Z2-Orbifolds of Affine Vertex Algebras and W-Algebras

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 VG (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 Vk(g) at level k. For any g, we shall find an explicit minimal strong generating set for the orbifold Vk(g)Z2, 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 != sl2, Vk(g)Z2 is of type W(1m,2d+(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)nr) if it has a minimal strong generating set consisting of ni fields in weight di, for i=1,...,r. In the case g = sl2, there is one extra field in weight 4, so that Vk(g)Z2 is of type W(1,23,3,42) for generic value of k. In the case g = sl2, 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 Wc3. 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 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, (Wc3)Z2 is of type W(2,6,8,10,12,14). The method introduced to study (Wc3)Z2 involves ideas from algebraic geometry and is applicable to a broad range of problems of this kind