## "Fermionic modular categories and the 16-fold way", Bruillard et al., 2017

#### Reviewed March 23, 2024

Citation: Bruillard, Paul, et al. "Fermionic modular categories and the 16-fold way." Journal of Mathematical Physics 58.4 (2017).

Tags: Foundational, Mathematical, Modular-tensor-categories, Fermionic-order

This is the first paper to study the general theory of fermionic topological order in the setting of anyon models. Namely, it is the first paper to study the general theory of super-modular tensor categories.

The general picture is as follows. In fermionic topological order, the physical fermion is an anyon which can interact with the other particles. At low temperatures this fermion will condense. This corresponds to gauging the fermion parity operator. Hence, fermionic topological order should correspond to anyon models with a transparent fermion along with a choice of gauging of fermion parity.

Here is the amazing fact: if there is a gauging of fermion parity, then there are exactly 16. This is a generalization of Kitaev's 16-fold way, which does this when the base category contains only a fermion. To get between the different choices of gauging one uses the mathematical procedure of zesting.

The above discussion gets at a very deep question: can fermion parity always be gauged? The physical conjecture, which can be viewed as the fundamental conjecture of fermionic topological order, is that fermion parity CAN always be gauged. It turns out that this fact is true, but is very deep. It took five years to prove, and was finally resoled by Theo Johnson-Freyd and David Reutter:

> Johnson-Freyd, Theo, and David Reutter. "Minimal nondegenerate extensions." Journal of the American Mathematical Society 37.1 (2024): 81-150.

Theo explained the difficulty to me as follows. In general with problems like these, the reason that you wouldn't be able to gauge a symmetry is that there would be an obstruction. When you are trying to prove that an obstruction always vanishes, then the typical thing to do is to prove that the space where the obstruction lives is always 0. In this case, the space can be computed and it is NOT zero. It is Z2. A-priori it seems like some super-modular category could have an obstruction in the non-trivial part of this Z2 space. It turns out that no, you can't.

Proving that this obstruction vanishes uses a clever argument filled with counting, contradictions, and constructions. It uses techniques from outside the immediate scope of modular tensor category theory. Theo says that it is still not clear to him why, on a philosophical level, this result is true.