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.


Talk Number PIRSA:18050002
Speaker Profile David Jordan