The Exoticness and Realisability of Twisted Haagerup–Izumi Modular Data
The Exoticness and Realisability of Twisted Haagerup–Izumi Modular Data
复制标题
Twisted Haagerup-Izumi 模块化数据的奇异性和可实现性
DOI：
10.1007/s00220-011-1329-3
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发表时间：
2010
影响因子：
2.4
通讯作者：
T. Gannon
中科院分区：
文献类型：
--
作者：
David E. Evans;T. Gannon
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 $${\mathcal{D}{\rm Hg}}$$ fits into a family $${\mathcal{D}^\omega {\rm Hg}_{2n+1}}$$ , where n ≥ 0 and $${\omega\in \mathbb{Z}_{2n+1}}$$ . We show $${\mathcal{D}^0 {\rm Hg}_{2n+1}}$$ is related to the subfactors Izumi hypothetically associates to the cyclic groups $${\mathbb{Z}_{2n+1}}$$ . 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 $${\mathbb{Z}_7, \mathbb{Z}_9}$$ and $${\mathbb{Z}_3^2}$$ , and find numerical evidence for 2, 1, 1, 1, 2 subfactors of Izumi type $${\mathbb{Z}_{11},\mathbb{Z}_{13},\mathbb{Z}_{15},\mathbb{Z}_{17},\mathbb{Z}_{19}}$$ (previously, Izumi had shown uniqueness for $${\mathbb{Z}_3}$$ and $${\mathbb{Z}_5}$$), and we identify their modular data. We explain how $${\mathcal{D}{\rm Hg}}$$ (more generally $${\mathcal{D}^\omega {\rm Hg}_{2n+1}}$$) is a graft of the quantum double $${\mathcal{D} Sym(3)}$$ (resp. the twisted double $${\mathcal{D}^\omega D_{2n+1}}$$) 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 $${\mathcal{D}^\omega {\rm Hg}_{2n+1}}$$ . 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.
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 $${\mathcal{D}{\rm Hg}}$$ fits into a family $${\mathcal{D}^\omega {\rm Hg}_{2n+1}}$$ , where n ≥ 0 and $${\omega\in \mathbb{Z}_{2n+1}}$$ . We show $${\mathcal{D}^0 {\rm Hg}_{2n+1}}$$ is related to the subfactors Izumi hypothetically associates to the cyclic groups $${\mathbb{Z}_{2n+1}}$$ . 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 $${\mathbb{Z}_7, \mathbb{Z}_9}$$ and $${\mathbb{Z}_3^2}$$ , and find numerical evidence for 2, 1, 1, 1, 2 subfactors of Izumi type $${\mathbb{Z}_{11},\mathbb{Z}_{13},\mathbb{Z}_{15},\mathbb{Z}_{17},\mathbb{Z}_{19}}$$ (previously, Izumi had shown uniqueness for $${\mathbb{Z}_3}$$ and $${\mathbb{Z}_5}$$), and we identify their modular data. We explain how $${\mathcal{D}{\rm Hg}}$$ (more generally $${\mathcal{D}^\omega {\rm Hg}_{2n+1}}$$) is a graft of the quantum double $${\mathcal{D} Sym(3)}$$ (resp. the twisted double $${\mathcal{D}^\omega D_{2n+1}}$$) 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 $${\mathcal{D}^\omega {\rm Hg}_{2n+1}}$$ . 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.