The exoticness and realisability of twisted Haagerup-Izumi modular data

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 $D^{\omega }H{g}_{2n+1}$, where $n\ge 0$ and $\omega \in {\mathbb{Z}}_{2n+1}$. We show $D^{0}H{g}_{2n+1}$ 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 $D^{\omega }H{g}_{2n+1}$) is a graft of the quantum double DSym(3) (resp. the twisted double $D^{\omega }{D}_{2n+1}$) 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 $D^{\omega }H{g}_{2n+1}$. 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.

52 pages, 4 figures

References in corpus (3)

• Vertex operator algebras, the Verlinde conjecture and modular tensor categories
• Subfactor realisation of modular invariants
• On conformal field theories with low number of primary fields

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