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

TL;DR

This paper investigates the modular functions arising in the genus-one low energy expansion of Type II superstring amplitudes, deriving exact relations and showing their structure involves Laplace eigenvalue equations and Eisenstein series.

Contribution

It provides exact differential and algebraic relations for a class of modular functions and characterizes their structure in the low energy expansion coefficients.

Findings

Modular functions satisfy Laplace eigenvalue equations with polynomial in Eisenstein series inhomogeneous terms.

Coefficients up to order D**10 R*4 are linear combinations of these functions and quadratic Eisenstein series.

Results show coefficients are rational multiples of monomials in odd Riemann zeta values.

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.