Abstract
The integral of an arbitrary two-loop modular graph function over the fundamental domain for SL(2, ℤ) in the upper half plane is evaluated using recent results on the Poincaré series for these functions.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. D’Hoker, M.B. Green and P. Vanhove, On the modular structure of the genus-one Type II superstring low energy expansion, JHEP 08 (2015) 041 [arXiv:1502.06698] [INSPIRE].
E. D’Hoker, M.B. Green, Ö. Gürdoğan and P. Vanhove, Modular Graph Functions, Commun. Num. Theor. Phys. 11 (2017) 165 [arXiv:1512.06779] [INSPIRE].
M.B. Green and P. Vanhove, The Low-energy expansion of the one loop type-II superstring amplitude, Phys. Rev. D 61 (2000) 104011 [hep-th/9910056] [INSPIRE].
M.B. Green, J.G. Russo and P. Vanhove, Low energy expansion of the four-particle genus-one amplitude in type-II superstring theory, JHEP 02 (2008) 020 [arXiv:0801.0322] [INSPIRE].
E. D’Hoker, M.B. Green and P. Vanhove, Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus, arXiv:1509.00363 [INSPIRE].
A. Basu, Poisson equation for the Mercedes diagram in string theory at genus one, Class. Quant. Grav. 33 (2016) 055005 [arXiv:1511.07455] [INSPIRE].
E. D’Hoker and M.B. Green, Identities between Modular Graph Forms, J. Number Theor. 189 (2018) 25 [arXiv:1603.00839] [INSPIRE].
A. Basu, Proving relations between modular graph functions, Class. Quant. Grav. 33 (2016) 235011 [arXiv:1606.07084] [INSPIRE].
E. D’Hoker and J. Kaidi, Hierarchy of Modular Graph Identities, JHEP 11 (2016) 051 [arXiv:1608.04393] [INSPIRE].
A. Kleinschmidt and V. Verschinin, Tetrahedral modular graph functions, JHEP 09 (2017) 155 [arXiv:1706.01889] [INSPIRE].
O. Ahlén and A. Kleinschmidt, D 6 R 4 curvature corrections, modular graph functions and Poincaré series, JHEP 05 (2018) 194 [arXiv:1803.10250] [INSPIRE].
D. Dorigoni and A. Kleinschmidt, Modular graph functions and asymptotic expansions of Poincaré series, arXiv:1903.09250 [INSPIRE].
E. D’Hoker and J. Kaidi, Modular graph functions and odd cuspidal functions. Fourier and Poincaré series, JHEP 04 (2019) 136 [arXiv:1902.04180] [INSPIRE].
F. Zerbini, Single-valued multiple zeta values in genus 1 superstring amplitudes, Commun. Num. Theor. Phys. 10 (2016) 703 [arXiv:1512.05689] [INSPIRE].
E. D’Hoker and W. Duke, Fourier series of modular graph functions, arXiv:1708.07998 [INSPIRE].
J. Blumlein, D.J. Broadhurst and J.A.M. Vermaseren, The Multiple Zeta Value Data Mine, Comput. Phys. Commun. 181 (2010) 582 [arXiv:0907.2557] [INSPIRE].
O. Schlotterer and S. Stieberger, Motivic Multiple Zeta Values and Superstring Amplitudes, J. Phys. A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].
J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, Multiple Zeta Values and Superstring Amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [INSPIRE].
J. Broedel, C.R. Mafra, N. Matthes and O. Schlotterer, Elliptic multiple zeta values and one-loop superstring amplitudes, JHEP 07 (2015) 112 [arXiv:1412.5535] [INSPIRE].
J. Broedel, O. Schlotterer and F. Zerbini, From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop, JHEP 01 (2019) 155 [arXiv:1803.00527] [INSPIRE].
F. Brown, A class of non-holomorphic modular forms I, arXiv:1707.01230 [INSPIRE].
F. Brown, A class of non-holomorphic modular forms II, equivariant iterated Eisenstein integrals, arXiv:1708.03354.
R. Rankin, Contributions to the theory of Ramanujan’s function τ(n) and similar arithmetic functions. I, Math. Proc. Camb. Philos. Soc. 35 (1939) 351.
A. Selberg, Bemerkungen über eine Dirichetsche Reihe, die mit der Theorie der Modulformen nahe verbunden ist, Arch. Math. Naturvid. 43 (1940) 47.
D. Zagier, The Rankin-Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Tokyo 28 (1982) 415.
E. D’Hoker and D.H. Phong, The Box graph in superstring theory, Nucl. Phys. B 440 (1995) 24 [hep-th/9410152] [INSPIRE].
E. D’Hoker and M.B. Green, Exploring transcendentality in superstring amplitudes, arXiv:1906.01652 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1905.06217
Research supported in part by the National Science Foundation under research grant PHY-16-19926.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
D’Hoker, E. Integral of two-loop modular graph functions. J. High Energ. Phys. 2019, 92 (2019). https://doi.org/10.1007/JHEP06(2019)092
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2019)092