$C_2$-cofiniteness of 2-cyclic permutation orbifold models

TL;DR

This paper proves that the property of $C_2$-cofiniteness is preserved under 2-cyclic permutation orbifold constructions for a broad class of vertex operator algebras, including lattice models.

Contribution

It establishes the $C_2$-cofiniteness of 2-cyclic permutation orbifold models for any $C_2$-cofinite simple vertex operator algebra of CFT type, extending known results.

Findings

Proves $C_2$-cofiniteness of $(V times V)^{S_2}$ for any $C_2$-cofinite $V$.

Shows $V_L^+$ is $C_2$-cofinite for rank one lattices.

Provides a new proof technique leveraging orbifold models and lattice VOAs.

Abstract

In this article, we consider permutation orbifold models of $C_2$-cofinite vertex operator algebras of CFT type. We show the $C_2$-cofiniteness of the 2-cyclic permutation orbifold model $(V\otimes V)^{S_2}$ for an arbitrary $C_2$-cofinite simple vertex operator algebra $V$ of CFT type. We also give a proof of the $C_2$-cofiniteness of a $\Z_2$-orbifold model $V_L^+$ of the lattice vertex operator algebra $V_L$ associated with a rank one positive definite even lattice $L$ by using our result and the $C_2$-cofiniteness of $V_L$.