Citation: Constantin Teleman, "Towards a universal target for TQFTs", 2022 Simons Collaboration on Global Categorical Symmetries Annual Meeting
Tags: Mathematical, TQFT, Modular-tensor-categories
In this talk, Constantin Teleman talks about his work-in-progress on fully extending all (2+1)D Reshetikhin-Turaev TQFTs. The idea is quite simple. Physically, we should expect the following 3-category to exist. The object are (2+1)D topological phases. The 1-morphisms are gapped boundaries between the phases. The 2-morphisms are defects in the gapped boundaries. The 3-morphisms are point-defects in spacetime. This leads to the natural question: what is this 3-category, mathematically? If we restrict our attention to doubled topological phases, then this is the 3-category of tensor categories, whose 1-morphisms are bimodules, 2-morphisms are bimodule functors, and 3-morphisms are natural transformations:
> Kitaev, Alexei, and Liang Kong. "Models for gapped boundaries and domain walls." Communications in Mathematical Physics 313.2 (2012): 351-373.
However, there are phases which are not doubled. These should be additional objects in the 3-category. There should be an object x_A corresponding to every modular category A (and choice of chiral central charge). The hom-space between any two objects should correspond to the space of gapped boundary theories between those two objects. For instance, whenever A is not doubled, Hom(1,x_A)=0 since x_A admits no gapped boundaries.
The goal of this project is to mathematically construct this 3-category. The work is still in progress so the construction is still shaky - I won't say anything about it. Abstractly you can consider this as adding some formal objects to the 3-category of tensor categories and formally impose the relations you want.
There's a lot of wonderful things about defining this 3-category. First off, it gives a good universal target for fully extended TQFTs. That is, The correct definition of TQFT in (2+1)D is a functor from the fully extended bordism category to this universal 3-category. This is the one that the authors of the work intended. However, there are a couple of other applications I have in mind. There's a bit too speculative to put here... Exciting stuff!
Dan Freed also has lecture notes about this work online, at his website, under the name "3-dimensional TQFTs through the lens of the cobordism hypothesis".