Citation: Galindo, Cesar, and Nicolas Jaramillo. "Solutions of the hexagon equation for abelian anyons." Revista Colombiana de Matemáticas 50.2 (2016): 277-298.
Web: https://arxiv.org/abs/1606.01414
Tags: Abelian-anyons, Mathematical, Monoidal-categories
Every abelian fusion system will be an abelian group. However, there can be multiple ways of choosing associativity/twist/hexagon solutions to give a ribbon fusion category.
This paper gives a detailed discussion of this problem, ending with a classification of abelian anyon theories.
This paper is also nice because of its discussion of quadratic forms. It details how to compute the size of quadratic forms on abelian groups with C^x coefficients, and connects these quadratic forms to the 3rd "abelian cohomology group".
A more standard reference for gauge equivalences of quantum doubles of quantum doubles of groups, with no special focus on the abelian case, is found in
> Mason, Geoffrey, and Siu-Hung Ng. "Group cohomology and gauge equivalence of some twisted quantum doubles." Transactions of the American Mathematical Society 353.9 (2001): 3465-3509.
and
> Etingof, Pavel, Dmitri Nikshych, and Victor Ostrik. "Fusion categories and homotopy theory." Quantum topology 1.3 (2010): 209-273.