Modular tensor categories, subcategories, and Galois orbits

Abstract

We establish a set of general results to study how the Galois action on modular tensor categories interacts with fusion subcategories. This includes a characterization of fusion subcategories of modular tensor categories which are closed under the Galois action, and a classification of modular tensor categories which factor as a product of pointed and transitive categories in terms of pseudoinvertible objects. As an application, we classify modular tensor categories with two Galois orbits of simple objects and a nontrivial grading group.

Pre-modular Fusion Categories of Small Global Dimensions

“…A finite-dimensional congruence representation ρ of the modular group SL(2, Z) is said to be non-degenerate if the eigenvalues of ρ(t) are distinct; non-degenerate finite-dimensional congruence representations are irreducible [9, Lemma 1]. In addition, the set of eigenvalues of ρ(t) is called the t-spectrum of ρ following [4,17,23]; we note that the t-spectrum of any finite-dimensional irreducible representation of SL(2, Zpm ) is produced in [23,Appendix].…”

“…2 . If ρC ∼ = ρ1 ⊕ ρ2 with dim(ρ1) = dim(ρ2) = p−1 2 , since the t-spectrums of ρ1 and ρ2 intersect non-trivially [4, Lemma 3.18], ρ1 = ρ2, which is impossible by [23,Lemma 5.2.2]. And the irreducible representations of dimension p+1 2 can't be realized as representations of modular fusion categories [8].…”

“…The S-matrix and T -matrix of modular fusion categories (see section 2), which reflect many important properties of modular fusion categories, also enjoy interesting algebraic and arithmetic properties [3,4,6]. Therefore, modular fusion categories are inseparable with algebraic number theory and representations of the modular group SL(2, Z) [3,4,10], in particular, one can peer into their properties by considering the number of Galois orbits of the simple objects [16,23] and the representation type of SL(2, Z) associated to a modular category [17], for example.…”