Algebra/Topology seminar

Speaker: David Reutter

Title: Semisimple topological quantum field theories and stable diffeomorphisms

Abstract:  A major open problem in quantum topology is the construction of an oriented 4-dimensional topological quantum field theory (TQFT) in the sense of Atiyah-Segal which is sensitive to exotic smooth structure. More generally, how much manifold topology can a TQFT see? 

In this talk, I will answer this question for `semisimple’ field theories in even dimensions — I will sketch a proof that such theories can at most see the stable diffeomorphism type of a manifold and conversely, that if two sufficiently finite manifolds are not stably diffeomorphic, then they can be distinguished by semisimple field theories. In this context, `semisimplicity' is a certain algebraic condition applying to all currently known examples of linear algebraic TQFTs, including `unitary field theories’, and `once-extended field theories' which assign algebras or linear categories to codimension 2 manifolds. I will discuss implications in dimension 4, such as the fact that oriented semisimple field theories cannot see smooth structure, while unoriented ones can. This is based on arXiv:2001.02288 and joint work with Christopher Schommer-Pries.