The exoticness and realisability of twisted Haagerup-Izumi modular data

TL;DR

This paper investigates the modular data of the quantum double of the Haagerup subfactor, exploring its relation to Izumi's subfactors, and discusses its potential realization in conformal field theory, revealing new classifications and numerical evidence.

Contribution

It introduces a family of modular data related to the Haagerup subfactor and computes new subfactor classifications and modular invariants, expanding understanding of exotic quantum symmetries.

Findings

Identifies a family of modular data $D^ ext{ω} Hg_{2n+1}$ related to the Haagerup subfactor.

Provides numerical evidence for multiple subfactors of Izumi type for various cyclic groups.

Shows the connection of these modular data to affine algebras and vertex operator algebras at level 2.

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 $D^\omega Hg_{2n+1}$, where $n\ge 0$ and $\omega\in \bbZ_{2n+1}$. 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 $D^\omega Hg_{2n+1}$) 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 $D^\omega Hg_{2n+1}$. 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.