Complex geometry in string perturbation theory

Duong H. Phong
Columbia U., USA
 

Abstract

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A basic principle of string theory is that the physical scattering amplitudes of N string states at a given perturbative order h should be given by the integrals of forms of maximal rank over the moduli space of Riemann surfaces of genus h with N punctures. These forms - sometimes called the "string measures" - depend heavily on the complex geometry of moduli space: in fact, they should arise as the norms squared of meromorphic sections of suitable vector bundles over moduli space. Despite many recent great advances in string theory, explicit and effective rules for constructing the string measures are still unavailable. Such rules can be viewed as the string analogues of the Feynman rules of quantum field theory. The main difficulty in finding them can be traced to the presence of "supermoduli", which is a manifestation of two fundamental competing symmetries of strings, namely conformal invariance and supersymmetry. In this talk, we shall describe these problems and recent joint work with Eric D'Hoker solving the case of genus h = 2, which is the first case where the supermoduli difficulty manifests itself for all scattering amplitudes.