Unitary solutions of the pentagon and hexagon equations

Given a fusion ring, one might wish to determine all possible (braided) fusion categorifications along with the skeletal data (F and R-symbols) by solving the pentagon and hexagon equations.  Approaching this task (even for small fusion rings) can be taxing. Below, you can find a small selection of handworked solutions for simple examples.  Please note that I'm only considering unitary solutions over the complex field here. 

Generalia

• Reference sheet: solving for the (multiplicity-free) pentagon equation

Worked solutions

Rank 2

• Fibonacci: pentagon

Related links

• A go-to reference for fusion rings and their categorifications: AnyonWiki.

• A catalog of modular tensor categories.

• This book containing lots of information on calculating F and R-symbols: Topological Quantum (S. Simon). 

• Tabulation of F and R-symbols for some fusion rings can be found in P. Bonderson's thesis (see Chapter 5).

• A list of prime modular braided fusion categories of rank <5, with F and R-symbols for some of these (see Section 5.3): Rowell, Stong, Wang (2009).

• More F and R-symbols: Ardonne, Slingerland (2010). Notably, the authors calculate the F-symbols for a fusion ring with multiplicity in Appendix B.

• Calculation of the F-symbols for the H_1 Haagerup fusion category: Barter, Bridgeman, Wolf (2022). The authors develop an algorithm that takes as input the F-symbols for a fusion category, and outputs the F-symbols for a Morita equivalent fusion category. This machinery is particuarly successful in that the H_1 example (which has multiplicity) is tackled using known data from a Morita equivalent fusion category which is multiplicity-free.

• A necessary condition for unitary categorification of fusion rings: Liu, Palcoux, Wu (2021); Etingof, Nikshych, Ostrik (2023).