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

TL;DR

This paper advances the understanding of fermion correlation functions in genus two surfaces by expressing cyclic products in terms of theta functions, building on previous work and addressing unresolved issues in the field.

Contribution

It introduces a framework fixing a branch point at infinity for genus two curves, generalizes genus one methods, and derives relations connecting fermion products to Pe-functions and theta functions.

Findings

Dependence of spin structures on Pe-function values at half-periods

Basis functions for decomposing fermion products

Relations between trilinear identities and genus two Pe-functions

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 simply setting the variables equal to the half-periods of the non-singular and even spin structures, as is the case for genus one. The focus of this manuscript is on the procedures for expressing the results of decomposed formulae in terms of the unique genus two theta function. At present we cannot provide a procedure for deriving the general form of the decomposed formula totally expressed in terms of the theta functions for an arbitrary number of the fermion correlation functions in the product, by the reason described in the text. We present some general results and demonstrate that concrete expressions of both the spin structure dependent and independent parts will be derived and simplified to analyze using the logic of the derivations of the classical solutions to Jacobi inversion problem and their modifications which will be given in this manuscript.