Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems

Abstract

We derive new Poincaré-series representations for infinite families of non-holomorphic modular invariant functions that include modular graph forms as they appear in the low-energy expansion of closed-string scattering amplitudes at genus one. The Poincaré series are constructed from iterated integrals over single holomorphic Eisenstein series and their complex conjugates, decorated by suitable combinations of zeta values. We evaluate the Poincaré sums over these iterated Eisenstein integrals of depth one and deduce new representations for all modular graph forms built from iterated Eisenstein integrals at depth two. In a companion paper, some of the Poincaré sums over depth-one integrals going beyond modular graph forms will be described in terms of iterated integrals over holomorphic cusp forms and their L-values.

90+25 pages. Part I of a series of two papers together with arXiv:2109.05018. Submission includes an ancillary data file. v2: expanded introduction. v3: JHEP version

References in corpus (12)

• Writing CFT correlation functions as AdS scattering amplitudes
• Matching the $D^6{\mathcal{R}}^4$ interaction at two-loops
• String scattering in flat space and a scaling limit of Yang-Mills correlators
• The One-Loop Five-Graviton Amplitude and the Effective Action
• $D^6{R}^4$ amplitudes in various dimensions
• All-order differential equations for one-loop closed-string integrals and modular graph forms
• Generating series of all modular graph forms from iterated Eisenstein integrals
• Tetrahedral modular graph functions
• Poincaré series for modular graph forms at depth two. II. Iterated integrals of cusp forms
• Low momentum expansion of one loop amplitudes in heterotic string theory
• Zero mode of the Fourier series of some modular graphs from Poincare series
• Poisson equation for genus two string invariants: a conjecture

Cited by in corpus (11)

• Harnessing S-Duality in $\mathcal{N}=4$ SYM & Supergravity as $SL(2,\mathbb{Z})$-Averaged Strings
• The SAGEX Review on Scattering Amplitudes, Chapter 10: Selected topics on modular covariance of type IIB string amplitudes and their $\mathcal{N}=4$ supersymmetric Yang-Mills duals
• Poincaré series for modular graph forms at depth two. II. Iterated integrals of cusp forms
• Modular graph forms from equivariant iterated Eisenstein integrals
• To the cusp and back: Resurgent analysis for modular graph functions
• Type II superstring amplitude at one-loop and transcendentality
• Integral of depth zero to three basis of Modular Graph Functions
• From Modular Graph Forms to Iterated Integrals
• The thermal representation of conformal ladder integrals
• Modular Features of Superstring Scattering Amplitudes: Generalised Eisenstein Series and Theta Lifts
• A construction of single-valued elliptic polylogarithms