This is the first of two papers in which we prove that a cell model of the moduli space of curves with marked points and tangent vectors at the marked points acts on the Hochschild co-chains of a Frobenius algebra. We also prove that a there is dg-PROP action of a version of Sullivan chord diagrams which acts on the normalized Hochschild co-chains of a Frobenius algebra. These actions lift to operadic correlation functions on the co-cycles. In particular, the PROP action gives an action on the homology of a loop space of a compact simply-connected manifold.
In this first part, we set up the topological operads/PROPs and their cell models. The main theorems of this part are: There is a cell model operad for the moduli space of genus g curves with n punctures and a tangent vector at each of these punctures, there exists a CW complex whose chains are isomorphic to a certain type of Sullivan chord diagrams and they form a PROP. Furthermore there exist weak versions of these structures on the topological level which all lie inside an all encompassing cyclic (rational) operad.
Cite this article
Ralph M. Kaufmann, Moduli space actions on the Hochschild co-chains of a Frobenius algebra I: cell operads. J. Noncommut. Geom. 1 (2007), no. 3, pp. 333–384DOI 10.4171/JNCG/10