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 $Z_G : HomCob \to Vect$, one for each finite group $G$, showing 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.

mathematical physicsalgebraic topologycategory theoryquantum algebra

Audience: researchers in the topic

( video )

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Topological Quantum Field Theory Club (IST, Lisbon)

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