The Structure of Sectors Associated With Longo–rehren Inclusions Ii: Examples

Izumi, Masaki

doi:10.1142/s0129055x01000818

Cited by 126 publications

(252 citation statements)

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“…By its nature, our work leads to many questions, including • What is the space of 2D CFTs we can generate from the anyonic chain procedure? Is it possible to associate an RCFT with the (extended) Haagerup subfactor (see, e.g., [46][47][48] for some important work in this direction) thus realizing an old dream of Vaughan Jones? • We saw that when we take k → ∞ in the C in = Rep su(2) int k case, we could end up with an irrational output theory [45].…”

Section: Discussionmentioning

confidence: 99%

Anyonic Chains, Topological Defects, and Conformal Field Theory

Buican

Gromov

2017

Commun. Math. Phys.

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Abstract:Motivated by the three-dimensional topological field theory/two-dimensional conformal field theory (CFT) correspondence, we study a broad class of one-dimensional quantum mechanical models, known as anyonic chains, which can give rise to an enormously rich (and largely unexplored) space of two-dimensional critical theories in the thermodynamic limit. One remarkable feature of these systems is the appearance of non-local microscopic "topological symmetries" that descend to topological defects of the resulting CFTs. We derive various model-independent properties of these theories and of this topological symmetry/topological defect correspondence. For example, by studying precursors of certain twist and defect fields directly in the anyonic chains, we argue that (under mild assumptions) the two-dimensional CFTs correspond to particular modular invariants with respect to their maximal chiral algebras and that the topological defects descending from topological symmetries commute with these maximal chiral algebras. Using this map, we apply properties of defect Hilbert spaces to show how topological symmetries give a handle on the set of allowed relevant deformations of these theories. Throughout, we give a unified perspective that treats the constraints from discrete symmetries on the same footing as the constraints from topological ones.

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Section: Discussionmentioning

confidence: 99%

Anyonic Chains, Topological Defects, and Conformal Field Theory

Buican

Gromov

2017

Commun. Math. Phys.

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“…A 3 G subfactor has principal graph consisting of |G| spokes of length 3, and the dual data is determined by the inverse law of the group G. In fact, Izumi has an unpublished construction of a 3 Z/4 subfactor using Cuntz algebras analogous to his treatment for odd order G in [Izu01]. Moreover, he shows such a subfactor is unique, which our approach does not attempt to show.…”

Section: Introductionmentioning

confidence: 92%

Calculating two-strand jellyfish relations

Penneys

Peters

2015

Pacific J. Math.

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We construct subfactors where one of the principal graphs is a spoke graph using an algorithm which computes two-strand jellyfish relations. One of the subfactors we construct is a 3 Z/4 subfactor known to Izumi, which has not previously appeared in the literature. To do so, we provide a systematic treatment of the space of second annular consequences, which is analogous to Jones' treatment of the space of first annular consequences in his quadratic tangles article.This article is the natural followup to two recent articles on spoke subfactor planar algebras and the jellyfish algorithm. Work of Bigelow-Penneys explains the connection between spoke subfactor planar algebras and the jellyfish algorithm, and work of Morrison-Penneys automates the construction of subfactors where both principal graphs are spoke graphs using one-strand jellyfish.

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“…Theorem 1.1. There are exactly ten subfactor planar algebras other than Temperley-Lieb with index between 4 and 5: the Haagerup planar algebra and its dual [AH99], the extended Haagerup planar algebra and its dual [BMPS09], the Asaeda-Haagerup planar algebra [AH99] and its dual, the 3311 Goodman-de la Harpe-Jones planar algebra [GdlHJ89] and its dual, and Izumi's self-dual 2221 planar algebra [Izu01] and its complex conjugate.…”

Section: Introductionmentioning

confidence: 99%

Subfactors of Index Less Than 5, Part 3: Quadruple Points

Izumi

Jones

Morrison

et al. 2012

Commun. Math. Phys.

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Abstract. One major obstacle in extending the classification of small index subfactors beyond 3 + √ 3 is the appearance of infinite families of candidate principal graphs with 4-valent vertices (in particular, the "weeds" Q and Q from Part 1 [MS10]). Thus instead of using triple point obstructions to eliminate candidate graphs, we need to develop new quadruple point obstructions. In this paper we prove two quadruple point obstructions. The first uses quadratic tangles techniques and eliminates the weed Q immediately. The second uses connections, and when combined with an additional number theoretic argument it eliminates both weeds Q and Q . Finally, we prove the uniqueness (up to taking duals) of the 3311 Goodman-de la Harpe-Jones subfactor using a combination of planar algebra techniques and connections.

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The Structure of Sectors Associated With Longo–rehren Inclusions Ii: Examples

Cited by 126 publications

References 20 publications

Anyonic Chains, Topological Defects, and Conformal Field Theory

Anyonic Chains, Topological Defects, and Conformal Field Theory

Calculating two-strand jellyfish relations

Subfactors of Index Less Than 5, Part 3: Quadruple Points

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