Title:

Automorphisms and twisted vertex operators

This work is an examination of various aspects of twisted vertex operator representations of KacMoody algebras. It starts with an introduction to KacMoody algebras and string theories, including a discussion of the propagation of strings on orbifolds. String interactions in a subclass of such models naturally involve twisted vertex operators. The centrally extended loop algebra realization of KacMoody algebras is used to explain why the inequivalent gradations of basic representations of KacMoody algebras g(^r) associated with g are in onetoone correspondence with the conjugacy classes of the automorphism group of the root system, aut Ф(_g).The structure of the automorphism groups of the simple Lie algebra root systems are examined. A method of classifying the conjugacy classes of the Weyl groups is explained and then extended to cover the whole automorphism group in cases where there are additional Dynkin diagram symmetries. All possible automorphisms, a, that have the property that det (1 – σ(^r)) ≠ 0, r = 1, ….. , n  1 where n is the order of a, are determined. Such automorphisms lead to interesting orbifold models in which some of the calculations are simplified. A thorough exposition of the twisted vertex operator representation is given including a detailed explanation of the zeromode Hilbert space and the construction of the required cocycle operators. The relation of the vacuum degeneracy to the number of fixed subspace singularities in the orbifold construction is discussed. Explicit examples of twisted vertex operators and their associated cocycles are given. Finally it is shown how the twisted and an alternative shifted vertex operator representation of the same gradation may be identified. This is used to determine the invariant subalgebras of the gradations along with the vacuum degeneracies and conformal weights of the representations. The results of calculations for inequivalent gradations of the simply laced exceptional algebras are given.
