Topological quantum field theories and homotopy cobordisms

Abstract: I will begin by explaining the construction of a category CofCos, whose objects are topological spaces and whose morphisms are cofibrant cospans. Here the identity cospan is chosen to be of the form $X\to X\times [0,1]\rightarrow X$, in contrast with the usual identity in the bicategory $Cosp(V)$ of cospans over a category $V$. The category $CofCos$ has a subcategory $HomCob$ in which all spaces are homotopically 1-finitely generated. There exist functors into HomCob from a number of categorical constructions which are potentially of use for modelling particle trajectories in topological phases of matter: embedded cobordism categories and motion groupoids for example. Thus, functors from HomCob into Vect give representations of the aforementioned categories.

I will also construct a family of functors $Z_G\colon HomCob\to Vect$, one for each finite group $G$, and show that topological quantum field theories previously constructed by Yetter, and an untwisted version of Dijkgraaf-Witten, generalise to functors from HomCob. I will construct this functor in such a way that it is clear the images are finite dimensional vector spaces, and the functor is explicitly calculable. I will also give example calculations throughout.

Mathematics

Audience: researchers in the topic

European Quantum Algebra Lectures (EQuAL)

Series comments: EQuAL is an online seminar series on quantum algebra and related topics such as topological and conformal field theory, operator algebra, representation theory, quantum topology, etc. sites.google.com/view/equalseminar/home