# Relation between cobordism and unitary fusion category classification of TQFT/ SPT phase

In the introduction part of Gaiotto's paper (https://arxiv.org/pdf/1712.07950.pdf), he says "in the context of topological field theory, homotopy-theoretic ideas also lead to the classification of invertible phases in terms of cobordism spectra." On the other hand, it's well known that we can construct (2+1)d TQFT by Turaev-Viro state sum from a unitary fusion category. Cobordism and unitary fusion category seem very different objects.

My question is: how are these two approaches related to each other? I am not a mathematician or an expert on this field. Any simple description/explanation will be very helpful to me.

## 1 Answer

The Turaev-Viro state sums are for constructing non-invertible 3d TQFTs, since there are no nontrivial invertible ones without adding extra symmetry. You can do that, and it leads to a G-equivariant notion of a fusion category. You can even get a nontrivial G-equivariant extension of a trivial fusion category, which means that the F-symbols (associators) becomes the group 3-cocycle. This gives a relationship between fusion categories and group cohomology. There is a nice discussion here https://arxiv.org/abs/1410.4540 .

As far as I know there is no direct extension of this to the cobordism theories, but you can write the cobordism groups as group cohomology like objects and do this same thing. That was basically the approach of these papers: https://arxiv.org/abs/1505.05856 and https://arxiv.org/abs/1701.08264 . There, the cobordism groups are associated with particular fusion categories that "bosonize" the fermionic SPT phases.

Note that while it seems possible to classify invertible TQFTs, classifying something like all 3d TQFTs even seems harder than classifying all finite groups. They are much more wild beasts. There is a really nice paper about the topological power of unitary TQFT: https://arxiv.org/abs/math/0503054

ps. why not ask anton to explain? :P