The exoticness and realisability of twisted Haagerup-Izumi modular data

TL;DR

The work places the Haagerup quantum double in a broad modular-data framework, extending it to twisted and dihedral-based families $D^ωHg_{2n+1}$ and linking Izumi-type subfactors to Haagerup-Izumi data. It clarifies how tube algebras, alpha-induction, and canonical modular invariants encode both subfactor and VOA structures, and it demonstrates close relationships to $S_3$ and SO$(13)$ at level 2 via grafting constructions. The authors compute explicit modular data, explore Galois twists, and provide detailed character-vector analyses for untwisted and twisted Haagerup doubles, producing strong evidence that Haagerup-Izumi modular data can be realized in VOAs and conformal nets rather than being exotic. The work also proposes a dihedral- Haagerup 'diamond' of interrelated VOAs and coset constructions, suggesting a rich web of conformal embeddings and subalgebra relations underpinning these exotic-looking modular data with practical implications for CFT, subfactor theory, and VOA classification.

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 DHg fits into a family $D^ωHg_{2n+1}$, where $n\ge 0$ and $ω\in \bbZ_{2n+1}$. We show $D^0 Hg_{2n+1}$ is related to the subfactors Izumi hypothetically associates to the cyclic groups $Z_{2n+1}$. 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 $Z_7$, $Z_9$ and $Z_3^2$, and find numerical evidence for 2, 1, 1, 1, 2 subfactors of Izumi type $Z_{11},Z_{13},Z_{15},Z_{17},Z_{19}$ (previously, Izumi had shown uniqueness for $Z_3$ and $Z_5$), and we identify their modular data. We explain how DHg (more generally $D^ωHg_{2n+1}$) is a graft of the quantum double DSym(3) (resp. the twisted double $D^ωD_{2n+1}$) by affine so(13) (resp. so(4n^2+4n+5)) at level 2. We discuss the vertex operator algebra (or conformal field theory) realisation of the modular data $D^ωHg_{2n+1}$. 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.