Functional determinants on punctured Riemann surfaces and their application to string theory

Abstract

We compute functional determinants on punctured Riemann surfaces by integrating the corresponding energy-momentum tensors. The results are applied to the bosonic closed string theory in two ways. We first give a rigorous proof of the relation between the scalar and ghost determinants for the singular metrics which are squares of meromorphic one-forms. This relation played an important role in the recent proof of the equivalence between the light-cone and Polyakov approaches given by D'Hoker and Giddings. Secondly we give explicit formulae for the scattering amplitudes as integrals over the moduli space of punctured Riemann surfaces.