Examples of Subfactors from Conformal Field Theory

Xu, Feng

doi:10.1007/s00220-017-2939-1

Cited by 11 publications

(15 citation statements)

References 55 publications

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“…However, the author was unable to find a suitable description of the category of modules in this case. Instead we have [52,Theorem 3.2] which proves the precise statement in this case. They show that C 5,5,5 is the even part of the 3 Z 2 ×Z 2 subfactor.…”

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confidence: 60%

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Type $II$ quantum subgroups of $\mathfrak{sl}_N$. $I$: Symmetries of local modules

Edie-Michell¹,

Gannon²

2021

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This paper is the first of a pair that aims to classify a large number of the type II quantum subgroups of the categories C(slr+1, k). In this work we classify the braided autoequivalences of the categories of local modules for all known type I quantum subgroups of C(slr+1, k), barring a single unresolved case for an orbifold of C(sl4, 8). We find that the symmetries are all non-exceptional except for four possible cases (up to level-rank duality). These exceptional cases are the orbifolds C(sl2, 16) 0Rep(Z 2 ) , C(sl3, 9) 0 Rep(Z 3 ) , C(sl4, 8) 0 Rep(Z 4 ) , and C(sl5, 5) 0Rep(Z 5 ) . We develop several technical tools in this work. We give a skein theoretic description of the orbifold quantum subgroups of C(slr+1, k). Our methods here are general, and the techniques developed will generalise to give skein theory for any orbifold of a braided tensor category. We also uncover an unexpected connection between quadratic categories and exceptional braided auto-equivalences of the orbifolds. We use this connection to construct two of the four exceptionals, and to reduce the C(sl4, 8) 0Rep(Z 4 ) case to a concrete finite computation. In the sequel to this paper we will use the classified braided auto-equivalences to construct the corresponding type II quantum subgroups of the categories C(slr+1, k). When paired with Gannon's type I classification for r ≤ 6, this will complete the type II classification for these same ranks, excluding the one exception at C(sl4, 8).

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confidence: 60%

“…We now deal with the case of C(sl 5 , 5) 0 Rep(Z 5 ) . This case has been examined in the literature previously [52,29]. Rep(Z 5 ) ) × S 3 has a subgroup isomorphic to A 4 .…”

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confidence: 99%

Type $II$ quantum subgroups of $\mathfrak{sl}_N$. $I$: Symmetries of local modules

Edie-Michell¹,

Gannon²

2021

Preprint

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“…It was shown in [Izu18] that there is a unique generalized Haagerup category C for G = Z 2 × Z 2 . This category is related to a conformal inclusion SU (5) 5 ⊂ Spin(24); see [Xu18;Edi21a]. It was shown in [Gro19] that the Brauer-Picard group of this category has order 360, and the group was identified as S 3 × A 5 in [Edi21a].…”

Section: 2mentioning

confidence: 99%

“…Our other application is to the generalized Haagerup category for the group Z 2 × Z 2 . This category is related to a conformal inclusion SU (5) 5 ⊂ Spin(24); see [Xu18;Edi21a]. This category is interesting because its Brauer-Picard group is unusually rich: it was shown in [Gro19] that this group has order 360, and it was identified as S 3 × A 5 in [Edi21a].…”

Section: Introductionmentioning

confidence: 99%

Graded extensions of generalized Haagerup categories

Grossman¹,

Izumi²,

Snyder³

2022

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We classify certain Z 2 -graded extensions of generalized Haagerup categories in terms of numerical invariants satisfying polynomial equations. In particular, we construct a number of new examples of fusion categories, including: Z 2 -graded extensions of Z 2n generalized Haagerup categories for all n ≤ 5; Z 2 × Z 2 -graded extensions of the Asaeda-Haagerup categories; and extensions of the Z 2 × Z 2 generalized Haagerup category by its outer automorphism group A 4 . The construction uses endomorphism categories of operator algebras, and in particular, free products of Cuntz algebras with free group C * -algebras.

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“…This framework was discovered by accident in the land of quantum field theory as we will now explain, see the following survey for more details [Bro19b]. Conformal field theories (in short CFT) à la Haag-Kastler provide subfactors and conversely certain subfactors provide CFT but the reconstruction is on a case by case basis and so far the most intriguing subfactors (with exotic representation theory uncaptured by groups and quantum groups) are not known to provide a CFT [EK92,JMS14,Bis17,Xu18]. By using the planar algebra of a subfactor, Jones created a lattice model approximating the desired CFT.…”

Section: Introductionmentioning

confidence: 99%

Classification of Thompson related groups arising from Jones technology I

Brothier¹

2020

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In the quest in constructing conformal field theories (CFT) Jones has discovered a beautiful and deep connection between CFT, Richard Thompson's groups and knot theory. This led to a powerful framework for constructing actions of particular groups arising from categories such as Richard Thompson's groups and braid groups. In particular, given a group and two of its endomorphisms one can construct a semidirect product. Those semidirect products have remarkable diagrammatic descriptions which were previously used to provide new examples of groups having the Haagerup property. Moreover, they naturally appear in certain field theories as being generated by local and global symmetries.We consider in this article the class of groups obtained in that way where one of the endomorphism is trivial leaving the case of two nontrivial endomorphisms to a second article. The groups obtained can be described in terms of permutational restricted twisted wreath products containing the larger Thompson group. We classify this class of groups up to isomorphisms and provide a thin description of their automorphism group thanks to an unexpected rigidity phenomena.À la mémoire de Vaughan Jones

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Examples of Subfactors from Conformal Field Theory

Cited by 11 publications

References 55 publications

Type $II$ quantum subgroups of $\mathfrak{sl}_N$. $I$: Symmetries of local modules

Type $II$ quantum subgroups of $\mathfrak{sl}_N$. $I$: Symmetries of local modules

Graded extensions of generalized Haagerup categories

Classification of Thompson related groups arising from Jones technology I

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