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Classical and quantum conformal field theory. (English) Zbl 0694.53074
In this paper a formalism is developed which treats the classification of all conformal field theories. The main objects of this formalism are chiral vertex operators, duality matrices and fundamental identities satisfied by them. The authors assign to every Riemann surface a vector space, spanned by the different conformal blocks. Since the Riemann surface can be formed by sewing a number of three holed spheres the different sewing procedures lead to different bases. Therefore the duality matrices are defined as linear transformations in this space. They have to satisfy some consistency conditions.
In order to obtain them the authors construct a simplicial complex and define the notations of “simple moves” and fundamental loops. The set of transformations on the simplicial complex is a duality groupoid. The simple moves are generators of the groupoid and the relations of the fundamental loops are its defining relations. It is shown that in the particular case of “classical conformal field theory”, when the conformal weights of all primary fields vanish, the meaning of these equations is well understood. Using results from category theory, the correspondence between classical conformal field theory and group theory is made more complete.
Reviewer: V.Abramov

MSC:
53C80 Applications of global differential geometry to the sciences
57R22 Topology of vector bundles and fiber bundles
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
20C35 Applications of group representations to physics and other areas of science
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