The ribbon quiver complex and operations on Hochschild invariants

The structure of a fully extended oriented 2d TQFT is given by a Frobenius algebra. If one wants to lift this structure to a cohomological field theory, the correct notion is of a Calabi-Yau algebra or category; the CohFT operations are then described by a certain graph complex. There are many different notions of categorical Calabi-Yau structure, all requiring some type of finiteness or dualizability. In this talk I will discuss a variation that works in non-dualizable cases as well; in this case the graphs get replaced by quivers.

The resulting complex admits an algorithmic description of orientations, and calculates the homology of certain moduli spaces of open-closed surfaces. This can be used to give a fully explicit description of these operations. In the second half of the talk I will describe some of these constructions, including relative versions of Calabi-Yau structures, and some appearances of these structures in Fukaya theory and string topology. This is joint work with M. Kontsevich and Y. Vlassopoulos.