Semisimple Field Theories and Stable Diffeomorphisms

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

In this talk, I will outline an answer to this question for even-dimensional `semisimple’ TFT: 
Such theories can at most see the stable diffeomorphism type of a manifold and conversely, if two sufficiently finite manifolds are not stably diffeomorphic, then they can be distinguished by a semisimple field theory. In this context, `semisimplicity' is a certain algebraic condition satisfied by all currently known examples of linear algebraic TFT in more than two dimensions, and two 2n-manifolds are said to be stably diffeomorphic if they become diffeomorphic after connected sum with sufficiently many copies of S^n x S^n.

Along the way, I will introduce a number of semisimple TFTs built from homotopy types acted on by the orthogonal group O(n), and will discuss various implications, such as the fact that 4d oriented semisimple TFT cannot see smooth structure, while unoriented ones can. 

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