Date of Award

1-1-2018

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

College/School/Department

Department of Mathematics and Statistics

Content Description

1 online resource (ii, v, 72 pages) : illustrations.

Dissertation/Thesis Chair

Antun Milas

Committee Members

Anupam Srivastav, Cristian Lenart, Alex Tchernev

Keywords

character, module, quantum dimension, superalgebra, verlinde formula, vertex algebra, Superalgebras, Vertex operator algebras, Lie superalgebras

Subject Categories

Physical Sciences and Mathematics | Quantum Physics

Abstract

In this work we study the vertex operator superalgebra known as the singlet vertex operator superalgebra, denoted $\overline{SM(1)}$. We are particularly interested in the number theoretic properties of the characters, and supercharacters, of irreducible modules of $\overline{SM(1)}$ as modules over the Neveu-Schwarz and Ramond Lie superalgebras, denoted $\mathfrak{ns}$ and $\mathfrak{R}$, respectively. These characters occur as modular-like forms, namely false and mock theta functions. The $S$- and $T$- transformations are computed and we demonstrate modular invariance of the vector space spanned by these characters and supercharacters. We use a continuous version of the Verlinde formula, introduced by Creutzig and Milas, to put an associative, commutative ring structure on the space of characters which we conjecture is isomorphic to the Grothendieck ring of the category $\overline{SM(1)}$-Mod. We compute the quantum dimensions of irreducible modules and show that relations shared by the characters are also shared by the quantum dimensions.

Download

Share

COinS