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

TL;DR

This paper explores the algebraic structures underlying string topology of manifolds, connecting Hochschild homology and cohomology of certain categories to the homology of free loop spaces, with implications for topological field theories.

Contribution

It introduces the string topology category enriched over chain complexes, relates Hochschild homology to free loop space homology, and connects these to Fukaya categories and topological field theories.

Findings

Hochschild homology of the string topology category equals the homology of the free loop space.

Hochschild cohomology of open string chain algebras relates to free loop space homology in simply connected cases.

Spectrum-level analogues extend the algebraic-topological relationships to a broader context.

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.