Relations between elliptic modular graphs

Abstract

We consider certain elliptic modular graph functions that arise in the asymptotic expansion around the non--separating node of genus two string invariants that appear in the integrand of the $D^8{R}^4$ interaction in the low momentum expansion of the four graviton amplitude in type II superstring theory. These elliptic modular graphs have links given by the Green function, as well its holomorphic and anti--holomorphic derivatives. Using appropriate auxiliary graphs at various intermediate stages of the analysis, we show that each graph can be expressed solely in terms of graphs with links given only by the Green function and not its derivatives. This results in a reduction in the number of basis elements in the space of elliptic modular graphs.

27 pages, LaTeX; 13 figures

References in corpus (8)

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• The One-Loop Five-Graviton Amplitude and the Effective Action
• All-order differential equations for one-loop closed-string integrals and modular graph forms
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• Tetrahedral modular graph functions
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Cited by in corpus (7)

• 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
• Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems
• Elliptic modular graph forms I: Identities and generating series
• To the cusp and back: Resurgent analysis for modular graph functions
• Poisson equation for genus two string invariants: a conjecture
• Integrating simple genus two string invariants over moduli space