Classification of integral modular data up to rank 13

Abstract

This paper classifies the modular data of integral modular fusion categories up to rank 13, and integral half-Frobenius fusion rings up to rank 12. We establish that every perfect case within these bounds is trivial. Furthermore, we refine the non-pointed odd-dimensional modular data at ranks below 25 to exactly three items, all of rank 17, FPdim 225, and type [[1,3],[3,8],[5,6]], filling existing literature gaps. For rank 25, we narrow the perfect case to three open types. Our core insight is that Egyptian fractions, typically used to list possible types, can be chosen with squared denominators. We develop several type criteria as initial filters. To construct the fusion rings, we solve dimension and associativity equations utilizing custom-built features in Normaliz. S-matrices are generated by self-transposing the character table, and T-matrices are derived by solving the Anderson-Moore-Vafa equations, concluding with the verification of extended modular data axioms. From rank 13 onward, types are restricted by modular-specific properties involving universal grading, congruence representations of the modular group, and Galois action. This establishes critical arithmetic constraints: up to rank 21, a prime divisor of the global FPdim cannot exceed the rank, and up to rank 15 (non-pointed case), it cannot exceed half the rank. Ultimately, we reduce the rank 14 classification to 35 possible types, 8 of which are non-perfect.