Citation: Ostrik, Victor. "Pre-modular categories of rank 3." arXiv preprint math/0503564 (2005).
Web: https://arxiv.org/abs/math/0503564
Tags: Mathematical, Modular-tensor-categories
In this paper, Viktor Ostrik classifies the pre-modular categories of rank 3. There are four fusion rings that admit categorification. Vec_{Z3}, Rep(S_3), Ising, and another one which is talked about much less. The quantum dimension of the simple objects are d=2cos(pi/7) and d^2-1. The category can be constructed from quantum groups several different ways, most directly as a subcategory of SO(3)_5.
These fusion rings have various numbers of categorifications to different levels of structure.
A neat comment this paper makes is at the bottom, where it brings attention to the 1/2 E6 fusion category. It also has three simple objects and admits a (spherical) categorification, and its fusion rules are symmetric, but it does NOT admit a braiding. This is an interesting possibility to be aware of! This paper can be treated as a primary source (proof!) of this fact.
This paper is also funny in how it fits into historical context. This paper was being written as the same time as Zhenghan and friends were writing the following paper:
> Rowell, Eric, Richard Stong, and Zhenghan Wang. "On classification of modular tensor categories." Communications in Mathematical Physics 292.2 (2009): 343-389.
So, Ostrik's paper was totally subsumed, and doesn't really add all that much to the literature now. It is cool to see that Ostrik and Zhenghan et al. both used Galois theory/number theory arguments, illustrating that Galois theory really is the natural tool to use in this situation.
A related paper which served as an inspiration for this one is a nice little work by Gepner and Kapustin where they classify low-rank fusion ranks. Since they don't work with categorification, they don't need any Galois theory:
> Gepner, Doron, and Anton Kapustin. "On the classification of fusion rings." Physics Letters B 349.1-2 (1995): 71-75.