Quantum Symmetries for Exceptional SU(4) Modular Invariants Associated with Conformal Embeddings

Coquereaux, Robert

doi:10.3842/sigma.2009.044

Cited by 14 publications

(27 citation statements)

References 47 publications

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“…Summarily, (13) sλ ⊗ tλ = t))λ). The first term is nonzero exactly as in the first case, but is negated by the second term when 2(ℓ − 1) − (u + t) = s − 2j for 0 ≤ j ≤ t. Hence (15) sλ ⊗ tλ = t j=s+t−(ℓ−1) (s + t − 2j)λ.…”

Section: The Categories C(g ℓ Q)mentioning

confidence: 89%

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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Section: The Categories C(g ℓ Q)mentioning

confidence: 89%

Lie theory for fusion categories: A research primer

Schopieray¹

2020

Topological Phases of Matter and Quantum Computation

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“…All exceptional connectedétale algebras in C(sl 2 , k) are succinctly listed in [21, Table 1], while all exceptional connectedétale algebras in C(sl 3 , k) are listed using modular invariants [17, Equations 2.7d,2.7e,2.7g] at levels k = 5, 9, 21. The theory of conformal embeddings provides examples of exceptional connectedétale algebras in C(sl 4 , k) at levels k = 4, 6, 8, which are described in detail in [7]. Although there is no explicit proof in the current literature that there are even finitely many exceptional connectedétale algebras in C(sl 4 , k), it is presumed based on computational evidence that the aforementioned list is exhaustive.…”

Section: Exceptional Algebrasmentioning

confidence: 99%

Level bounds for exceptional quantum subgroups in rank two

Schopieray

2018

Int. J. Math.

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There is a long-standing belief that the modular tensor categories C(g, k), for k ∈ Z ≥1 and finite-dimensional simple complex Lie algebras g, contain exceptional connectedétale algebras at only finitely many levels k. This premise has known implications for the study of relations in the Witt group of nondegenerate braided fusion categories, modular invariants of conformal field theories, and the classification of subfactors in the theory of von Neumann algebras. Here we confirm this conjecture when g has rank 2, contributing proofs and explicit bounds when g is of type B2 or G2, adding to the previously known positive results for types A1 and A2.fusion categories whose braidings are entirely non-degenerate, or furthest from symmetric as possible.Definition 2.1.1. A pre-modular category C is a modular tensor category if 1 ∈ C is the only simple transparent object.Lastly we recall that each pre-modular category has a family of natural isomorphisms θ(X) : X ∼ → X for all X ∈ C known as twists (or ribbon structure), compatible with the inherent braiding isomorphisms [12, Definition 8.10.1]. We will abuse this notation when it suits our purposes by denoting the complex number α such that θ(X) = α · id X simply as θ(X) for any simple X ∈ C.

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“…In the last section, we shall illustrate these techniques (the chiral method) on a particular example. The reader interested in seeing how the general machinery works (the general method using the full collection of modular splitting equations and the determination of the graph O) is referred for instance to the non trivial examples studied in [7].…”

Section: General Methodsmentioning

confidence: 99%

“…Therefore we fix K and look for "overgroups" G such that the pair (K, G) is conformal. The fact that each such pair gives rise to a quantum subgroup of K results from investigations carried out more recently (in the last ten years) but we should stress than few of them have been worked out explicitly: only the SU (N ) cases with N = 2, 3, 4 are described (their associated graphs and algebras of quantum symmetries are known) in the available literature [23,4,26,24,5,14,15,6,7]. We always assume that G is simple.…”

Section: Quantum Subgroups Of Lie Groups From Conformal Embeddings 4mentioning

confidence: 99%

“…The SU (4) family includes the A k (SU (4)) series and its conjugate for all k, two kinds of orbifolds, the D (2) k = A k /2 series for all k (with self-fusion when k is even) and the D (4) k = A k /4 series for k = 0, 2, 6 mod 8 (with self-fusion when k is divisible by 8), their corresponding conjugated series, one exceptional case obtained by twisting, D (4)t 8 , without selffusion (a generalization of E 7 ) and finally three exceptional quantum graphs with self-fusion, at levels 4, 6 and 8, denoted E 4 , E 6 and E 8 , together with one exceptional module for each of the last two. These exceptional quantum subgroups at levels 4, 6, 8 can be respectively obtained from the study of the conformal embeddings of SU (4) into Spin(15), SU (10) and Spin (20), see [7]; their modules (for the last two, since E 4 doesn't have any) are obtained as a by-product of the determination of their algebras of quantum symmetries. We should also mention the existence of a conformal embedding of SU (4), at level 2, in SU (6), but it gives rise to D (2) 2 = A 2 /2, the first member of the D (2) k series, which can itself be obtained by conformal embedding of the non simple SU (4) × SU (k) in SU (4k), followed by reduction.…”

Section: The Classification Problemmentioning

confidence: 99%

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