A Cyclic Orbifold Theory for Holomorphic Vertex Operator Algebras and Applications

TL;DR

This thesis develops a cyclic orbifold theory for holomorphic vertex operator algebras, classifies their fusion structures, and constructs new examples with applications to algebra classification and automorphic forms.

Contribution

It introduces a new orbifold framework for cyclic groups acting on holomorphic VOAs, solves the extension problem, and constructs five new holomorphic VOAs of central charge 24.

Findings

Fusion algebra of fixed-point subalgebras is group-like.

Constructed five new holomorphic VOAs of central charge 24.

Presented BRST construction of ten Borcherds-Kac-Moody algebras.

Abstract

In this thesis we develop an orbifold theory for a finite, cyclic group $G$ acting on a suitably regular, holomorphic vertex operator algebra $V$. To this end we describe the fusion algebra of the fixed-point vertex operator subalgebra $V^G$ and show that $V^G$ has group-like fusion. Then we solve the extension problem for vertex operator algebras with group-like fusion. We use these results to construct five new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds, contributing to the classification of the $V_1$-structures of suitably regular, holomorphic vertex operator algebras of central charge 24. As another application we present the BRST construction of ten Borcherds-Kac-Moody algebras whose denominator identities are completely reflective automorphic products of singular weight.