Two string theory flavours of generalised Eisenstein series

TL;DR

This paper unifies two classes of generalised Eisenstein series arising in string theory using a Poincaré series approach, revealing algebraic relations, non-perturbative expansions, and connections to the Riemann zeta function zeros.

Contribution

It introduces a unified framework for different string theory related Eisenstein series and explores their algebraic, differential, and asymptotic properties, including links to the Riemann zeta zeros.

Findings

Unified classes of Eisenstein series via Poincaré series

Derived non-perturbative expansions at the cusp

Discovered connections between asymptotics and Riemann zeta zeros

Abstract

Generalised Eisenstein series are non-holomorphic modular invariant functions of a complex variable, $\tau$, subject to a particular inhomogeneous Laplace eigenvalue equation on the hyperbolic upper-half $\tau$-plane. Two infinite classes of such functions arise quite naturally within different string theory contexts. A first class can be found by studying the coefficients of the effective action for the low-energy expansion of type IIB superstring theory, and relatedly in the analysis of certain integrated four-point functions of stress tensor multiplet operators in $\mathcal{N} = 4$ supersymmetric Yang-Mills theory. A second class of such objects is known to contain all two-loop modular graph functions, which are fundamental building blocks in the low-energy expansion of closed-string scattering amplitudes at genus one. In this work, we present a Poincar\'e series approach that unifies both classes of generalised Eisenstein series and manifests certain algebraic and differential relations amongst them. We then combine this technique with spectral methods for automorphic forms to find general and non-perturbative expansions at the cusp $\tau \to i \infty$. Finally, we find intriguing connections between the asymptotic expansion of these modular functions as $\tau \to 0$ and the non-trivial zeros of the Riemann zeta function.