BatalinVilkovisky algebra structures on Hochschild Cohomology
Abstract
Let $M$ be any compact simplyconnected $d$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 BatalinVilkovisky algebra. Such BatalinVilkovisky algebra was conjecturated to exist and is expected to be isomorphic to the BatalinVilkovisky 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 ChasSullivan string bracket on the equivariant homology $H_*^{S^1}(LM)$.
 Publication:

arXiv eprints
 Pub Date:
 November 2007
 arXiv:
 arXiv:0711.1946
 Bibcode:
 2007arXiv0711.1946M
 Keywords:

 Mathematics  Quantum Algebra;
 Mathematics  Algebraic Topology
 EPrint:
 minor corrections, essentially typos