FUNCTIONAL INTEGRAL METHODS FOR FERMIONIC STRING AND SUPER RIEMANN SURFACES

Abstract

We investigate the light-cone gauge formulation of fermionic string of Mandelstam in a superspace. To formulate the fermionic string in a superspace, we use the theory of super Riemann surfaces (SRS). We define the Neumann functions and the Mandelstam mappings in a superspace by means of the so-called Abelian differentials of the first and the third kinds on SRS. In the super Schottky parametrization of SRS these superdifferentials are constructed at the multiloop level. The functional integral measure of a super light-cone diagram, which consists of 6g−6+2N even moduli parameters and 4g−4+2N odd ones, are specified in our formulation. In one-loop case with even spin structures we explicitly evaluate the superdeterminants on the super light-cone diagrams and calculate the one-loop amplitudes with N massless vector states. It is shown that the result is written in the form of a correlation function of the vertex operators. Furthermore, we evaluate the 3- and 4-particle amplitudes explicitly, which agree exactly with those calculated in the Green-Schwarz formulation.