On the pole structures of the disconnected part of hyper elliptic g loop M point super string amplitudes

Structures of the disconnected part of higher genus superstring amplitudes restricted to the hyper elliptic cases are investigated in the NSR formalism, based on the DHoker Phong and recent results. A set of equations, which we can regard as a basic tool to sum over the spin structures of any of g loop, M point amplitudes systematically, is shown by using a classical result of Abelian functions. We discuss structures of g loop, M point massless external boson superstring amplitudes by assuming that the spin structure dependence of any of the disconnected amplitudes is only on one kind of constants, the genus g Weierstrass Pe function valued at the summation of g number of half periods chosen out of 2g+1 half periods. This is a natural generalization of the case of genus 1. This assumption will be validated by a conjectured theorem which states that the spin structure dependent part of any string amplitude will be naturally decomposed into two parts. One is composed of manifestly modular invariant functions of positions of inserting operators, and the other is the polynomial of Pe function constants related to the moduli of Riemann surfaces only. It is shown that this is actually the case for any M for g=1, and M=1,2,3 for any g. Due to a technical problem, our consideration is at present restricted to the case that g(g+1)divided by 2 is odd. Example calculations are shown for the genus 2 by the method described here. In particular, our method correctly reproduces biholomorphic 1 form of DHoker Phong result as for the four point amplitudes of the disconnected parts.