
Luuk Stehouwer
Dalhousie University
Title
The unitary cobordism hypothesis
Abstract
Atiyah gave us a definition of topological quantum field theories (TQFTs) in terms of category theory. The cobordism hypothesis classifies (extended) 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 generalization of dagger categories. I explain joint work in progress with Theo Johnson-Freyd, Cameron Krulewski and Lukas Müller in which we use these new techniques to prove a version of the cobordism hypothesis for unitary TQFTs. I will explicitly spell out the result in low dimensions, where we observe a connection with unitary representation theory.