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Intersection theory, integrable hierarchies and topological field theory. (English) Zbl 1221.32005

Fröhlich, J. (ed.) et al., New symmetry principles in quantum field theory. Proceedings of the NATO Advanced Study Institute held in Cargèse, July 16–27, 1991. New York, NY: Plenum Press (ISBN 0-306-44240-X). NATO Advanced Science Institutes Series B: Physics, 295, 95-158 (1992).
Summary: We review the various relations between intersection theory on the moduli space of Riemann surfaces, integrable hierarchies of KdV type, matrix models, and topological quantum field theories. We explain in particular why matrix integrals of the type considered by Kontsevich naturally appear as tau-functions associated to minimal models. Our starting point is the extremely simple form of the string equation for the topological \((p,1)\) models, where the so-called Baker-Akhiezer function is given by a (generalized) Airy function.
For the entire collection see [Zbl 1222.81031].

MSC:

32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
32G81 Applications of deformations of analytic structures to the sciences
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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