# Connections between mock modular forms and vertex operator algebras Open Access

## Beneish, Lea (Spring 2020)

Permanent URL: https://etd.library.emory.edu/concern/etds/mk61rh66j?locale=en
Published

## Abstract

The results in this dissertation come in two flavors, first we aim to strengthen the analogy between monstrous and umbral moonshine using vertex operator algebras, and second we derive structural results on vertex operator algebras using mock modular forms.

Towards strengthening the analogy between umbral and monstrous moonshine, we reframe Mathieu moonshine by repackaging the Mathieu moonshine mock modular forms in a few different ways, verifying the existence of corresponding modules, and giving various applications including connections with arithmetic. We produce vertex operator algebra constructions of some of these modules.

Using results from orbifold theory and from the theory of mock modular forms, we derive new structural results on vertex operator algebras. In joint work with Victor Manuel Aricheta, we study the asymptotic structure of sequences of G-modules where G are finite automorphism groups of certain vertex operator algebras (in particular this holds for umbral moonshine modules). And in joint work with Michael Mertens, we use Weierstrass mock modular forms to relate a dimension formula for certain vertex operator algebras to the arithmetic of modular curves.

1 Introduction

1.1  On the analogy between monstrous and umbral moonshine . . . . . . 1

1.2  Structural results on vertex operator algebras . . . . . . . . . . . . . 10

1.2.1 On a dimension formula ..................... 10

1.2.2 On the asymptotic structure of modules . . . . . . . . . . . . 15

2  Background–mock modular forms 23

2.1 Harmonic Maass forms and mock modular forms. . . . . . . . . . . . 23

2.2 Operators on (mock) modular forms................... 25

2.3 Mock modular forms as Rademacher sums . . . . . . . . . . . . . . . 28

2.4 Weierstrass mock modular forms..................... 30

3 Background–vertex operator algebras 36

3.1 Vertex operator algebras: basics and definitions . . . . . . . . . . . . 36

3.2 Orbifold theory .............................. 38

3.3 Modular invariance of characters .................... 39

4 Quasimodular moonshine and arithmetic connections 42

4.1 Quasimodular M24 forms......................... 42

4.2 More general framework ......................... 45

4.3 Arithmetic/geometric connections.................... 55

4.4 An explicit module construction..................... 59

5 Module constructions for certain subgroups of the largest Mathieu group 68

5.1 The functions ............................... 69

5.2 Module construction I .......................... 77

5.3 Module construction II.......................... 85

6 On Weierstrass mock modular forms and a dimension formula for certain vertex operator algebras 96

6.1 The space of harmonic Maass forms in terms of Weierstrass mock modular forms................................. 96

6.2 Dimension formulas............................ 103

7 Moonshine modules and a question of Griess 112

7.1 Asymptotic structure of homogeneous subspaces . . . . . . . . . . . . 112

7.2 Partial orders on irreducible representations . . . . . . . . . . . . . . 114

7.3 Application to umbral moonshine .................... 119

7.4 Asymptotic regularity for vertex operator algebras . . . . . . . . . . . 122

Bibliography 125