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 \( \mathcal{N} \) = 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 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.
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Acknowledgments
We would like to thank Nathan Benjamin, Cyuan-Han Chang, Ksenia Fedosova, Axel Kleinschmidt, Kim Klinger-Logan, Eric Perlmutter, Oliver Schlotterer, and Don Zagier for useful discussions. In particular we would like to thank Nathan Benjamin and Cyuan-Han Chang for helping us correct one of our results and Axel Kleinschmidt for comments on the draft. We are grateful to the organisers of the Pollica Summer Workshop “New Connections between Physics and Number Theory” supported by the Regione Campania, Università degli Studi di Salerno, Università degli Studi di Napoli “Federico II”, the Physics Department “Ettore Pancini” and “E.R. Caianiello”, and Istituto Nazionale di Fisica Nucleare. DD would also like to thank the Albert Einstein Institute, Golm, for the hospitality during the final stages of this project.
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Dorigoni, D., Treilis, R. Two string theory flavours of generalised Eisenstein series. J. High Energ. Phys. 2023, 102 (2023). https://doi.org/10.1007/JHEP11(2023)102
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DOI: https://doi.org/10.1007/JHEP11(2023)102