# Dimension Formulae in Genus Zero and Uniqueness of Vertex Operator Algebras

## TL;DR

This paper establishes a dimension formula for orbifold vertex operator algebras of central charge 24 with specific automorphisms, and proves the uniqueness of certain Lie algebra structures as the weight-one space of these VOAs.

## Contribution

It introduces a new dimension formula for orbifold VOAs and uses inverse orbifold techniques to classify unique VOA structures with specified Lie algebra weight-one spaces.

## Key findings

- Dimension formula for orbifold VOAs with genus zero automorphisms
- Uniqueness of 15 specific Lie algebra structures as VOA weight-one spaces
- Complete classification of certain holomorphic VOAs of central charge 24

## Abstract

We prove a dimension formula for orbifold vertex operator algebras of central charge 24 by automorphisms of order $n$ such that $\Gamma_0(n)$ is a genus zero group. We then use this formula together with the inverse orbifold construction for automorphisms of orders 2, 4, 5, 6 and 8 to establish that each of the following fifteen Lie algebras is the weight-one space $V_1$ of exactly one holomorphic, $C_2$-cofinite vertex operator algebra $V$ of CFT-type of central charge 24: $A_5C_5E_{6,2}$, $A_3A_{7,2}C_3^2$, $A_{8,2}F_{4,2}$, $B_8E_{8,2}$, $A_2^2A_{5,2}^2B_2$, $C_8F_4^2$, $A_{4,2}^2C_{4,2}$, $A_{2,2}^4D_{4,4}$, $B_5E_{7,2}F_4$, $B_4C_6^2$, $A_{4,5}^2$, $A_4A_{9,2}B_3$, $B_6C_{10}$, $A_1C_{5,3}G_{2,2}$ and $A_{1,2}A_{3,4}^3$.