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String theory and algebraic geometry of moduli spaces. (English) Zbl 0658.14014

The bulk of this paper consists of a tour through the algebraic geometry of Riemann surfaces in order to define the partition function of string theory in terms of a natural holomorphic volume form on the moduli space. At the end, there is an interpretation in terms of algebraic number theory of the partition function, using the arithmetic geometry of Faltings.
Although much of the material also appears explicitly or otherwise in the works of other physicists, this is a good exposition for the mathematician. It gives an ab initio treatment of the essentials of holomorphic line bundles, divisor classes, Cauchy-Riemann operators, determinant bundles, the Arakelov metric, Quillen metric, and Faltings metric, the Grothendieck-Riemann-Roch theorem, Siegel upper half space and many other topics, to culminate in a formula for the partition function.
Reviewer: N.J.Hitchin

MSC:

14H15 Families, moduli of curves (analytic)
14H10 Families, moduli of curves (algebraic)
81T99 Quantum field theory; related classical field theories
14H25 Arithmetic ground fields for curves
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