A Cyclic Orbifold Theory for Holomorphic Vertex Operator Algebras and Applications

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.

276 pages, LaTeX; Ph.D. thesis, some typos corrected

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• The Moonshine Anomaly
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• Haploid algebras in $C^{\ast }$-tensor categories and the Schellekens list
• Dimension Formulae in Genus Zero and Uniqueness of Vertex Operator Algebras
• Reverse orbifold construction and uniqueness of holomorphic vertex operator algebras
• On orbifold constructions associated with the Leech lattice vertex operator algebra
• On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras
• Inertia groups and uniqueness of holomorphic vertex operator algebras
• Lattices, Vertex Algebras and Modular Categories
• Orbifold Vertex Operator Algebras and the Positivity Condition
• Extensions of tensor products of ${\mathbb{Z}}_p$-orbifold models of the lattice vertex operator algebra $V_{\sqrt{2}{A}_{p-1}}$
• Frobenius Monoidal Functors of Dijkgraaf-Witten Categories and Rigid Frobenius Algebras
• Simple current extensions of tensor products of vertex operator algebras
• Beauty And The Beast Part 2: Apprehending The Missing Supercurrent
• Representation theory of vertex operator algebras and orbifold conformal field theory