A ${\mathbb{Z}}_3$-orbifold theory of lattice vertex operator algebra and ${\mathbb{Z}}_3$-orbifold constructions

Abstract

Let $V$ be a simple VOA of CFT-type satisfying $V^{\prime }\cong V$ and $\sigma$ a finite automorphism of $V$. We prove that if all $V$-modules are completely reducible and a fixed point subVOA $V^{\sigma }$ is $C_2$-cofinite, then all $V^{\sigma }$-modules are completely reducible and every simple $V^{\sigma }$-module appears in some twisted or ordinary $V$-modules as a $V^{\sigma }$-submodule. We also prove that $V_L^{\sigma }$ is $C_2$-cofinite for any lattice VOA $V_L$ and $σ\in \Aut(V_L)$ lifted from any triality automorphism of $L$. Using these results, we present two $Z_3$-orbifold constructions as examples. One is the moonshine VOA $V^{♮}$ and the other is a new CFT No.32 in Schellekens' list.

24 pages, we have revised some part of the proof

Cited by in corpus (1)

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