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The category of modules of the triplet W-algebras associated to the Virasoro minimal models

Nakano, Hiromu 東北大学

2023.03.24

概要

We have studied the structure of the category of modules of the family of vertex operator
algebras called triplet W -algebras. In this thesis, we mainly discuss the triplet W -algebras
Wp+ ,p− constructed by Feigin, Gainutdinov, Semikhatov and Tipunin and the super triplet
W -algebras SW(m) constructed by Adamovi´c and Milas. These triplet W -algebras are
one of the few examples of non-rational vertex operator algebras satisfying the C2 -cofinite
condition. In general, for any rational vertex operator algebra, the abelian category of
modules is semisimple, but for any non-rational vertex operator algebra, the abelian category of modules of the vertex operator algebra is not semisimple and contains logarithmic
modules whose L0 nilpotent rank n ≥ 2, where L0 is the zero mode of the Virasoro algebra. Furthermore, if the C2 -cofinite condition is satisfied, the number of simple modules
is finite, and the category of modules has braided tensor category structure as developed
in the series of papers by Huang, Lepowsky and Zhang [37, 38, 39, 40, 41, 42, 43, 44].
Thus, the triplet W -algebras are mathematically tractable among the non-rational vertex
operator algebras, but specific aspects such as the structure of logarithmic modules and
tensor products among logarithmic modules are still not fully understood.
In the following, we will give a brief description of the research background and problems related to triplet W -algebras.
First let us review triplet W -algebras associated to Virasoro minimal models. Let
p ∈ Z≥1 and let p− > p+ ≥ 2 be coprime integers. ...

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