On slightly degenerate fusion categories

Yu, Zhiqiang

doi:10.48550/arxiv.1903.06345

Cited by 1 publication

(2 citation statements)

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“…Super-modular categories (or slight variations) have been studied from several perspectives, see [7,19,20,10,5,33,13,48] for a few examples. An algebraic motivation for studying these categories is the following: any unitary braided fusion category is the equivariantization [22] of either a modular or super-modular category (see [44,Theorem 2]).…”

Section: 2mentioning

confidence: 99%

“…The classification of braided fusion categories (BFCs) stands as a formidable, yet enticing problem. There are many approaches to this problem, with varying levels of preciseness and corresponding degrees of difficulty-as examples, one might try to classify by categorical dimension [27,39,12,14,14,11,48], by Witt class [19,20], by dimension of a generating object [1,23,24], or by rank [43,42]. Each of these approaches have different motivations and have seen some measure of success.…”

Section: Introductionmentioning

confidence: 99%

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Classification of Super-Modular Categories by Rank

Bruillard

Galíndo

et al. 2019

Algebr Represent Theor

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We develop categorical and number theoretical tools for the classification of super-modular categories. We apply these tools to obtain a partial classification of super-modular categories of rank 8. In particular we find three distinct families of prime categories in rank 8 in contrast to the lower rank cases for which there is only one such family. 2 2. PRELIMINARIES In this section, we first introduce the notion of super-modular categories and some of its properties. Most of the results can be found in ([10, 13]) and the references therein. Then we discuss the Galois symmetry for super-modular categories.2.1. Centralizers. Whereas one may always define an S-matrix for any ribbon fusion category B, it may be degenerate. This failure of modularity is encoded it the subcategory of transparent objects called the Müger center B ′ . Here an object X is called transparent if all the double braidings with X are trivial:Generally, we have the following notion of the centralizer of the braiding.Definition 2.1. The Müger centralizer of a subcategory D of a pre-modular category B is the full fusion subcategoryWhile the notation D ′ is slightly ambiguous as it is relative to an ambient category, the context will always make it clear. By a theorem of Bruguières [8], the simple objects inSymmetric fusion categories have been classified by Deligne in terms of representations of supergroups [21]. In the case that B ′ ∼ = Rep(G) (i.e., B ′ is Tannakian), the de-equivariantization procedure of Bruguières [8] and Müger [36] yields a modular category B G of dimension dim(B)/|G|. Otherwise, by taking a maximal Tannakian subcategory Rep(G) ⊂ B ′ , the deequivariantization B G has Müger center (B G ) ′ ∼ = sVec, the symmetric fusion category of super-vector spaces. Generally, a braided fusion category B with B ′ ∼ = sVec as symmetric fusion categories is called slightly degenerate [22], while if B ′ ∼ = Vec, B is non-degenerate. The symmetric fusion category sVec has a unique spherical structure compatible with unitarity and has Sand T -matrices: S sVec = 1 √ 2 and T sVec = 1 0 0 −1 . Definition 2.4.Ŝ andT are called the Sand T -matrix of the fermionic quotient. By the following proposition, pointed super-modular categories always splits.4

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