Construction and Classification of Holomorphic Vertex Operator Algebras

TL;DR

The paper develops an orbifold framework for holomorphic VOAs under finite cyclic automorphisms and proves that Schellekens' $V_1$-structure classification for central charge $24$ is a theorem within VOA theory. It leverages modular invariance, the Verlinde formula, and abelian intertwining algebras to analyze fixed-point subalgebras and twisted sectors, enabling rigid constraints on possible $V_1$-structures and the construction of new examples. By applying lattice and Niemeier lattice orbifolds, it realizes five new holomorphic VOAs of central charge $24$ and determines their $V_1$-structures, contributing to the realization of Schellekens' list and connecting to broader moonshine phenomena.

Abstract

We develop an orbifold theory for finite, cyclic groups acting on holomorphic vertex operator algebras. Then we show that Schellekens' classification of $V_1$-structures of meromorphic conformal field theories of central charge 24 is a theorem on vertex operator algebras. Finally we use these results to construct some new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds.

Theorems & Definitions (38)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 3.1
• Proposition 3.2
• Proposition 3.3
• Theorem 3.4
• Proposition 3.5
• Theorem 3.6
• Theorem 4.1