From authors’ introduction: “It is thus urgent to improve our understanding of the principles and techniques of string theory in particular to formulate Feynman rules for loop amplitudes. For scattering of pure particle states the Feynman diagram for closed oriented strings at the h-loop level is a compact surface M with h handles, with insertion of the corresponding vertex operators. In the Polyakov formulation of string theory, the amplitudes are given by functional integrals over all (super) geometries of M. Conformal and reparametrization invariances in the critical dimensions suggest that the amplitudes should reduce to integrals over the (finite-dimensional) moduli space of M, and Feynman rules in the string case should correspond to an explicit identification of the integrand as well as of the measure on moduli space occuring in the amplitude.
We shall show that the Polyakov quantization of strings leads to the Weil-Petersson measure on moduli space and that the integrand is a combination of determinants and propagators for the Laplacian and Dirac operators with respect to constant curvature metrics. In the superstring case, zero modes of the gravitation field also appear, which can be viewed as the superanalogues of quadratic differentials. These ingredients in turn can be evaluated in terms of fundamental notions in the mathematics of Riemann surfaces and number theory, such as Poincaré series, prime forms, theta functions and special values of Selberg zeta- functions.”