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

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

2009-06-29

arxiv.org/abs/0906.5198v1

Mathematics

Algebraic Topology

25 pages, 11 figures. To appear in, "Alpine perspectives on algebraic topology", edited by C. Ausoni, K. Hess, and J. Scherer,

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Open-closed field theories, string topology, and Hochschild homology does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Open-closed field theories, string topology, and Hochschild homology, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Open-closed field theories, string topology, and Hochschild homology will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-247226