Speaker: Dr. Hendryk Pfeiffer
Title: Two-dimensional open-closed TQFT from a state sum model
Abstract: Two-dimensional closed TQFTs can be described either `globally' by a functor from 1+1)-dimensional cobordisms to vector spaces, or `locally' by the state sum of Fukuma-Hosono-Kawai (FHK). The essence of the global description is the vector space associated with the circle which is automatically finite-dimensional and equipped with the structure of a commutative Frobenius algebra. The state sum, on the contrary, employs a symmetric, but not necessarily commutative Frobenius algebra which is not directly related to the former. We present a generalization of the FHK construction to the open-closed case, i.e. to 2-manifolds with corners (open strings without any structure on the world-sheet if you wish), and show that the algebraic structures of the global and of the local construction agree. Taking into account the corners thus clarifies the relationship between the algebra used in the state sum construction and the topology of 2-manifolds. This work can be seen as a warm-up exercise in order to understand to what extent modular categories (and thereby 3d quantum gravity) know the topology of 3-manifolds. (joint work with Aaron Lauda, DPMMS, University of Cambridge)
Talk: Transparencies, Recorded Talk