Citation: Ostrik, Viktor. "Fusion categories of rank 2." arXiv preprint math/0203255 (2002)."
Web: https://arxiv.org/abs/math/0203255
Tags: Mathematical, Modular-tensor-categories, Monoidal-categories, Property-F
In this lovely little paper, the authors characterize the space of all fusion categories of rank 2. The conclusion is that there are four categories - Vec_{Z2}, Vec_{Z2} with twisted associativity (the semion), Fibonacci, and Yang-Lee.
The proof is nicely split into three step is the classify Vec_G fusion rules by H3(G,U(1)), which gives the two solutions to Vec_{Z2} fusion rules. The next is to classify fusion categories with Fiboancci fusion rules, as was done in
> Moore, Gregory, and Nathan Seiberg. "Classical and quantum conformal field theory." Communications in Mathematical Physics 123 (1989): 177-254.
It's a neat historical point to see that the first two examples of fusion rules whose F-matrices were abstractly studied were Ising and Fibonacci. Shows how natural the Fibonacci fusion rules are.
The last step is to show that all of the other possible fusion rules admit no solutions. This is the main content of the present paper. The authors argue this in a way which is roundabout using modern machinery, but is impressive with what was available at the time. Broadly it seems like what they're doing is showing that the fusion category C admits a braiding and that Z(C) looks like C \times \overline{C}. Moreover, they show that C has a spherical structure. So, since C is a pre-modular category which is factorizable, it must be modular. C cannot be a modular category with any of the exotic fusion rules by some simple number-theoretical concerns.