Abstracts Category : Other

Z2-Orbifolds of Affine Vertex Algebras and W-Algebras

by Masoumah Abdullah Al-Ali

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 <em>VG</em> (called G-orbifold of V), and its extensions. For example, the Moonshine vertex algebra arises as an extension of the Z2-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 g called the Cartan involution, which lifts to a Z2-action on the universal affine vertex algebra <em>Vk(g)</em> at level k. For any g, we shall find an explicit minimal strong generating set for the orbifold <em>Vk(g)Z2</em>, for generic values of k. Let l = rank(g) and let m be the number of positive roots of g, so that dim(g) = 2m+ l. We will prove that for g != <em>sl2</em>, <em>Vk(g)Z2</em> is of type W(1m,2<em>d</em>+(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((d1)n1,... (dr)<em>nr</em>) if it has a minimal strong generating set consisting of <em>ni</em> fields in weight <em>di</em>, for i=1,...,r. In the case g = sl2, there is one extra field in weight 4, so that <em>Vk(g)Z2</em> is of type W(1,23,3,42) for generic value of k. In the case g = <em>sl</em>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 Z2-orbifold of the Zamolodchikov W3-algebra with central charge c, which we denote by <em>Wc3</em>. It was conjectured over 20 years ago in the physics literature that (Wc3)Z2 should be of type W(2,4,6,8,10) for generic values ofAdvisors/Committee Members: Andrew R. Linshaw, Ph.D..