Fusion rules for Z/2Z permutation gauging

Edie-Michell, Cain; Jones, Corey; Plavnik, Julia Yael

doi:10.1063/1.5117843

Cited by 8 publications

(6 citation statements)

References 26 publications

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“…which in the case n = 2 is also true by construction for the examples considered in [EMJP18]. In general, for the full symmetric group, we get that rk((C ≀ S n ) g ) depends only on the conjugacy class C g of g ∈ S n .…”

Section: Thensupporting

confidence: 51%

The rank of 𝐺-crossed braided extensions of modular tensor categories

Bischoff¹

2020

Topological Phases of Matter and Quantum Computation

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Section: Thensupporting

confidence: 51%

The rank of 𝐺-crossed braided extensions of modular tensor categories

Bischoff¹

2020

Topological Phases of Matter and Quantum Computation

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“…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%

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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“…In the physics literature, defect fusion is often synonymous with symmetry gauging [17,25,[41][42][43]. In this work, we have computed the fusions without consideration of gauging.…”

Section: Discussionmentioning

confidence: 99%

Fusing binary interface defects in topological phases: The Z/pZ case

Bridgeman

Barter

Jones

2019

Journal of Mathematical Physics

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A binary interface defect is any interface between two (not necessarily invertible) domain walls. We compute all possible binary interface defects in Kitaev's Z/pZ model and all possible fusions between them. Our methods can be applied to any Levin-Wen model. We also give physical interpretations for each of the defects in the Z/pZ model. These physical interpretations provide a new graphical calculus which can be used to compute defect fusion. *

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Fusion rules for Z/2Z permutation gauging

Cited by 8 publications

References 26 publications

The rank of 𝐺-crossed braided extensions of modular tensor categories

The rank of 𝐺-crossed braided extensions of modular tensor categories

Classification of Super-Modular Categories by Rank

Fusing binary interface defects in topological phases: The Z/pZ case

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