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 D8ℛ4 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.
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, O. Gürdogan 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].
D.M. Richards, The One-Loop Five-Graviton Amplitude and the Effective Action, JHEP 10 (2008) 042 [arXiv:0807.2421] [INSPIRE].
M.B. Green, C.R. Mafra and O. Schlotterer, Multiparticle one-loop amplitudes and S-duality in closed superstring theory, JHEP 10 (2013) 188 [arXiv:1307.3534] [INSPIRE].
E. D’Hoker and M.B. Green, Exploring transcendentality in superstring amplitudes, JHEP 07 (2019) 149 [arXiv:1906.01652] [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, J. Number Theor. 196 (2019) 381 [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].
A. Basu, Non-BPS interactions from the type-II one loop four graviton amplitude, Class. Quant. Grav. 33 (2016) 125028 [arXiv:1601.04260] [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, Poisson equation for the three loop ladder diagram in string theory at genus one, Int. J. Mod. Phys. A 31 (2016) 1650169 [arXiv:1606.02203] [INSPIRE].
A. Basu, Proving relations between modular graph functions, Class. Quant. Grav. 33 (2016) 235011 [arXiv:1606.07084] [INSPIRE].
A. Basu, Simplifying the one loop five graviton amplitude in type IIB string theory, Int. J. Mod. Phys. A 32 (2017) 1750074 [arXiv:1608.02056] [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].
A. Basu, Eigenvalue equation for the modular graph Ca,b,c,d, JHEP 07 (2019) 126 [arXiv:1906.02674] [INSPIRE].
J.E. Gerken, A. Kleinschmidt and O. Schlotterer, All-order differential equations for one-loop closed-string integrals and modular graph forms, JHEP 01 (2020) 064 [arXiv:1911.03476] [INSPIRE].
J.E. Gerken, A. Kleinschmidt and O. Schlotterer, Generating series of all modular graph forms from iterated Eisenstein integrals, JHEP 07 (2020) 190 [arXiv:2004.05156] [INSPIRE].
J.E. Gerken, Basis Decompositions and a Mathematica Package for Modular Graph Forms, arXiv:2007.05476 [INSPIRE].
E. D’Hoker, M. Gutperle and D.H. Phong, Two-loop superstrings and S-duality, Nucl. Phys. B 722 (2005) 81 [hep-th/0503180] [INSPIRE].
E. D’Hoker and M.B. Green, Zhang-Kawazumi Invariants and Superstring Amplitudes, arXiv:1308.4597 [INSPIRE].
E. D’Hoker, M.B. Green, B. Pioline and R. Russo, Matching the D6 R4 interaction at two-loops, JHEP 01 (2015) 031 [arXiv:1405.6226] [INSPIRE].
B. Pioline, A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces, J. Number Theor. 163 (2016) 520 [arXiv:1504.04182] [INSPIRE].
A. Basu, Perturbative type-II amplitudes for BPS interactions, Class. Quant. Grav. 33 (2016) 045002 [arXiv:1510.01667] [INSPIRE].
E. D’Hoker, M.B. Green and B. Pioline, Higher genus modular graph functions, string invariants, and their exact asymptotics, Commun. Math. Phys. 366 (2019) 927 [arXiv:1712.06135] [INSPIRE].
E. D’Hoker, M.B. Green and B. Pioline, Asymptotics of the D8ℛ4 genus-two string invariant, Commun. Num. Theor. Phys. 13 (2019) 351 [arXiv:1806.02691] [INSPIRE].
A. Basu, Eigenvalue equation for genus two modular graphs, JHEP 02 (2019) 046 [arXiv:1812.00389] [INSPIRE].
E. D’Hoker, C.R. Mafra, B. Pioline and O. Schlotterer, Two-loop superstring five-point amplitudes II: Low energy expansion and S-duality, arXiv:2008.08687 [INSPIRE].
E. D’Hoker and O. Schlotterer, Identities among higher genus modular graph tensors, arXiv:2010.00924 [INSPIRE].
E. D’Hoker and D.H. Phong, Two-loop superstrings VI: Non-renormalization theorems and the 4-point function, Nucl. Phys. B 715 (2005) 3 [hep-th/0501197] [INSPIRE].
N. Berkovits, Super-Poincaré covariant two-loop superstring amplitudes, JHEP 01 (2006) 005 [hep-th/0503197] [INSPIRE].
N. Berkovits and C.R. Mafra, Equivalence of two-loop superstring amplitudes in the pure spinor and RNS formalisms, Phys. Rev. Lett. 96 (2006) 011602 [hep-th/0509234] [INSPIRE].
E. D’Hoker, C.R. Mafra, B. Pioline and O. Schlotterer, Two-loop superstring five-point amplitudes. Part I. Construction via chiral splitting and pure spinors, JHEP 08 (2020) 135 [arXiv:2006.05270] [INSPIRE].
W. Lerche, B.E.W. Nilsson, A.N. Schellekens and N.P. Warner, Anomaly Cancelling Terms From the Elliptic Genus, Nucl. Phys. B 299 (1988) 91 [INSPIRE].
A. Basu, Poisson equations for elliptic modular graph functions, arXiv:2009.02221 [INSPIRE].
E.P. Verlinde and H.L. Verlinde, Chiral Bosonization, Determinants and the String Partition Function, Nucl. Phys. B 288 (1987) 357 [INSPIRE].
E. D’Hoker and D.H. Phong, The Geometry of String Perturbation Theory, Rev. Mod. Phys. 60 (1988) 917 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2010.08331
Rights and permissions
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.
About this article
Cite this article
Basu, A. Relations between elliptic modular graphs. J. High Energ. Phys. 2020, 195 (2020). https://doi.org/10.1007/JHEP12(2020)195
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2020)195