Let M be a closed oriented smooth manifold, and LM be the free loop space of M. It has been known since the seminal work of Chas and Sullivan that loop operations produce rich algebraic structures on the ordinary homology and S^1-equivariant homology of LM, and there are similar structures on Hochschild cohomology and cyclic cohomology of algebras. In this talk, I will describe a chain level refinement of the gravity algebra structure on the (negative) S^1-equivariant homology of LM, and relate it to structures on the algebraic side, in the spirit of Deligne’s conjecture. This result is literally an application of a theorem of Ben Ward to a construction of Kei Irie; the key ingredient is a Jones’ type isomorphism between the S^1-equivariant homology of an S^1-space X and the cyclic homology of certain cocyclic chain complex associated to X.

Chain level gravity algebra structure in string topology