The Unitary Cobordism Hypothesis

Abstract: The cobordism hypothesis classifies extended topological quantum field theories (TQFTs) in terms of algebraic information in the target category. One of the core principles in quantum field theory - unitarity - says that state spaces are not just vector spaces, but Hilbert spaces. Recently in joint work with many others, we have defined unitarity for extended TQFTs, inspired by Freed and Hopkins. Our main technical tool is a higher-categorical version of dagger categories; categories $C$ equipped with a strict anti-involution $\dagger: C \to C^{op}$ which is the identity on objects. I explain joint work in progress with Theo Johnson-Freyd, Cameron Krulewski and Lukas Müller in which we prove a version of the cobordism hypothesis for unitary TQFTs. The main observation is that the stably framed bordism $n$-category is freely generated as a symmetric monoidal dagger $n$-category with unitary duals by a single object: the point.

mathematical physicsalgebraic topologycategory theory

Audience: researchers in the topic

Topology and Geometry Seminar (Texas, Kansas)