Classification of integral modular data up to rank 13

TL;DR

This paper classifies all integral modular data and fusion rings up to rank 13 and 12 respectively, revealing their triviality in these ranges and refining classifications at higher ranks using new mathematical techniques.

Contribution

It introduces novel methods for classifying integral modular data and fusion rings, including the use of squared denominators in Egyptian fractions and new features in Normaliz for solving equations.

Findings

All perfect integral modular fusion categories up to rank 13 are trivial.

Refined classification of non-pointed odd-dimensional modular data below rank 25.

Narrowed down the classification at rank 14 to 35 possible types.

Abstract

This paper classifies all modular data of integral modular fusion categories up to rank 13. Furthermore, it also classifies all integral half-Frobenius fusion rings up to rank 12. We find that each perfect integral modular fusion category up to rank 13, as well as every perfect integral half-Frobenius fusion ring up to rank 12, is trivial. We have also refined the non-pointed odd-dimensional modular data at ranks below 25 to three items, all of rank 17, FPdim 225, and type [[1,3],[3,8],[5,6]], filling gaps in the literature. For rank 25, we have narrowed down the perfect case to 3 open types. Our initial key insight is that the Egyptian fractions, which are typically employed to list possible types, can be chosen with squared denominators. We then develop several type criteria as initial filters. To obtain the fusion rings, we solve the dimension and associativity equations using new features on Normaliz created specifically for this purpose. The S-matrices (if they exist) are obtained by self-transposing the character table, while the T-matrices are derived by solving the Anderson-Moore-Vafa equations. Finally, we verify the extended axioms of modular data. From rank 13 onward, the types were further restricted by additional properties unique to the modular case, which involved the universal grading, congruence representations of the modular group and Galois action, leading to critical arithmetic constraints. In particular, we get that, up to rank 21, a prime divisor of the global FPdim does not exceed the rank, and more strongly up to rank 15 in the non-pointed case, does not exceed half the rank. Ultimately, we narrowed down the classification at rank 14 to 35 possible types, 8 of which are non-perfect.