Citation: Ng, Siu-Hung, Yilong Wang, and Qing Zhang. "Modular categories with transitive Galois actions." Communications in Mathematical Physics 390.3 (2022): 1271-1310.
Web: https://arxiv.org/abs/2007.01366
Tags: Mathematical, Modular-tensor-categories, Property-F
In this paper, the authors prove a lovely classification result for modular categories with transitive Galois actions. Every character of the Verlinde algebra (Grothendieck group) is of the form Y -> S_(X,Y)/S_(X,0) for some fixed object X. The absolute Galois group of the rationals acts on the space of characters, and thus acts on the space of simple objects. Sometimes, this action is transitive. In this special case the authors give a shockingly restrictive characterization - the category is necessarily a Deligne tensor product of copies of C(p) for some primes p>3, where C(p) is the adjoint subcategory of the quantum group SL(2)_(p-2). Going further, if if we require that the Galois action is sufficiently transitive (such as having two orbits, or having a pointed object in every orbit) then strong statements can be made:
> Plavnik, Julia, et al. "Modular tensor categories, subcategories, and Galois orbits." Transformation Groups (2023): 1-26.
A nice lecture-based discussion by Julia Plavnik is found here.
The proof of this paper is quite technical and involves a lot of interesting number theory. A key step is showing that the modular representation associated to these categories is always irreducible.