BV-algebra in string topology.(English)Zbl 1160.55006

For a smooth orientable manifold $$M$$ of dimension $$m$$, let $$LM$$ denote the space of free loops on $$M$$, and let $$HH^*(A,R)$$ denote the Hochschild cohomology of a differential graded algebra $$A$$ with coefficients in the differential graded $$A$$-module $$R$$. Recall that Chas-Sullivan showed that the shifted homology $${\mathbb H}_*(LM,{\mathbf k}):=H_{*+m}(LM,{\mathbf k})$$ has the structure of a BV-algebra. If $${\mathbf k}$$ is a field, it is also known (due to a result of J. Jones) that there is an isomorphism of graded vector spaces $${\mathbb H}_*(LM,{\mathbf k})\cong HH^*(C^*(M);C^*(M))$$. In this paper, the authors prove that if $$M$$ is $$1$$-connected and $$ch({\mathbf k})=0$$, then Poincaré duality induces a $$BV$$-algebra structure on $$HH^*(C^*(M);C^*(M))$$ and that there is an isomorphism of BV-algebras, $${\mathbb H}_*(LM,{\mathbf k})\cong HH^*(C^*(M);C^*(M))$$. Moreover, they prove that the Chas-Sullivan product and the BV-operator behave well with respect to a Hodge decomposition of $${\mathbb H}_*(LM)$$.

### MSC:

 55P35 Loop spaces 55N33 Intersection homology and cohomology in algebraic topology 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
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