Level bounds for exceptional quantum subgroups in rank two

Schopieray, Andrew

doi:10.1142/s0129167x18500349

Cited by 14 publications

(17 citation statements)

References 32 publications

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“…One computes (6.2) ξ 1 (C) = exp(−πi/3) = ξ 1 (D), but C and D are not Witt equivalent. To see this, note that ord(T C ) = 36, and ord(T D ) = 45 [47]. By direct computation, we have ξ 13 (C) = 1 and ξ 13 (D) = −1.…”

Section: Witt Relations and Central Chargesmentioning

confidence: 92%

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Higher Gauss sums of modular categories

Schopieray

Wang

2019

Sel. Math. New Ser.

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The definitions of the n th Gauss sum and the associated n th central charge are introduced for premodular categories C and n ∈ Z. We first derive an expression of the n th Gauss sum of a modular category C, for any integer n coprime to the order of the Tmatrix of C, in terms of the first Gauss sum, the global dimension, the twist and their Galois conjugates. As a consequence, we show for these n, the higher Gauss sums are d-numbers and the associated central charges are roots of unity. In particular, if C is the Drinfeld center of a spherical fusion category, then these higher central charges are 1. We obtain another expression of higher Gauss sums for de-equivariantization and local module constructions of appropriate premodular and modular categories. These expressions are then applied to prove the Witt invariance of higher central charges for pseudounitary modular categories.

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Section: Witt Relations and Central Chargesmentioning

confidence: 92%

“…Example 6.3. In this example, we use the formulas for the modular data of C(g 2 , k) as in [47,Sections 2.3.4]. Set C := C(g 2 , 8) ⊠5 and D := C(g 2 , 11) ⊠10 .…”

Section: Witt Relations and Central Chargesmentioning

confidence: 99%

Higher Gauss sums of modular categories

Schopieray

Wang

2019

Sel. Math. New Ser.

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“…When r is even, all twists are 1 so we have (C r ) pt ∼ = Rep(Z/2Z × Z/2Z), therefore we can again do the condensation and denote the resulting modular category by D r . The pseudo-unitarity of D r follows from [35,Lem. 2.4].…”

Section: Braided Fusion Categoriesmentioning

confidence: 99%

The Witt classes of $SO(2r)_{2r}$

Rowell¹,

Ruan²,

Wang³

2021

Preprint

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We study the Witt classes of the modular categories SO(2r) 2r associated with quantum groups of type Dr at (4r − 2)-th roots of unity. From these classes we derive infinitely many Witt classes of order 2 that are linearly independent modulo the subgroup generated by the pointed modular categories. In particular we produce an example of a simple, completely anisotropic modular category that is not pointed whose Witt class has order 2, answering a question of Davydov, Müger, Nikshych and Ostrik. Our resultsshow that the trivial Witt class [Vec] has infinitely many square roots modulo the pointed classes, in analogy with the recent construction of infinitely many square roots of the Ising Witt classes modulo the pointed classes constructed in a similar way from certain type Br modular categories. We compare the subgroups generated by the Ising square roots and [Vec] square roots and provide evidence that they also generate linearly independent subgroups.

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“…Numerous examples of this geometric interpretation can be found in Sections 4-6 of [70] for the rank 2 Lie algebras. In rank greater than 2 this task is substantially tedious and one may be satisfied deriving a coarser set of conclusions from the quantum Racah formula, as an explicit expression akin to Example 13 is unrealistic at this time.…”

Section: The Categories C(g ℓ Q)mentioning

confidence: 99%

Lie theory for fusion categories: A research primer

Schopieray¹

2020

Topological Phases of Matter and Quantum Computation

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A diverse collection of fusion categories, in the language of [22], may be realized by the representation theory of quantum groups. There is substantial literature where one will find detailed constructions of quantum groups, and proofs of the representation-theoretic properties these algebras possess. Here we will forego technical intricacy as a growing number of researchers study fusion categories disjoint from Lie theory, representation theory, and a laundry list of other obstacles to understanding the mostly combinatorial, geometric, and numerical descriptions of the examples of fusion categories arising from quantum groups. Our expository piece aims to create a self-contained guide for researchers to study from a computational standpoint with only the prerequisite knowledge of fusion categories. A multitude of figures and worked examples are included to elucidate the material, and additional references are abundant for those readers looking to delve deeper. Note that in general our chosen references are intended to provide useable resources for the reader and do not always indicate provenance. Lastly we have included several open and approachable questions of general interest throughout the final sections.The organization of this paper is as follows: Sections 1 and 2 summarize the classical representation theory of semisimple Lie algebras in the spirit of [37] to introduce the chosen language and notation used extensively in what follows. Those unfamiliar with Lie algebras are encouraged to work through the provided examples themselves, while readers who possesses this prerequisite knowledge can safely begin reading in Section 3 referring back to earlier sections as needed. Terminology most relevant to future explanation is italicized for this purpose. Section 3 explains computationally relevant subtleties of the modern generalization of the representation theory of quantum groups including quantum dimensions and the affine Weyl group, followed by Section 4 which defines our primary objects of study: the fusion categories C(g, ℓ, q) where g is a finite-dimensional simple complex Lie algebra and q is a root of unity such that q 2 has order ℓ ∈ Z ≥1 . Section 5 discusses the fusion rules of C(g, ℓ, q), the classification of fusion subcategories, and simple factorizations. Modular data and the Galois symmetry thereof is covered in Section 6, while tensor autoequivalences, module categories, and commutative algebras are contained in Section 7.

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Level bounds for exceptional quantum subgroups in rank two

Cited by 14 publications

References 32 publications

Higher Gauss sums of modular categories

Higher Gauss sums of modular categories

The Witt classes of $SO(2r)_{2r}$

Lie theory for fusion categories: A research primer

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