Abstract

A topological field theory is an invariant of oriented manifolds, valued in some category C, with many pleasant properties.  According to the cobordism hypothesis, a fully extended— a.k.a. fully local— TFT is uniquely determined by a single object of C, which we may think of as the invariant assigned by the theory to the point.  This object must have strong finiteness properties, called dualizability, and strong symmetry properties, called orientability.

In this talk I'd like to give an expository discussion of several recent works “in dimension 1,2, and 3” —of Schommer-Pries, Douglas—Schommer-Pries—Snyder, Brandenburg-Chivrasitu-Johnson-Freyd, Calaque-Scheimbauer —which unwind the abstract notions of dualizability and orientability into notions very familiar to the assembled audience:  things like Frobenius algebras, fusion categories, pivotal fusion categories, modular tensor categories.  Finally in this context, I'll discuss some work in progress with Adrien Brochier and Noah Snyder, which finds a home on these shelves for arbitrary tensor and pivotal tensor categories (no longer finite, or semi-simple), and for braided and ribbon braided tensor categories.