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ADE subalgebras of the triplet vertex algebra W(p): A-series

Motivated by D. Adamovic, A. Milas, On the triplet vertex algebra W(p), Adv. Math.217 (2008) 2664– 2699 for every finite subgroup Γ of PSL(2, C) we investigate the fixed point subalgebra W(p) ^Γ of the triplet vertex W(p), of central charge c_{; ; ; ; ; 1, p}; ; ; ; ; . This part deals with the A-series in the ADE classification of finite subgroups of PSL(2, C). First, we prove the C2-cofiniteness of the A_m- fixed subalgebra W(p) ^{; ; ; ; ; A_m}; ; ; ; ; . Then we construct a family of W(p) ^{; ; ; ; ; A_m}; ; ; ; ; - modules, which are expected to form a complete set of irreps. As a strong support to our conjecture, we prove modular invariance of (generalized) characters of the relevant (logarithmic) modules. Further evidence is provided by calculations in Zhu's algebra for m=2. We also present a rigorous proof of the fact that the full automorphism group of W(p) is PSL(2, C).

Vertex operator algebras ; orbifold models ; automorphism groups ; C2-cofiniteness ; Zhu's algebras

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