#### Date of Award

1-1-2017

#### Document Type

Dissertation

#### Degree Name

Ph.D.

#### Department

Mathematics

#### First Advisor

Andrew R. Linshaw, Ph.D.

#### Keywords

Algebraic Geometry, Vertex Algebras, Leech Lattice

#### Abstract

Vertex algebras arose in conformal field theory and were first defined axiomatically by Borcherds in his famous proof of the Moonshine Conjecture in 1986. The orbifold construction is a standard way to construct new vertex algebras from old ones. Starting with a vertex algebra *V* and a group *G* of automorphisms, one considers the invariant subalgebra *V ^{G}* (called

*G*-orbifold of

*V*), and its extensions. For example, the Moonshine vertex algebra arises as an extension of the

*Z*

_{2}-orbifold of the lattice vertex algebra associated to the Leech lattice.

In this thesis we consider two problems. First, given a simple, finite-dimensional Lie algebra ** g**, there is an involution on

**called the**

*g**Cartan involution*, which lifts to a

*Z*

_{2}-action on the universal affine vertex algebra

*V*at level

^{k}(**g**)*k*. For any

**, we shall find an explicit minimal strong generating set for the orbifold**

*g**V*, for generic values of

^{k}(**g**)^{Z2}*k*. Let

*l*= rank(

**) and let**

*g**m*be the number of positive roots of

**, so that dim(**

*g***) = 2**

*g**m*+

*l*. We will prove that for

**!=**

*g**,*

**s**l_{2}*V*is of type

^{k}(**g**)^{Z2}*W*(1

^{m},2

^{d}^{+(binomial{d}{2})},3

^{(binomial{d}{2})}, 4),

*d*=

*m*+

*l*, for generic values of

*k*. In this notation, a vertex algebra is said to be of type

*W*((

*d*

_{1})

^{n1},... (

*d*

_{r})

*) if it has a minimal strong generating set consisting of*

^{nr}*n*fields in weight

_{i}*d*, for

_{i}*i*=1,...,

*r*. In the case

**=**

*g*

*s**l*

_{2}, there is one extra field in weight 4, so that

*V*is of type

^{k}(**g**)^{Z2}*W*(1,2

^{3},3,4

^{2}) for generic value of

*k*. In the case

**=**

*g*

**s**l_{2}, we explicitly determine the set of

*nongeneric*values of

*k*where this set does not strongly generate the orbifold; it consists only of {0, +-(32/3), 16, 48}. Second, we consider the

*Z*

_{2}-orbifold of the Zamolodchikov

*W*

_{3}-algebra with central charge

*c*, which we denote by

*W*. It was conjectured over 20 years ago in the physics literature that (W

^{c}_{3}^{c}

_{3})

^{Z2}should be of type

*W*(2,4,6,8,10) for generic values of

*c*. We prove this conjecture for all values of

*c*!= (559 +- 7*sqrt(76657))/95, and we show that for these two values of

*c*, (W

^{c}

_{3})

^{Z2}is of type

*W*(2,6,8,10,12,14). The method introduced to study (W

^{c}

_{3})

^{Z2}involves ideas from algebraic geometry and is applicable to a broad range of problems of this kind.

#### Recommended Citation

Al-Ali, Masoumah Abdullah, "Z2-Orbifolds of Affine Vertex Algebras and W-Algebras" (2017). *Electronic Theses and Dissertations*. 1313.

https://digitalcommons.du.edu/etd/1313

#### Provenance

Received from ProQuest

#### Rights holder

Masoumah Abdullah Al-Ali

#### File size

123 p.

#### Copyright date

2017

#### File format

application/pdf

#### Language

en

#### Discipline

Mathematics

## Comments

Copyright is held by the author.