Open-closed field theories, string topology, and Hochschild homology

Abstract

In this expository paper we discuss a project regarding the string topology of a manifold, that was inspired by recent work of Moore-Segal, Costello, and Hopkins and Lurie, on "open-closed topological conformal field theories". Given a closed, oriented manifold M, we describe the "string topology category" S_M, which is enriched over chain complexes over a fixed field k. The objects of S_M are connected, closed, oriented submanifolds N of M, and the complex of morphisms between N_1 and N_2 is a chain complex homotopy equivalent to the singular chains C_*(P_{N_1, N_2}), where C_*(P_{N_1, N_2}) is the space of paths in M that start in N_1 and end in N_2. The composition pairing in this category is a chain model for the open string topology operations of Sullivan and expanded upon by Harrelson, and Ramirez. We will describe a calculation yielding that the Hochschild homology of the category S_M is the homology of the free loop space, LM. Another part of the project is to calculate the Hochschild cohomology of the open string topology chain algebras C_*(P_{N,N}) when M is simply connected, and relate the resulting calculation to H_*(LM). We also discuss a spectrum level analogue of the above results and calculations, as well as their relations to various Fukaya categories of the cotangent bundle T^*M.