The exoticness and realisability of twisted Haagerup-Izumi modular data

Evans, David Emrys and Gannon, Terry 2011. The exoticness and realisability of twisted Haagerup-Izumi modular data. Communications in Mathematical Physics 307 (2) , pp. 463-512. 10.1007/s00220-011-1329-3

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Abstract

The quantum double of the Haagerup subfactor, the first irreducible finite depth subfactor with index above 4, is the most obvious candidate for exotic modular data. We show that its modular data DHgHg fits into a family Dw Hg2n+1Hg2n+1 , where n ≥ 0 and w Î \mathbbZ2n+1Z2n+1 . We show D0 Hg2n+10Hg2n+1 is related to the subfactors Izumi hypothetically associates to the cyclic groups \mathbbZ2n+1Z2n+1 . Their modular data comes equipped with canonical and dual canonical modular invariants; we compute the corresponding alpha-inductions, etc. In addition, we show there are (respectively) 1, 2, 0 subfactors of Izumi type \mathbbZ7, \mathbbZ9Z7Z9 and \mathbbZ32Z23 , and find numerical evidence for 2, 1, 1, 1, 2 subfactors of Izumi type \mathbbZ11,\mathbbZ13,\mathbbZ15,\mathbbZ17,\mathbbZ19Z11Z13Z15Z17Z19 (previously, Izumi had shown uniqueness for \mathbbZ3Z3 and \mathbbZ5Z5), and we identify their modular data. We explain how DHgHg (more generally Dw Hg2n+1Hg2n+1 ) is a graft of the quantum double D Sym(3)Sym(3) (resp. the twisted double Dw D2n+1D2n+1 ) by affine so(13) (resp. so(4n2+4n+5)(4n2+4n+5)) at level 2. We discuss the vertex operator algebra (or conformal field theory) realisation of the modular data Dw Hg2n+1Hg2n+1 . For example we show there are exactly 2 possible character vectors (giving graded dimensions of all modules) for the Haagerup VOA at central charge c = 8. It seems unlikely that any of this twisted Haagerup-Izumi modular data can be regarded as exotic, in any reasonable sense.

Item Type:	Article
Date Type:	Publication
Status:	Published
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Publisher:	Springer
ISSN:	0010-3616
Last Modified:	04 Jun 2017 03:10
URI:	https://orca.cardiff.ac.uk/id/eprint/17671

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