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

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.

💡 Research Summary

This paper addresses the long‑standing problem of efficiently evaluating cyclic products of fermion correlation functions that appear in two‑loop (genus‑2) superstring amplitudes with massless external bosons. Building on the authors’ previous works (arXiv:2211.09069 and arXiv:2209.14633), the authors adopt a geometric framework in which one of the five branch points of the hyper‑elliptic genus‑2 curve is sent to infinity. This choice mirrors the familiar “one branch point at infinity” trick used in genus‑1 calculations and greatly simplifies the Abel‑Jacobi map and the associated period matrix.

The central observation is that, for any number N of fermion correlators multiplied together under a cyclic constraint, the dependence on spin structures collapses to the values of the genus‑2 Weierstrass ℘‑function (the Pe‑function) evaluated at the half‑periods corresponding to non‑singular even spin structures. Consequently, the spin‑structure‑dependent part of the product can be expressed solely in terms of these ℘‑values, while the spin‑independent part can be expanded in a finite basis of ℘‑functions and their derivatives. The basis functions are determined by the well‑known differential equations satisfied by the genus‑2 ℘‑function; by substituting the half‑periods for the generic variables, the trilinear relations previously derived in arXiv:2211.09069 are reproduced in a straightforward manner.

A key technical tool introduced in the paper is a modified version of the Jacobi inversion theorem. In the classical setting, Jacobi inversion provides the inverse of the Abel map for hyper‑elliptic curves. Here the authors adapt the theorem to relate ℘‑functions directly to the unique genus‑2 theta function (θ‑function) with even, non‑singular characteristics. This modification allows the spin‑independent part of the cyclic product to be written as a rational combination of theta functions, while the spin‑dependent part remains a polynomial in ℘‑values at half‑periods. The authors demonstrate the method explicitly for N = 3 and N = 4, presenting detailed formulae that separate the two contributions and verify that the resulting expressions satisfy the known differential identities.

Despite these advances, the paper acknowledges a significant limitation: a general algorithm that produces a fully theta‑function‑based expression for arbitrary N is still missing. The difficulty stems from the increasingly intricate algebraic relations between higher‑order ℘‑derivatives and theta constants as N grows. The authors suggest that future work should focus on a deeper algebraic understanding of these relations or on developing computer‑algebra implementations that can automate the expansion process.

In summary, the manuscript provides a concrete procedure for decomposing cyclic products of fermion correlators in genus‑2 superstring theory into a combination of ℘‑functions (capturing spin‑structure dependence) and a unique theta function (capturing the spin‑independent part). By fixing a branch point at infinity and exploiting the half‑period values of the ℘‑function, the authors achieve a substantial simplification of the spin‑sum, extending the elegance of genus‑1 techniques to the more complex genus‑2 setting. The work lays the groundwork for systematic higher‑loop calculations in superstring perturbation theory and points toward future developments that could render the method fully general for any number of fermionic insertions.