Exotic Subfactors of Finite Depth with Jones Indices $(5+\sqrt{13})/2$ and $(5+\sqrt{17})/2$

Asaeda, Marta; Haagerup, Uffe

doi:10.1007/s002200050574

Cited by 104 publications

(263 citation statements)

References 4 publications

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“…In terms of sectors, the Z n rotation σ corresponds to [λ kΛ (1) ] and the Z m rotation σ q to [λ kΛ (q) ] which realize the rotations σ respectively σ q as fusion rules. In fact as the vacuum block of Eq.…”

Section: Z M Orbifold Inclusions Of Su (N)mentioning

confidence: 99%

“…However, we briefly show that we also have Z =Z in this case. Example E (24) : SU (3) 21 ⊂ (E 7 ) 1 : The corresponding modular invariant reads Z E (24) = |χ (0,0) + χ (21,0) + χ (21,21) + χ (8,4) + χ (17,4) + χ (17,13) +χ (11,1) + χ (11,10) + χ (20,10) + χ (12,6) + χ (15,6) + χ (15,9) | 2 +|χ (6,0) + χ (21,6) + χ (15,15) + χ (15,0) + χ (21,15) + χ (6,6) +χ (11,4) + χ (17,7) + χ (14,10) + χ (11,7) + χ (14,4) + χ (17,10) (21,21) ] ⊕ [λ (8,4) ] ⊕ [λ (17,4) ] ⊕ [λ (17,13) ]…”

Section: Non-degenerate Braidings On Orbifold Graphsmentioning

confidence: 99%

“…In concrete examples, in particular for the conformal embeddings SU (2) 4 ⊂ SU (3) 1 , SU (2) 10 ⊂ SO(5) 1 , SU (2) 28 ⊂ (G 2 ) 1 , SU (3) 3 ⊂ SO (8) 1 , SU (3) 5 ⊂ SU (6) 1 , SU (3) 9 ⊂ (E 6 ) 1 , SU (3) 21 ⊂ (E 7 ) 1 and SU (4) 4 ⊂ SO (15) 1 , such conditions can be shown to be satisfied, and thus V + and V − only intersect on the marked vertices or we recover the modular invariant matrix Z from the induced α + Λ , α − Λ ′ M . The completeness of the induced system has another important aspect.…”

Section: Introductionmentioning

confidence: 99%

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Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

Böckenhauer¹,

Evans²

1999

Commun. Math. Phys.

113

321

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In this paper we further develop the theory of α-induction for nets of subfactors, in particular in view of the system of sectors obtained by mixing the two kinds of induction arising from the two choices of braiding. We construct a relative braiding between the irreducible subsectors of the two "chiral" induced systems, providing a proper braiding on their intersection. We also express the principal and dual principal graphs of the local subfactors in terms of the induced sector systems. This extended theory is again applied to conformal or orbifold embeddings of SU (n) WZW models. A simple formula for the corresponding modular invariant matrix is established in terms of the two inductions, and we show that it holds if and only if the sets of irreducible subsectors of the two chiral induced systems intersect minimally on the set of marked vertices, i.e. on the "physical spectrum" of the embedding theory, or if and only if the canonical endomorphism sector of the conformal or orbifold inclusion subfactor is in the full induced system. We can prove either condition for all simple current extensions of SU (n) and many conformal inclusions, covering in particular all type I modular invariants of SU (2) and SU (3), and we conjecture that it holds also for any other conformal inclusion of SU (n) as well. As a by-product of our calculations, the dual principal graph for the conformal inclusion SU (3) 5 ⊂ SU (6) 1 is computed for the first time.

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Section: Z M Orbifold Inclusions Of Su (N)mentioning

confidence: 99%

Section: Non-degenerate Braidings On Orbifold Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

Böckenhauer¹,

Evans²

1999

Commun. Math. Phys.

113

321

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“…Indeed subfactors of index at most 4 were classified [Ocn88,GdlHJ89,Pop94,Izu91]. This approach has been extremely successful in work of Haagerup [Haa94] and others [AH99,Bis98,Izu91,SV93,BMPS12]. Recently the classification has been extended up to index 5, and even beyond.…”

Section: Introductionmentioning

confidence: 99%

Singly generated planar algebras of small dimension, Part III

Bisch

Jones

Liu

2016

Trans. Amer. Math. Soc.

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Abstract. The first two authors classified subfactor planar algebra generated by a non-trivial 2-box subject to the condition that the dimension of 3-boxes is at most 12 in Part I; 13 in Part II of this series. They are the group planar algebra for Z 3 , the Fuss-Catalan planar algebra ; and the group/subgroup planar algebra for Z 2 ⊂ Z 5 Z 2 . In the present paper, we extend the classification to 14 dimensional 3-boxes. They are all BMW. Precisely it contains a depth 3 one from quantum SO(3), and a one-parameter family from quantum Sp(4).

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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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Exotic Subfactors of Finite Depth with Jones Indices $(5+\sqrt{13})/2$ and $(5+\sqrt{17})/2$

Cited by 104 publications

References 4 publications

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

Modular Invariants, Graphs and α-Induction¶for Nets of Subfactors. III

Singly generated planar algebras of small dimension, Part III

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

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