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Journal of High Energy Physics

, 2016:51 | Cite as

Hierarchy of modular graph identities

  • Eric D’HokerEmail author
  • Justin Kaidi
Open Access
Regular Article - Theoretical Physics

Abstract

The low energy expansion of Type II superstring amplitudes at genus one is organized in terms of modular graph functions associated with Feynman graphs of a conformal scalar field on the torus. In earlier work, surprising identities between two-loop graphs at all weights, and between higher-loop graphs of weights four and five were constructed. In the present paper, these results are generalized in two complementary directions. First, all identities at weight six and all dihedral identities at weight seven are obtained and proven. Whenever the Laurent polynomial at the cusp is available, the form of these identities confirms the pattern by which the vanishing of the Laurent polynomial governs the full modular identity. Second, the family of modular graph functions is extended to include all graphs with derivative couplings and worldsheet fermions. These extended families of modular graph functions are shown to obey a hierarchy of inhomogeneous Laplace eigenvalue equations. The eigenvalues are calculated analytically for the simplest infinite sub-families and obtained by Maple for successively more complicated sub-families. The spectrum is shown to consist solely of eigenvalues s(s − 1) for positive integers s bounded by the weight, with multiplicities which exhibit rich representation-theoretic patterns.

Keywords

Discrete Symmetries Integrable Hierarchies 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]
    M.B. Green and M. Gutperle, Effects of D instantons, Nucl. Phys. B 498 (1997) 195 [hep-th/9701093] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    M.B. Green, M. Gutperle and P. Vanhove, One loop in eleven-dimensions, Phys. Lett. B 409 (1997) 177 [hep-th/9706175] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    M.B. Green and S. Sethi, Supersymmetry constraints on type IIB supergravity, Phys. Rev. D 59 (1999) 046006 [hep-th/9808061] [INSPIRE].ADSMathSciNetGoogle Scholar
  4. [4]
    N.A. Obers and B. Pioline, Eisenstein series and string thresholds, Commun. Math. Phys. 209 (2000) 275 [hep-th/9903113] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    E. D’Hoker and D.H. Phong, Lectures on two loop superstrings, Conf. Proc. C 0208124 (2002) 85 [hep-th/0211111] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  6. [6]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    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
  8. [8]
    H. Gomez and C.R. Mafra, The closed-string 3-loop amplitude and S-duality, JHEP 10 (2013) 217 [arXiv:1308.6567] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    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
  10. [10]
    B. Pioline, \( {D}^6{\mathrm{\mathcal{R}}}^4 \) amplitudes in various dimensions, JHEP 04 (2015) 057 [arXiv:1502.03377] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  11. [11]
    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
  12. [12]
    E. D’Hoker and M.B. Green, Zhang-Kawazumi Invariants and Superstring Amplitudes, arXiv:1308.4597 [INSPIRE].
  13. [13]
    S.W. Zhang, Gross-Schoen Cycles and Dualising Sheaves, Invent. Math. 179 1 [arXiv:0812.0371].
  14. [14]
    N. Kawazumi, Johnson’s homomorphisms and the Arakelov Green function, arXiv:0801.4218.
  15. [15]
    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
  16. [16]
    E. D’Hoker, M.B. Green, O. Gurdogan and P. Vanhove, Modular Graph Functions, arXiv:1512.06779 [INSPIRE].
  17. [17]
    E. D’Hoker and M.B. Green, Identities between Modular Graph Forms, arXiv:1603.00839 [INSPIRE].
  18. [18]
    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].MathSciNetCrossRefGoogle Scholar
  19. [19]
    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, arXiv:1509.00363 [INSPIRE].
  20. [20]
    F. Zerbini, Single-valued multiple zeta values in genus 1 superstring amplitudes, arXiv:1512.05689 [INSPIRE].
  21. [21]
    E. D’Hoker, M.B. Green and P. Vanhove, unpublished notes (2016).Google Scholar
  22. [22]
    D. Zagier, Values of zeta functions and their application, First European Congress of Mathematics, Paris France (1992) [Progr. Math. 120 (1994) 497].Google Scholar
  23. [23]
    M.E. Hoffmann, Multiple harmonic series, Pacific J. Math. 152 (1992) 275.MathSciNetCrossRefGoogle Scholar
  24. [24]
    M. Waldschmidt, Valeurs zêta multiples: une introduction, J. Théor. Nombres Bordeaux 12 (2000) 581.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    J.M. Borwein, D.M. Bradley, D.J. Broadhurst and P. Lisonek, Special values of multiple polylogarithms, Trans. Am. Math. Soc. 353 (2001) 907 [math/9910045] [INSPIRE].
  26. [26]
    V.V. Zudilin, Algebraic relations for multiple zeta values, Uspekhi Mat. Nauk 58 (2003) 3 [Russ. Math. Surv. 58 (2003) 1].Google Scholar
  27. [27]
    J. Blumlein, D.J. Broadhurst and J.A.M. Vermaseren, The Multiple Zeta Value Data Mine, Comput. Phys. Commun. 181 (2010) 582 [arXiv:0907.2557] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    F. Brown, Single-valued Motivic Periods and Multiple Zeta Values, SIGMA 2 (2014) e25 [arXiv:1309.5309] [INSPIRE].
  29. [29]
    J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, Multiple Zeta Values and Superstring Amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    S. Stieberger, Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator, J. Phys. A 47 (2014) 155401 [arXiv:1310.3259] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  31. [31]
    J. Broedel, N. Matthes and O. Schlotterer, Relations between elliptic multiple zeta values and a special derivation algebra, J. Phys. A 49 (2016) 155203 [arXiv:1507.02254] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  32. [32]
    O. Schlotterer and S. Stieberger, Motivic Multiple Zeta Values and Superstring Amplitudes, J. Phys. A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  33. [33]
    E. D’Hoker and D.H. Phong, Momentum analyticity and finiteness of the one loop superstring amplitude, Phys. Rev. Lett. 70 (1993) 3692 [hep-th/9302003] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    E. D’Hoker and D.H. Phong, The Box graph in superstring theory, Nucl. Phys. B 440 (1995) 24 [hep-th/9410152] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    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
  36. [36]
    D. Zagier, Notes on Lattice Sums, unpublished.Google Scholar
  37. [37]
    A. Basu, Poisson equation for the three loop ladder diagram in string theory at genus one, arXiv:1606.02203 [INSPIRE].
  38. [38]
    A. Basu, Proving relations between modular graph functions, arXiv:1606.07084 [INSPIRE].
  39. [39]
    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
  40. [40]
    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
  41. [41]
    A. Basu, Simplifying the one loop five graviton amplitude in type IIB string theory, arXiv:1608.02056 [INSPIRE].

Copyright information

© The Author(s) 2016

Authors and Affiliations

  1. 1.Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and AstronomyUniversity of CaliforniaLos AngelesU.S.A.

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