On the modular structure of the genus-one Type II superstring low energy expansion

TL;DR

This work uncovers a coherent modular structure in the genus-one Type II superstring low energy expansion. By introducing and analyzing the three-chain modular functions C_{a,b,c} and related D-functions, the authors derive exact Laplace-eigenvalue equations with inhomogeneous Eisenstein-series sources, and show that weight up to 5 contributions to the four-graviton amplitude reduce to linear combinations of these C-functions, Eisenstein series, and odd zeta-values after tau-integration. A generating function W is developed to organize the C-functions and prove the spectrum structure of the modular Laplacian, while explicit weight-4 and weight-5 relations (and supporting cusp asymptotics) constrain the higher-derivative interactions D^8 R^4 and D^{10} R^4. The results provide an analytical framework for evaluating tau-integrals and reveal deep connections between modular forms, Poincaré series, and string-theory amplitudes, with implications for open-closed dualities and higher-genus generalizations.

Abstract

The analytic contribution to the low energy expansion of Type II string amplitudes at genus-one is a power series in space-time derivatives with coefficients that are determined by integrals of modular functions over the complex structure modulus of the world-sheet torus. These modular functions are associated with world-sheet vacuum Feynman diagrams and given by multiple sums over the discrete momenta on the torus. In this paper we exhibit exact differential and algebraic relations for a certain infinite class of such modular functions by showing that they satisfy Laplace eigenvalue equations with inhomogeneous terms that are polynomial in non-holomorphic Eisenstein series. Furthermore, we argue that the set of modular functions that contribute to the coefficients of interactions up to order D**10 R*4 are linear sums of functions in this class and quadratic polynomials in Eisenstein series and odd Riemann zeta values. Integration over the complex structure results in coefficients of the low energy expansion that are rational numbers multiplying monomials in odd Riemann zeta values.