Abstract
We obtain a second order differential equation on moduli space satisfied by certain modular graph functions at genus two, each of which has two links. This eigenvalue equation is obtained by analyzing the variations of these graphs under the variation of the Beltrami differentials. This equation involves seven distinct graphs, three of which appear in the integrand of the D8\( \mathrm{\mathcal{R}} \)4 term in the low momentum expansion of the four graviton amplitude at genus two in type II string theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.R. Ellis, P. Jetzer and L. Mizrachi, One loop string corrections to the effective field theory, Nucl. Phys. B 303 (1988) 1 [INSPIRE].
M. Abe, H. Kubota and N. Sakai, Loop corrections to the E 8 × E 8 heterotic string effective Lagrangian, Nucl. Phys. B 306 (1988) 405 [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].
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].
B. Pioline, Rankin-Selberg methods for closed string amplitudes, Proc. Symp. Pure Math. 88 (2014) 119 [arXiv:1401.4265] [INSPIRE].
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].
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, M.B. Green, Ö. Gürdogan and P. Vanhove, Modular graph functions, Commun. Num. Theor. Phys. 11 (2017) 165 [arXiv:1512.06779] [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].
F. Brown, A class of non-holomorphic modular forms I, arXiv:1707.01230 [INSPIRE].
A. Basu, Low momentum expansion of one loop amplitudes in heterotic string theory, JHEP 11 (2017) 139 [arXiv:1708.08409] [INSPIRE].
A. Basu, A simplifying feature of the heterotic one loop four graviton amplitude, Phys. Lett. B 776 (2018) 182 [arXiv:1710.01993] [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. Zerbini, Modular and holomorphic graph function from superstring amplitudes, in KMPB conference: elliptic integrals, elliptic functions and modular forms in quantum field theory, Zeuthen, Germany, 23–26 October 2017 [arXiv:1807.04506] [INSPIRE].
J.E. Gerken and J. Kaidi, Holomorphic subgraph reduction of higher-valence modular graph forms, arXiv:1809.05122 [INSPIRE].
J.E. Gerken, A. Kleinschmidt and O. Schlotterer, Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings, JHEP 01 (2019) 052 [arXiv:1811.02548] [INSPIRE].
M.B. Green, H.-H. Kwon and P. Vanhove, Two loops in eleven-dimensions, Phys. Rev. D 61 (2000) 104010 [hep-th/9910055] [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].
R. Wentworth, The asymptotics of the Arakelov-Green’s function and Faltings’ delta invariant, Commun. Math. Phys. 137 (1991) 427.
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].
R. De Jong, Asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces, Asian J. Math. 18 (2014) 507 [arXiv:1207.2353].
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 D 6 R 4 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].
B. Pioline and R. Russo, Infrared divergences and harmonic anomalies in the two-loop superstring effective action, JHEP 12 (2015) 102 [arXiv:1510.02409] [INSPIRE].
E. D’Hoker, M.B. Green and B. Pioline, Higher genus modular graph functions, string invariants and their exact asymptotics, arXiv:1712.06135 [INSPIRE].
E. D’Hoker, M.B. Green and B. Pioline, Asymptotics of the D 8 R 4 genus-two string invariant, arXiv:1806.02691 [INSPIRE].
N. Kawazumi, Johnson’s homomorphisms and the Arakelov Green function, arXiv:0801.4218.
S.-W. Zhang, Gross-Schoen cycles and dualising sheaves, Invent. Math. 179 (2009) 1 [arXiv:0812.0371].
A. Basu, Supergravity limit of genus two modular graph functions in the worldline formalism, Phys. Lett. B 782 (2018) 570 [arXiv:1803.08329] [INSPIRE].
E. D’Hoker and D.H. Phong, The geometry of string perturbation theory, Rev. Mod. Phys. 60 (1988) 917 [INSPIRE].
E.P. Verlinde and H.L. Verlinde, Chiral bosonization, determinants and the string partition function, Nucl. Phys. B 288 (1987) 357 [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: 1812.00389
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
Basu, A. Eigenvalue equation for genus two modular graphs. J. High Energ. Phys. 2019, 46 (2019). https://doi.org/10.1007/JHEP02(2019)046
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2019)046