On theta function expressions of cyclic products of fermion correlation functions in genus two

Abstract

In arXiv:2211.09069, significant progress was made in decomposing simple products of fermion correlation functions, and in summing over spin structures of superstring amplitudes in genus two under cyclic constraints. In this manuscript we consider part of the same subject using a framework in which one of the branch points of the genus two curve is fixed at infinity. This framework is a direct generalization of the popular one in the case of genus one. We address some of the issues that remained unresolved in our previous paper arXiv:2209.14633. We show that the spin structures of the simple products of fermion correlation functions with cyclic conditions depend only on the Pe-function values at the half-periods of the genus two surface, for any number of factors in the products. Similar to the genus one case, we can provide basis functions to decompose the product. Consequently, the trilinear relations found in arXiv:2211.09069 can be derived from the known set of differential equations of genus two Pe-functions by setting the variables equal to the half-periods of the non-singular and even spin structures, as is the case for genus one. Based on these considerations, we present a procedure for expressing the results of decomposed formulae in terms of the unique genus two theta function for two, three, and four point cases, and discuss a realistic approach for calculating six point function. A general formula for the expression of the results in terms of the theta function for the product of an arbitrary number of the fermion correlation functions is not yet derivable.