Functional integral methods for fermionic string and super Riemann surfaces

The authors investigate the light-cone gauge formulation string of Mandelstam in a superspace. To formulate the fermionic string in a superspace, they use the theory of super Riemann surfaces (SRS). They 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 moduloi parameters and 4g - 4 + 2N odd ones, are specified in their formulation. In one-loop case with even spin structures they 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