Modular Features of Superstring Scattering Amplitudes: Generalised Eisenstein Series and Theta Lifts

TL;DR

This paper explores the mathematical structure of modular functions arising in superstring theory and supersymmetric Yang-Mills theory, revealing a unifying framework involving theta lifts and Eisenstein series that simplifies their complex relationships.

Contribution

It introduces a unifying description of modular functions in superstring amplitudes using theta lifts of Maass functions, highlighting the cancellation of L-values of cusp forms.

Findings

Modular functions can be expressed as theta lifts of local Maass functions.

L-values of holomorphic cusp forms cancel in certain Eisenstein series.

Unified framework links superstring amplitudes and gauge theory correlators.

Abstract

In previous papers it has been shown that the coefficients of terms in the large-$N$ expansion of a certain integrated four-point correlator of superconformal primary operators in $\mathcal{N}=4$ supersymmetric Yang-Mills theory are rational sums of real-analytic Eisenstein series and "generalised Eisenstein series''. The latter are novel modular functions first encountered in the context of graviton amplitudes in type IIB superstring theory. Similar modular functions, known as two-loop modular graph functions, are also encountered in the low-energy expansion of the integrand of genus-one closed superstring amplitudes. In this paper we further develop the mathematical structure of such generalised Eisenstein series emphasising, in particular, the occurrence of $L$-values of holomorphic cusp forms in their Fourier mode decomposition. We show that both the coefficients in the large-$N$ expansion of the integrated correlator and two-loop modular graph functions admit a unifying description in terms of four-dimensional lattice sums generated by theta lifts of local Maass functions, which generalise the structure of real-analytic Eisenstein series. Through the theta lift representation, we demonstrate that elements belonging to these two families of non-holomorphic modular functions can be expressed as rational linear combinations of generalised Eisenstein series for which all the $L$-values of holomorphic cusp forms precisely cancel.