# Bosonic Construction of Vertex Operator Para-Algebras from Symplectic Affine Kac-Moody Algebras

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*Michael David Weiner*

Inspired by mathematical structures found by theoretical
physicists and by the desire to understand the "monstrous moonshine" of
the Monster group, Borcherds, Frenkel, Lepowsky, and Meurman
introduced the definition of vertex operator algebra (VOA). An
important part of the theory of VOAs concerns their modules and
intertwining operators between modules. Feingold, Frenkel, and Ries
defined a structure, called a vertex operator para-algebra (VOPA),
where a VOA, its modules and their intertwining operators are
unified.

In this work, for each \(n \geq 1\), the author uses the
bosonic construction (from a Weyl algebra) of four level \(-
1/2\) irreducible representations of the symplectic affine
Kac-Moody Lie algebra \(C_n^{(1)}\). They define intertwining
operators so that the direct sum of the four modules forms a
VOPA. This work includes the bosonic analog of the fermionic
construction of a vertex operator superalgebra from the four level 1
irreducible modules of type \(D_n^{(1)}\) given by Feingold,
Frenkel, and Ries. While they get only a VOPA when \(n = 4\)
using classical triality, the techniques in this work apply to any
\(n \geq 1\).

#### Table of Contents

# Table of Contents

## Bosonic Construction of Vertex Operator Para-Algebras from Symplectic Affine Kac-Moody Algebras

- Contents vii8 free
- Preface viii9 free
- Chapter 1. Introduction 110 free
- Chapter 2. Bosonic Construction of Symplectic Affine Kac-Moody Algebras 1120 free
- Chapter 3. Bosonic Construction of Symplectic Vertex Operator Algebras and Modules 1625
- Chapter 4. Bosonic Construction of Vertex Operator Para-Algebras 4251
- Appendix 100109
- Bibliography 106115