Pointwise vanishing of two-loop contributions to the one, two, three-point functions in the Neveu-Schwarz-Ramond formalism

Abstract

i. It is often argued that in a properly defined superstring theory the one, two, three-point functions for massless particles must, together with the partition functions, vanish in all perturbation orders (in flat spacetime) [i]. However, explicit realization of such a theory outside the single-loop approximation is as yet unknown. It is widely believed that a correct theory will arise from a fermion string after a special summation over the spinor structures, this being analogous to the Gliozzi-Scherk-Olive projection. Unfortunately, the situation here is not so simple. The main problem is that the theory of a fermionic string depends on the choice of the odd moduli [2,3]. If they are chosen arbitrarily, nothing like a superstring is obtained. Moreover, even if the arbitrariness can be eliminated (it actually reduces to arbitrariness in the choice of the odd moduli on the boundary of the space of the ordinary moduli, where the ideas of physical factorization are valid) integration over the moduli space in an arbitrary coordinate system need not necessarily lead to simple expressions (see [3] in connection with a general analysis of this kind). Fortunately, there is as yet no reason not to believe that the formalism corresponding to a superstring can be simple if one uses a special coordinate system on the supermoduli space of the fermionic string, i.e., in this coordinate system the moduli space for the superstring and for the fermionic string have the simplest connection. Moreover, it was shown in [4] that the partition function at the two-loop level is determined pointwise by the zero measure on the moduli space if one of the odd moduli is placed at the branch point. There has since been a successful generalization of this coordinate system [5]: The odd moduli must be placed at the zeros of the holomorphic l-differential W, which must then be made to tend to ~ (similar arguments are also given in [6]). It is readily verified that for such a choice of the coordinates the expressions for the one, two, three-point functions in two loops also vanish pointwise on the moduli space. This prescription also gives promising results for higher genuses. However, the experience gained from the calculation of the two-loop partition functions gives a hint that the prescription could be too restrictive -for the partition functions it was sufficient to fix only one of the odd moduli at the branch point. We shall show below that for pointwise vanishing of the one, two, three-point functions the position of the second modulus must be positioned at the same point, in accordance with the general prescription of [5]. In addition, several of the expressions used in this paper may be helpful for the further development of the technique of calculations on Riemann surfaces of genus 2. We shall also give a more accurate formulation and proof of the important identities in the Appendix to [2], which synthesize Riemann's identities and Riemann's zero theorem.