Open-closed field theories, string topology, and Hochschild homology [unknown]

Andrew J. Blumberg, Ralph L. Cohen, Constantin Teleman
2009 Contemporary Mathematics   unpublished
In this expository paper we discuss a project regarding the string topology of a manifold, that was inspired by recent work of Moore-Segal [28] and Costello [17] on "open-closed topological conformal field theories". In particular, given a closed, oriented manifold M , we describe the "string topology category" SM , which is enriched over chain complexes over a fixed field k. The objects of SM are closed, oriented submanifolds N of M , and the space of morphisms between N1 and N2 is a chain
more » ... d N2 is a chain complex homotopy equivalent to the singular chains C * (PN 1 ,N 2 ) where PN 1 ,N 2 , is the space paths in M that start in N1 and end in N2. The composition pairing in this category is a chain model for the open string topology operations of Sullivan [35], and expanded upon by Harrelson [23] and Ramirez [31]. We will describe a calculation yielding that the Hochschild homology of the category SM 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 * (PN,N ) when M is simply connected, and relate the resulting calculation to H * (LM ). These calculations generalize known results for the extreme cases of N = point and N = M , in which case the resulting Hochschild cohomologies are both isomorphic 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 with its canonical symplectic structure. Contents
doi:10.1090/conm/504/09875 fatcat:wjjzuza7izg7xco2vcmxqsn5le