Cyclic homology, graph homology and maps between them

The ribbon graph complex has been invented by Kontsevich as the chain complex of a certain orbi-cellular decomposition of the moduli space of decorated Riemann surfaces. It is also part of a modular operad whose algebras are A-infinity algebras with an invariant inner product. As a consequence of that, such an A-infinity algebra determines cohomology classes on the moduli spaces of Riemann surfaces; this is the so-called 'direct' construction of Kontsevich. There is some resemblance between the ribbon graph complex and the Connes-Tsygan cyclic complex of an associative (more generally, A-infinity) algebra. I will explain how to make this connection precise by constructing a family of maps from cyclic homology to graph homology. This could be viewed as a higher analogue of the Kontsevich construction. I will then describe some applications to derived Morita theory. If time permits, I will also describe another, superficially similar, family of maps from cyclic cohomology to the graph homology, generalizing the construction of Penkava-Schwarz. Both constructions point to a rather mysterious relationship between cyclic (co)homology and ribbon graph homology. This is a joint work with J. Chuang.