BATALIN-VILKOVISKY ALGEBRA STRUCTURES ON HOCHSCHILD COHOMOLOGY

摘要

Let M be any compact simply-connected oriented d-dimensional smooth manifold and let 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 conjectured 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*(S1) (LM).