String theory is currently the most promising theory to explain the spectrum
of the elementary particles and their interactions. One of its most important
features is its large symmetry group, which contains the conformal
transformations in two dimensions as a subgroup. At quantum level, the symmetry
group of a theory gives rise to differential equations between correlation
functions of observables. We show that these Ward-identities are equivalent to
Operator Product Expansions (OPEs), which encode the short-distance
singularities of correlation functions with symmetry generators. The OPEs allow
us to determine algebraically many properties of the theory under study. We
analyse the calculational rules for OPEs, give an algorithm to compute OPEs,
and discuss an implementation in Mathematica. There exist different string
theories, based on extensions of the conformal algebra to so-called W-algebras.
These algebras are generically nonlinear. We study their OPEs, with as main
results an efficient algorithm to compute the beta-coefficients in the OPEs,
the first explicit construction of the WB_2-algebra, and criteria for the
factorisation of free fields in a W-algebra. An important technique to
construct realisations of W-algebras is Drinfel'd- Sokolov reduction. The
method consists of imposing certain constraints on the elements of an affine
Lie algebra. We quantise this reduction via gauged WZNW-models. This enables us
in a theory with a gauged W-symmetry, to compute exactly the correlation
functions of the effective theory. Finally, we investigate the critical
W-string theories based on an extension of the conformal algebra with one
symmetry generator of dimension N. We clarify how the spectrum of this theory
forms a minimal model of the W_N-algebra.Comment: 127 pages, LaTex, shar-file including readme.txt, 12 latex files, 6
eps files and 6 pcx files, PhD. thesis KU Leuve
