According to the cobordism hypothesis, a fully local n-dimensional TFT with values in some (∞,n) category is determined by its “fully dualizable” and “SO(n)-fixed” objects. I will recall this general framework, and then explain recent works of R. Haugseng, T. Johnson-Freyd and C. Scheimbauer, which builds a 4-category called the Morita theory of braided tensor categories. Finally, I'll describe recent work of mine with A. Brochier and N. Snyder identifying natural 3- and 4-dualizable subcategories thereof: the rigid (3-dualizable) and fusion (4-dualizable) braided tensor categories. I'll also report on work in progress, also joint with Brochier and Snyder, constructing SO(3)- and SO(4)-fixed points, from ribbon and pre-modular categories, respectively.

Applying the cobordism hypothesis, we obtain new fully local 3- and 4-dimensional topological field theories. I'll outline expected relationships of these to: Crane-Yetter-Kauffman invariants, quantum A-polynomials, and DAHA-Jones polynomials.