Journal of High Energy Physics

, Volume 2015, Issue 12, pp 1–22 | Cite as

Infrared divergences and harmonic anomalies in the two-loop superstring effective action

Open Access
Regular Article - Theoretical Physics


We analyze the pertubative contributions to the \( {D}^4{\mathrm{\mathcal{R}}}^4 \) and \( {D}^6{\mathrm{\mathcal{R}}}^4 \) couplings in the low-energy effective action of type II string theory compactified on a torus T d , with particular emphasis on two-loop corrections. In general, it is necessary to introduce an infrared cut-off Λ to separate local interactions from non-local effects due to the exchange of massless states. We identify the degenerations of the genus-two Riemann surface which are responsible for power-like dependence on Λ, and give an explicit prescription for extracting the Λ-independent effective couplings. These renormalized couplings are then shown to be eigenmodes of the Laplace operator with respect to the torus moduli, up to computable anomalous source terms arising in the presence of logarithmic divergences, in precise agreement with predictions from U-duality. Our results for the two-loop \( {D}^6{\mathrm{\mathcal{R}}}^4 \) contribution also probe essential properties of the Kawazumi-Zhang invariant.


Superstrings and Heterotic Strings Supersymmetry and Duality String Duality 


Open Access

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  1. [1]
    M.B. Green and M. Gutperle, Effects of D instantons, Nucl. Phys. B 498 (1997) 195 [hep-th/9701093] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  2. [2]
    M.B. Green and P. Vanhove, D instantons, strings and M-theory, Phys. Lett. B 408 (1997) 122 [hep-th/9704145] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  3. [3]
    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].CrossRefMATHADSGoogle Scholar
  4. [4]
    B. Pioline and E. Kiritsis, U duality and D-brane combinatorics, Phys. Lett. B 418 (1998) 61 [hep-th/9710078] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  5. [5]
    B. Pioline, A note on nonperturbative R 4 couplings, Phys. Lett. B 431 (1998) 73 [hep-th/9804023] [INSPIRE].CrossRefADSGoogle 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].MathSciNetADSGoogle Scholar
  7. [7]
    N.A. Obers and B. Pioline, Eisenstein series and string thresholds, Commun. Math. Phys. 209 (2000) 275 [hep-th/9903113] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  8. [8]
    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].MathSciNetADSGoogle Scholar
  9. [9]
    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
  10. [10]
    A. Basu and S. Sethi, Recursion relations from space-time supersymmetry, JHEP 09 (2008) 081 [arXiv:0808.1250] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  11. [11]
    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].MathSciNetADSGoogle Scholar
  12. [12]
    B. Pioline, R 4 couplings and automorphic unipotent representations, JHEP 03 (2010) 116 [arXiv:1001.3647] [INSPIRE].CrossRefMATHADSGoogle Scholar
  13. [13]
    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].MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    M.B. Green, S.D. Miller and P. Vanhove, Small representations, string instantons and Fourier modes of Eisenstein series (with an appendix by D. Ciubotaru and P. Trapa), arXiv:1111.2983 [INSPIRE].
  15. [15]
    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].CrossRefMATHADSGoogle Scholar
  16. [16]
    G. Bossard and V. Verschinin, Minimal unitary representations from supersymmetry, JHEP 10 (2014) 008 [arXiv:1406.5527] [INSPIRE].CrossRefADSGoogle Scholar
  17. [17]
    G. Bossard and V. Verschinin, \( \mathbf{\mathcal{E}} \)4 R 4 type invariants and their gradient expansion, JHEP 03 (2015) 089 [arXiv:1411.3373] [INSPIRE].CrossRefGoogle Scholar
  18. [18]
    M.B. Green, J.G. Russo and P. Vanhove, String theory dualities and supergravity divergences, JHEP 06 (2010) 075 [arXiv:1002.3805] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  19. [19]
    Y. Wang and X. Yin, Constraining higher derivative supergravity with scattering amplitudes, Phys. Rev. D 92 (2015) 041701 [arXiv:1502.03810] [INSPIRE].MathSciNetADSGoogle Scholar
  20. [20]
    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].MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    B. Pioline, Rankin-Selberg methods for closed string amplitudes, Proc. Symp. Pure Math. 88 (2014) 119 [arXiv:1401.4265] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    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
  23. [23]
    B. Pioline, D 6 R 4 amplitudes in various dimensions, JHEP 04 (2015) 057 [arXiv:1502.03377] [INSPIRE].CrossRefADSGoogle Scholar
  24. [24]
    G. Bossard and A. Kleinschmidt, Supergravity divergences, supersymmetry and automorphic forms, JHEP 08 (2015) 102 [arXiv:1506.00657] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  25. [25]
    M.B. Green and P. Vanhove, Duality and higher derivative terms in M-theory, JHEP 01 (2006) 093 [hep-th/0510027] [INSPIRE].MathSciNetCrossRefADSGoogle Scholar
  26. [26]
    G. Bossard and V. Verschinin, The two6 R 4 type invariants and their higher order generalisation, JHEP 07 (2015) 154 [arXiv:1503.04230] [INSPIRE].CrossRefADSGoogle Scholar
  27. [27]
    E. D’Hoker and M.B. Green, Zhang-Kawazumi invariants and superstring amplitudes, arXiv:1308.4597 [INSPIRE].
  28. [28]
    N. Kawazumi, Johnsons homomorphisms and the Arakelov-Green function, arXiv:0801.4218.
  29. [29]
    S.W. Zhang, Gross-Schoen cycles and dualising sheaves, Invent. Math. 179 (2010) 1.MathSciNetCrossRefMATHADSGoogle Scholar
  30. [30]
    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].MathSciNetADSGoogle Scholar
  31. [31]
    M.B. Green, S.D. Miller and P. Vanhove, \( SL\left(2,\mathbb{Z}\right) \) -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
  32. [32]
    A. Basu, The D 6 R 4 term from three loop maximal supergravity, Class. Quant. Grav. 31 (2014) 245002 [arXiv:1407.0535] [INSPIRE].CrossRefMATHADSGoogle Scholar
  33. [33]
    B. Pioline, A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces, arXiv:1504.04182 [INSPIRE].
  34. [34]
    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].MathSciNetCrossRefADSGoogle Scholar
  35. [35]
    P. Tourkine, Tropical amplitudes, arXiv:1309.3551 [INSPIRE].
  36. [36]
    D. Zagier, The Rankin-Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981) 415.MathSciNetMATHGoogle Scholar
  37. [37]
    M.B. Green, H.-h. Kwon and P. Vanhove, Two loops in eleven-dimensions, Phys. Rev. D 61 (2000) 104010 [hep-th/9910055] [INSPIRE].MathSciNetADSGoogle Scholar
  38. [38]
    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].MathSciNetCrossRefADSGoogle Scholar
  39. [39]
    R. Wentworth, The asymptotics of the Arakelov-Greens function and Faltingsdelta invariant, Comm. Math. Phys. 137 (1991) 427.MathSciNetCrossRefMATHADSGoogle Scholar
  40. [40]
    R. De Jong, Asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann surfaces, Asian J. Math. 18 (2014) 507.MathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    H. Gomez and C.R. Mafra, The closed-string 3-loop amplitude and S-duality, JHEP 10 (2013) 217 [arXiv:1308.6567] [INSPIRE].MathSciNetCrossRefMATHADSGoogle Scholar
  42. [42]
    Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, Supersymmetry constraints and string theory on K3, arXiv:1508.07305 [INSPIRE].
  43. [43]
    A. Basu, Perturbative type-II amplitudes for BPS interactions, arXiv:1510.01667 [INSPIRE].

Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.CERN PH-TH, Case C01600, CERNGeneva 23Switzerland
  2. 2.Sorbonne Universités, UPMC Université Paris 6, UMR 7589ParisFrance
  3. 3.Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589Université Pierre et Marie CurieParis cedex 05France
  4. 4.Centre for Research in String Theory, School of Physics and AstronomyQueen Mary University of LondonLondonU.K.

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