Batalin-Vilkovisky algebra structures on Hochschild Cohomology

By Luc Menichi. 2007

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Let $M$ be any compact simply-connected $d$-dimensional smooth manifold and let $\mathbb{F}$ be any field. We show that the Gerstenhaber algebra structure on the Hochschild cohomology on the singular cochains of $M$, $HH^*(S^*(M);S^*(M))$, extends to a Batalin-Vilkovisky algebra. Such Batalin-Vilkovisky algebra was conjecturated to exist and is expected to be isomorphic to the Batalin-Vilkovisky algebra on the free loop space homology on $M$, $H_{*+d}(LM)$ introduced by Chas and Sullivan. We also show that the negative cyclic cohomology $HC^*_-(S^*(M))$ has a Lie bracket. Such Lie bracket is expected to coincide with the Chas-Sullivan string bracket on the equivariant homology $H_*^{S^1}(LM)$.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f239b57fdbd14440b2c0ecd151ffac16e9dd76bf