On the classification of modular categories

Title: On the classification of modular categories

Abstract: A modular tensor category (MTC) is a fusion category with braiding and ribbon structures, which satisfy a non-degeneracy condition. They are of interest for a variety of mathematical subjects, such as topological quantum field theory, representation theory of quantum groups, von Neumann algebras, conformal field theory and vertex operator algebras. In particular, modular tensor categories sit at the crossroads of quantum algebra and condensed-matter physics, providing an algebraic model for anyon systems arising from topological phases of matter, which are currently viewed as potential hardware for fault-tolerant quantum computing. Although these categories play a central role in both mathematics and physics, their overall landscape remains only partially understood, making their classification a rich and challenging problem.

A major breakthrough showed that for any fixed rank, defined as the number of isomorphism classes of simple objects, there exist only finitely many MTCs  [BNRW]. This finiteness result has led to a systematic classification program by rank, with complete or partial results now known in small ranks and for various restricted classes. Recently, [NRW] presented a classification of modular data up to rank 11.

One particularly tractable and well-motivated subclass consists of integral MTCs, where all simple objects have integer Frobenius–Perron dimension; these categories are in correspondence with the categories of representations of modular finite-dimensional semisimple quasi-Hopf algebras.

Within the integral setting, MTCs of odd Frobenius–Perron dimension form a distinguished family. These categories are maximally non-self-dual, a property that significantly reduces the complexity of classification and leads to strong structural constraints. In joint work, we show that all such categories of ranks 13 and 15 are pointed, and that those of ranks 19–23 are necessarily pointed as well, completing their classification. More broadly, our methods yield structural results for odd-dimensional MTCs in substantially higher ranks, showing that in many cases such categories must be either pointed or perfect [CP, CGP]. On the other hand, it turns out that odd-dimensional MTCs and MTCs whose Frobenius–Perron dimension is congruent to $2$ modulo $4$ share many properties. In [CCP], we use these similarities to extend the classification work of such categories. By contrast, MTCs whose Frobenius–Perron dimension are divisible by $4$ are still relatively unexplored.

In this talk, I will give an overview of the classification program for modular categories, focusing on key invariants such as rank, quantum dimensions, and modular data, and on the construction techniques that allow new examples to be built from known ones.