Celebratio Mathematica — Schwarz

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Works connected to Ichirō Satake

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A. Schwarz: “Geometry of fermionic string,” pp. 1377–1386 in Proceedings of the International Congress of Mathematicians (Kyoto, Japan, 21–29 August 1990), vol. 2. Edited by I. Satake. Springer (Tokyo), 1991. MR 1159322 Zbl 0751.53024 incollection

Physicists hope that the Green–Schwarz superstring theory describes all interactions existing in the Nature. However the verification of this conjecture is connected with very difficult and very interesting mathematical problems. We consider here only some problems arising in the Polyakov approach to the fermionic string. (Fermionic string is closely related with the Green–Schwarz superstring.) We explain the connection between string theory and superconformal geometry, the origin of string measure on superconformal moduli space and analytic properties of this measure, the construction of universal moduli space and the expression of string measure in terms of super \( \tau \)-function etc. The lecture is based on the papers [Baranov and Schwarz 1985; Baranov, Frolov and Schwarz 1987; Baranov and Schwarz 1987; Schwarz 1988; Rosly, Schwarz and Voronov 1988; Rosly, Schwarz and Voronov 1989; Schwarz 1989; Dolgikh and Schwarz 1990; Dolgikh, Rosly and Schwarz 1990; Baranov, Frolov and Schwarz 1989]. The results concerning the measure on the moduli space of \( N = 2 \) superconformal manifolds are new.