Advertisement

D6R4 curvature corrections, modular graph functions and Poincaré series

  • Olof Ahlén
  • Axel KleinschmidtEmail author
Open Access
Regular Article - Theoretical Physics

Abstract

In this note we study the U-duality invariant coefficient functions of higher curvature corrections to the four-graviton scattering amplitude in type IIB string theory compactified on a torus. The main focus is on the D6R4 term that is known to satisfy an inhomogeneous Laplace equation. We exhibit a novel method for solving this equation in terms of a Poincaré series ansatz and recover known results in D = 10 dimensions and find new results in D < 10 dimensions. We also apply the method to modular graph functions as they arise from closed superstring one-loop amplitudes.

Keywords

M-Theory String Duality Superstrings and Heterotic Strings 

Notes

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.

References

  1. [1]
    D.J. Gross and E. Witten, Superstring Modifications of Einsteins Equations, Nucl. Phys. B 277 (1986) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    M.B. Green and M. Gutperle, Effects of D instantons, Nucl. Phys. B 498 (1997) 195 [hep-th/9701093] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    M.B. Green, M. Gutperle and P. Vanhove, One loop in eleven-dimensions, Phys. Lett. B 409 (1997) 177 [hep-th/9706175] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  4. [4]
    E. Kiritsis and B. Pioline, On R 4 threshold corrections in IIB string theory and (p, q) string instantons, Nucl. Phys. B 508 (1997) 509 [hep-th/9707018] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    B. Pioline, A Note on nonperturbative R 4 couplings, Phys. Lett. B 431 (1998) 73 [hep-th/9804023] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    M.B. Green and S. Sethi, Supersymmetry constraints on type IIB supergravity, Phys. Rev. D 59 (1999) 046006 [hep-th/9808061] [INSPIRE].ADSMathSciNetGoogle Scholar
  7. [7]
    N.A. Obers and B. Pioline, Eisenstein series and string thresholds, Commun. Math. Phys. 209 (2000) 275 [hep-th/9903113] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    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
  9. [9]
    B. Pioline, H. Nicolai, J. Plefka and A. Waldron, R 4 couplings, the fundamental membrane and exceptional theta correspondences, JHEP 03 (2001) 036 [hep-th/0102123] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    M.B. Green and P. Vanhove, Duality and higher derivative terms in M-theory, JHEP 01 (2006) 093 [hep-th/0510027] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    A. Basu, The D 4 R 4 term in type IIB string theory on T 2 and U-duality, Phys. Rev. D 77 (2008) 106003 [arXiv:0708.2950] [INSPIRE].ADSGoogle Scholar
  12. [12]
    A. Basu, The D 6 R 4 term in type IIB string theory on T 2 and U-duality, Phys. Rev. D 77 (2008) 106004 [arXiv:0712.1252] [INSPIRE].ADSGoogle Scholar
  13. [13]
    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
  14. [14]
    M.B. Green, J.G. Russo and P. Vanhove, Automorphic properties of low energy string amplitudes in various dimensions, Phys. Rev. D 81 (2010) 086008 [arXiv:1001.2535] [INSPIRE].ADSMathSciNetGoogle Scholar
  15. [15]
    B. Pioline, R 4 couplings and automorphic unipotent representations, JHEP 03 (2010) 116 [arXiv:1001.3647] [INSPIRE].ADSCrossRefGoogle Scholar
  16. [16]
    F. Gubay, N. Lambert and P. West, Constraints on Automorphic Forms of Higher Derivative Terms from Compactification, JHEP 08 (2010) 028 [arXiv:1002.1068] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    M.B. Green, S.D. Miller, J.G. Russo and P. Vanhove, Eisenstein series for higher-rank groups and string theory amplitudes, Commun. Num. Theor. Phys. 4 (2010) 551 [arXiv:1004.0163] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  18. [18]
    A. Basu, Supersymmetry constraints on the R 4 multiplet in type IIB on T 2, Class. Quant. Grav. 28 (2011) 225018 [arXiv:1107.3353] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    M.B. Green, S.D. Miller and P. Vanhove, Small representations, string instantons and Fourier modes of Eisenstein series, J. Number Theor. 146 (2015) 187 [arXiv:1111.2983] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  20. [20]
    F. Gubay and P. West, Parameters, limits and higher derivative type-II string corrections, JHEP 11 (2012) 027 [arXiv:1204.1403] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  21. [21]
    P. Fleig and A. Kleinschmidt, Eisenstein series for infinite-dimensional U-duality groups, JHEP 06 (2012) 054 [arXiv:1204.3043] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  22. [22]
    M.R. Garousi, S-duality invariant dilaton couplings at order α ′3, JHEP 10 (2013) 076 [arXiv:1306.6851] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  23. [23]
    G. Bossard and V. Verschinin, Minimal unitary representations from supersymmetry, JHEP 10 (2014) 008 [arXiv:1406.5527] [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    G. Bossard and V. Verschinin, ε4 R 4 type invariants and their gradient expansion, JHEP 03 (2015) 089 [arXiv:1411.3373] [INSPIRE].CrossRefGoogle Scholar
  25. [25]
    B. Pioline, D 6 R 4 amplitudes in various dimensions, JHEP 04 (2015) 057 [arXiv:1502.03377] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    Y. Wang and X. Yin, Constraining Higher Derivative Supergravity with Scattering Amplitudes, Phys. Rev. D 92 (2015) 041701 [arXiv:1502.03810] [INSPIRE].ADSMathSciNetGoogle Scholar
  27. [27]
    G. Bossard and V. Verschinin, The two6 R 4 type invariants and their higher order generalisation, JHEP 07 (2015) 154 [arXiv:1503.04230] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    Y. Wang and X. Yin, Supervertices and Non-renormalization Conditions in Maximal Supergravity Theories, arXiv:1505.05861 [INSPIRE].
  29. [29]
    G. Bossard and A. Kleinschmidt, Loops in exceptional field theory, JHEP 01 (2016) 164 [arXiv:1510.07859] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  30. [30]
    G. Bossard and A. Kleinschmidt, Cancellation of divergences up to three loops in exceptional field theory, JHEP 03 (2018) 100 [arXiv:1712.02793] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  31. [31]
    E. Cremmer and B. Julia, The SO(8) Supergravity, Nucl. Phys. B 159 (1979) 141 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  32. [32]
    C.M. Hull and P.K. Townsend, Unity of superstring dualities, Nucl. Phys. B 438 (1995) 109 [hep-th/9410167] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  33. [33]
    M.B. Green, J.G. Russo and P. Vanhove, String theory dualities and supergravity divergences, JHEP 06 (2010) 075 [arXiv:1002.3805] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  34. [34]
    G. Bossard and A. Kleinschmidt, Supergravity divergences, supersymmetry and automorphic forms, JHEP 08 (2015) 102 [arXiv:1506.00657] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  35. [35]
    P. Fleig, H.P.A. Gustafsson, A. Kleinschmidt and D. Persson, Eisenstein series and automorphic representationswith applications to string theory, Cambridge University Press, to appear (2018) [arXiv:1511.04265] [INSPIRE].
  36. [36]
    M.B. Green, J.G. Russo and P. Vanhove, Modular properties of two-loop maximal supergravity and connections with string theory, JHEP 07 (2008) 126 [arXiv:0807.0389] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  37. [37]
    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].ADSMathSciNetCrossRefGoogle Scholar
  38. [38]
    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].MathSciNetCrossRefGoogle Scholar
  39. [39]
    A. Basu, Proving relations between modular graph functions, Class. Quant. Grav. 33 (2016) 235011 [arXiv:1606.07084] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  40. [40]
    A. Kleinschmidt and V. Verschinin, Tetrahedral modular graph functions, JHEP 09 (2017) 155 [arXiv:1706.01889] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    M.B. Green, S.D. Miller and P. Vanhove, SL(2, ℤ)-invariance and D-instanton contributions to the D 6 R 4 interaction, Commun. Num. Theor. Phys. 09 (2015) 307 [arXiv:1404.2192] [INSPIRE].CrossRefGoogle Scholar
  42. [42]
    A. Basu, The D 6 R 4 term from three loop maximal supergravity, Class. Quant. Grav. 31 (2014) 245002 [arXiv:1407.0535] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    E. D’Hoker and M.B. Green, Zhang-Kawazumi Invariants and Superstring Amplitudes, arXiv:1308.4597 [INSPIRE].
  44. [44]
    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].ADSCrossRefGoogle Scholar
  45. [45]
    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].MathSciNetCrossRefGoogle Scholar
  46. [46]
    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
  47. [47]
    K. Klinger-Logan, Differential equations in automorphic forms, arXiv:1801.00838.
  48. [48]
    R.P. Langlands, On the Functional Equations Satisfied by Eisenstein Series, Lecture Notes in Mathematics, vol. 544, Springer-Verlag, New York, Berlin-Heidelberg (1976).CrossRefGoogle Scholar
  49. [49]
    D. Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1997).Google Scholar
  50. [50]
    B. Pioline and D. Persson, The Automorphic NS5-brane, Commun. Num. Theor. Phys. 3 (2009) 697 [arXiv:0902.3274] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  51. [51]
    H. Gomez and C.R. Mafra, The closed-string 3-loop amplitude and S-duality, JHEP 10 (2013) 217 [arXiv:1308.6567] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  52. [52]
    C. Angelantonj, I. Florakis and B. Pioline, A new look at one-loop integrals in string theory, Commun. Num. Theor. Phys. 6 (2012) 159 [arXiv:1110.5318] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  53. [53]
    H. Iwaniec, Spectral methods of automorphic forms, Graduate Studies in Mathematics, vol. 53, 2nd edition, American Mathematical Society, Providence, (2002).zbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut)PotsdamGermany
  2. 2.International Solvay InstitutesBrusselsBelgium

Personalised recommendations