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An Algorithmic Approach to Operator Product Expansions, W-Algebras and W-Strings
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String theory is currently the most promising theory to explain the spectrum of the elementary particles and their interactions. One of the 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 analyze the calculational rules for OPEs, give an algorithm to computer 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 computer the B-coefficients in the OPEs, the first explicit construction of the WB2-algebra, and criteria for the factorization of free fields in a W-algebra. An important technique to construct realizations of W-algebras is Drinfel'd-Sokolov reduction. The method consists of imposing certain constraints on the elements of an affine Lie algebra. We quantize this reduction via gauged WXNW-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 extra symmetry generator of dimension N. We clarify how the spectrum of this theory forms a minimal model of the WN-algebra.
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http://www.arxiv.org/abs/hep-th/9506159
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