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 field 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 vector-space-valued 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.

Throughout, I will use the Crane-Yetter field theory associated to a ribbon fusion category as a guiding example.

This is based on arXiv:2001.02288 and joint work in progress with Chris Schommer-Pries.