Citation: Etingof, Pavel, Dmitri Nikshych, and Victor Ostrik. "Fusion categories and homotopy theory." Quantum topology 1.3 (2010): 209-273.
Web: https://arxiv.org/abs/0909.3140
Tags: Mathematical, SPT/SETs, Foundational, Monoidal-categories
In this paper, the authors introduce and prove the pure mathematical results which underlie the theory of gauging symmetry in topological phases. One noteworthy feature of this paper, though, is that it makes no reference to physics or quantum phases. The only stated motivation for this work is to classify G-graded extensions of fusion categories, as well as giving a computation-free derivation of the classification of Tambara-Yamagami fusion categories.
The process of gauging symmetry can be broken into two different steps:
In general, both of these steps can fail. That is, there can be obstructions to the existence of solutions. These obstructions are explained and are shown to live in certain cohomology classes. In physics literature, these obstructions are known as anomalies. If there are solutions, then the solutions form a torsor under a cohomology class living in one dimension lower. Abstractly, the elements of this torsor are related by zesting:
> Delaney, Colleen, et al. "Braided zesting and its applications." Communications in Mathematical Physics 386 (2021): 1-55.
This paper also developed a lot of general fusion category theory, which was necessary for the application. This theory had to do with symmetries of fusion categories and modules over fusion categories.