On a formula of spin sums, Eisenstein-Kronecker series in higher genus Riemann surfaces

Abstract: We discuss a decomposition formula of simple products of fermion correlation functions with cyclic constrains and its applications to spin sums of super string amplitudes. Based on some facts which are noted or derived in this paper, we propose a candidate of the form of this decomposition formula for some of higher genus cases which includes genus two case. Although we had to use several conjectures and assumptions due to unsolved mathematical difficulties, the method described in the text may be an efficient way to obtain the decomposition formula in higher genus cases. In particular, for those cases, we propose a concrete method to sum over non singular even spin structures for the product of arbitrary number of the fermion correlation functions with cyclic constraints in super string amplitudes. We also propose an explicit generalization of Eisenstein-Kronecker series to the higher genus cases in the process of considerations above.