Braided tensor categories and the cobordism hypothesis
APA
Jordan, D. (2018). Braided tensor categories and the cobordism hypothesis. Perimeter Institute. https://pirsa.org/18050002
MLA
Jordan, David. Braided tensor categories and the cobordism hypothesis. Perimeter Institute, May. 07, 2018, https://pirsa.org/18050002
BibTex
@misc{ pirsa_PIRSA:18050002, doi = {10.48660/18050002}, url = {https://pirsa.org/18050002}, author = {Jordan, David}, keywords = {Mathematical physics}, language = {en}, title = {Braided tensor categories and the cobordism hypothesis}, publisher = {Perimeter Institute}, year = {2018}, month = {may}, note = {PIRSA:18050002 see, \url{https://pirsa.org}} }
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.