On slightly degenerate fusion categories

Yu, Zhiqiang

doi:10.48550/arxiv.1903.06345

Cited by 1 publication

(2 citation statements)

References 21 publications

(59 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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%

See 1 more Smart Citation

Classification of Super-Modular Categories by Rank

Bruillard

Galíndo

et al. 2019

Algebr Represent Theor

View full text Add to dashboard Cite

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

show abstract