Abstract
Modular Graph Functions (MGFs) are SL(2,ℤ)-invariant functions that emerge in the study of the low-energy expansion of the one-loop closed string amplitude. To find the string scattering amplitude, we must integrate MGFs over the moduli space of the torus. In this paper, we use the iterated integral representation of MGFs to establish a depth-dependent basis for them, where “depth” refers to the number of iterations in the integral. This basis has a suitable Laplace equation. We integrate this basis from depth zero to depth three over the fundamental domain of SL(2,ℤ) with a cut-off.
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Acknowledgments
I am grateful to Axel Kleinschmidt for the extensive discussions and insights that greatly influenced this work and for many fruitful comments on the draft. I also thank Oliver Schlotterer, Daniele Dorigoni, Guillaume Bossard, Federico Zerbini and Emiel Claasen for valuable discussions. Special thanks to Oliver Schlotterer and Federico Zerbini for their comments on the first version of the paper and I thank the anonymous referee for valuable comments and suggestions. My appreciation goes to École Polytechnique for its hospitality during the early stages of this research. This work was supported by the IMPRS for Mathematical and Physical Aspects of Gravitation, Cosmology, and Quantum Field Theory.
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Doroudiani, M. Integral of depth zero to three basis of Modular Graph Functions. J. High Energ. Phys. 2024, 29 (2024). https://doi.org/10.1007/JHEP07(2024)029
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DOI: https://doi.org/10.1007/JHEP07(2024)029