Journal of High Energy Physics

, 2019:46 | Cite as

Eigenvalue equation for genus two modular graphs

  • Anirban BasuEmail author
Open Access
Regular Article - Theoretical Physics


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.


Superstrings and Heterotic Strings Extended Supersymmetry Supersymmetric Effective Theories 


Open Access

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  1. [1]
    J.R. Ellis, P. Jetzer and L. Mizrachi, One loop string corrections to the effective field theory, Nucl. Phys. B 303 (1988) 1 [INSPIRE].ADSMathSciNetGoogle Scholar
  2. [2]
    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].ADSCrossRefGoogle Scholar
  3. [3]
    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].ADSMathSciNetGoogle Scholar
  4. [4]
    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].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    B. Pioline, Rankin-Selberg methods for closed string amplitudes, Proc. Symp. Pure Math. 88 (2014) 119 [arXiv:1401.4265] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    A. Basu, Poisson equation for the Mercedes diagram in string theory at genus one, Class. Quant. Grav. 33 (2016) 055005 [arXiv:1511.07455] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    A. Basu, Non-BPS interactions from the type-II one loop four graviton amplitude, Class. Quant. Grav. 33 (2016) 125028 [arXiv:1601.04260] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    E. D’Hoker and M.B. Green, Identities between modular graph forms, J. Number Theor. 189 (2018) 25 [arXiv:1603.00839] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    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].ADSCrossRefzbMATHGoogle Scholar
  13. [13]
    A. Basu, Proving relations between modular graph functions, Class. Quant. Grav. 33 (2016) 235011 [arXiv:1606.07084] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    E. D’Hoker and J. Kaidi, Hierarchy of modular graph identities, JHEP 11 (2016) 051 [arXiv:1608.04393] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    A. Kleinschmidt and V. Verschinin, Tetrahedral modular graph functions, JHEP 09 (2017) 155 [arXiv:1706.01889] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    F. Brown, A class of non-holomorphic modular forms I, arXiv:1707.01230 [INSPIRE].
  18. [18]
    A. Basu, Low momentum expansion of one loop amplitudes in heterotic string theory, JHEP 11 (2017) 139 [arXiv:1708.08409] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    A. Basu, A simplifying feature of the heterotic one loop four graviton amplitude, Phys. Lett. B 776 (2018) 182 [arXiv:1710.01993] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    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].CrossRefGoogle Scholar
  21. [21]
    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].
  22. [22]
    J.E. Gerken and J. Kaidi, Holomorphic subgraph reduction of higher-valence modular graph forms, arXiv:1809.05122 [INSPIRE].
  23. [23]
    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].ADSCrossRefGoogle Scholar
  24. [24]
    M.B. Green, H.-H. Kwon and P. Vanhove, Two loops in eleven-dimensions, Phys. Rev. D 61 (2000) 104010 [hep-th/9910055] [INSPIRE].ADSMathSciNetGoogle Scholar
  25. [25]
    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].ADSCrossRefzbMATHGoogle Scholar
  26. [26]
    N. Berkovits, Super-Poincaré covariant two-loop superstring amplitudes, JHEP 01 (2006) 005 [hep-th/0503197] [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    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].ADSMathSciNetCrossRefGoogle Scholar
  28. [28]
    R. Wentworth, The asymptotics of the Arakelov-Green’s function and Faltings’ delta invariant, Commun. Math. Phys. 137 (1991) 427.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    R. De Jong, Asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces, Asian J. Math. 18 (2014) 507 [arXiv:1207.2353].MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    E. D’Hoker and M.B. Green, Zhang-Kawazumi invariants and superstring amplitudes, arXiv:1308.4597 [INSPIRE].
  32. [32]
    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].CrossRefGoogle Scholar
  33. [33]
    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].MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    A. Basu, Perturbative type-II amplitudes for BPS interactions, Class. Quant. Grav. 33 (2016) 045002 [arXiv:1510.01667] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  35. [35]
    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].ADSMathSciNetzbMATHGoogle Scholar
  36. [36]
    E. D’Hoker, M.B. Green and B. Pioline, Higher genus modular graph functions, string invariants and their exact asymptotics, arXiv:1712.06135 [INSPIRE].
  37. [37]
    E. D’Hoker, M.B. Green and B. Pioline, Asymptotics of the D 8 R 4 genus-two string invariant, arXiv:1806.02691 [INSPIRE].
  38. [38]
    N. Kawazumi, Johnson’s homomorphisms and the Arakelov Green function, arXiv:0801.4218.
  39. [39]
    S.-W. Zhang, Gross-Schoen cycles and dualising sheaves, Invent. Math. 179 (2009) 1 [arXiv:0812.0371].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    A. Basu, Supergravity limit of genus two modular graph functions in the worldline formalism, Phys. Lett. B 782 (2018) 570 [arXiv:1803.08329] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    E. D’Hoker and D.H. Phong, The geometry of string perturbation theory, Rev. Mod. Phys. 60 (1988) 917 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    E.P. Verlinde and H.L. Verlinde, Chiral bosonization, determinants and the string partition function, Nucl. Phys. B 288 (1987) 357 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Harish-Chandra Research InstituteHBNIPrayagrajIndia

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