# Braided tensor categories and the cobordism hypothesis

The cobordism hypothesis gives a functorial bijection between oriented

n-dimensional fully local topological field theories, valued in some

higher category C, and the fully dualizable object of C equipped with

the structure of SO(n)-fixed point.  In this talk I'll explain recent

works of Haugseng, Johnson-Freyd and Scheimbauer which construct a

Morita 4-category of braided tensor categories, and I'll report on joint

work with Brochier and Snyder which identifies two natural subcategories

therein which are 3- and 4-dualizable.  These are the rigid braided

tensor categories with enough compact projectives, and the braided

fusion categories, respectively.  I'll also explain work in progress by

us to construct SO(3)- and SO(4)-fixed point structures in each case,

starting from ribbon and pre-modular categories, respectively.

Applying the cobordism hypothesis, we obtain 3- and 4-dimensional fully

local TFT's, which extend the 2-dimensional TFT's we constructed with

Ben-Zvi and Brochier, and which conjecturally relate to a number of

constructions in the literature, including: skein modules, quantum

A-polynomials, Crane-Kauffmann-Yetter invariants; hence our construction

puts these on firm foundational grounds as fully local TFT's.  A key

feature of our construction in dimension 3 is that we require the input

braided tensor category neither to be finite, nor semi-simple, so this

opens up new examples -- such as non-modularized quantum groups at roots

of unity -- which were not obtainable by earlier methods.

Collection/Series:
Event Type:
Seminar
Scientific Area(s):
Speaker(s):
Event Date:
Monday, May 7, 2018 - 14:00 to 15:30
Location:
Space Room
Room #:
400