Two string theory flavours of generalised Eisenstein series

Dorigoni, Daniele; Treilis, Rudolfs

Abstract

Generalised Eisenstein series are non-holomorphic modular invariant functions of a complex variable, τ, subject to a particular inhomogeneous Laplace eigenvalue equation on the hyperbolic upper-half τ-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 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 Poincaré 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 τ → i∞. Finally, we find intriguing connections between the asymptotic expansion of these modular functions as τ → 0 and the non-trivial zeros of the Riemann zeta function.

Citation

Dorigoni, D., & Treilis, R. (2023). Two string theory flavours of generalised Eisenstein series. Journal of High Energy Physics, 2023(11), Article 102. https://doi.org/10.1007/jhep11%282023%29102

• Journal Article Type: Article
• Acceptance Date: Oct 31, 2023
• Online Publication Date: Nov 17, 2023
• Publication Date: 2023
• Deposit Date: Jan 9, 2024
• Publicly Available Date: Jan 9, 2024
• Journal: Journal of High Energy Physics
• Print ISSN: 1126-6708
• Electronic ISSN: 1029-8479
• Publisher: Scuola Internazionale Superiore di Studi Avanzati (SISSA)
• Peer Reviewed: Peer Reviewed
• Volume: 2023
• Issue: 11
• Article Number: 102
• DOI: https://doi.org/10.1007/jhep11%282023%29102
• Public URL: https://durham-repository.worktribe.com/output/1949279