Advertisement

Journal of High Energy Physics

, 2019:155 | Cite as

From elliptic multiple zeta values to modular graph functions: open and closed strings at one loop

  • Johannes BroedelEmail author
  • Oliver Schlotterer
  • Federico Zerbini
Open Access
Regular Article - Theoretical Physics
  • 18 Downloads

Abstract

We relate one-loop scattering amplitudes of massless open- and closed-string states at the level of their low-energy expansion. The modular graph functions resulting from integration over closed-string punctures are observed to follow from symmetrized open-string integrals through a tentative generalization of the single-valued projection known from genus zero.

Keywords

Gauge-gravity correspondence 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 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
  2. [2]
    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
  3. [3]
    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
  4. [4]
    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].
  5. [5]
    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
  6. [6]
    E. D’Hoker, M.B. Green, O. Gürdogan and P. Vanhove, Modular Graph Functions, Commun. Num. Theor. Phys. 11 (2017) 165 [arXiv:1512.06779] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    E. D’Hoker and M.B. Green, Identities between Modular Graph Forms, J. Number Theor. 189 (2018) 25 [arXiv:1603.00839] [INSPIRE].
  8. [8]
    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
  9. [9]
    A. Basu, Proving relations between modular graph functions, Class. Quant. Grav. 33 (2016) 235011 [arXiv:1606.07084] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    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
  11. [11]
    E. D’Hoker and J. Kaidi, Hierarchy of Modular Graph Identities, JHEP 11 (2016) 051 [arXiv:1608.04393] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    A. Kleinschmidt and V. Verschinin, Tetrahedral modular graph functions, JHEP 09 (2017) 155 [arXiv:1706.01889] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    E. D’Hoker and W. Duke, Fourier series of modular graph functions, arXiv:1708.07998 [INSPIRE].
  14. [14]
    D. Zagier, The Bloch-Wigner-Ramakrishnan polylogarithm function, Math. Ann. 286 (1990) 613.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    A. Levin, Elliptic polylogarithms: An analytic theory, Compos. Math. 106 (1997) 267.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    F. Brown and A. Levin, Multiple elliptic polylogarithms, arXiv:1110.6917.
  17. [17]
    O. Schlotterer and S. Stieberger, Motivic Multiple Zeta Values and Superstring Amplitudes, J. Phys. A 46 (2013) 475401 [arXiv:1205.1516] [INSPIRE].ADSMathSciNetzbMATHGoogle Scholar
  18. [18]
    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
  19. [19]
    S. Stieberger and T.R. Taylor, Closed String Amplitudes as Single-Valued Open String Amplitudes, Nucl. Phys. B 881 (2014) 269 [arXiv:1401.1218] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    O. Schnetz, Graphical functions and single-valued multiple polylogarithms, Commun. Num. Theor. Phys. 08 (2014) 589 [arXiv:1302.6445] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    F. Brown, Single-valued Motivic Periods and Multiple Zeta Values, SIGMA 2 (2014) e25 [arXiv:1309.5309] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  22. [22]
    B. Enriquez, Analogues elliptiques des nombres multizétas, Bull. Soc. Math. Fr. 144 (2016) 395 [arXiv:1301.3042].CrossRefzbMATHGoogle Scholar
  23. [23]
    J. Broedel, C.R. Mafra, N. Matthes and O. Schlotterer, Elliptic multiple zeta values and one-loop superstring amplitudes, JHEP 07 (2015) 112 [arXiv:1412.5535] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    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
  25. [25]
    J. Broedel, N. Matthes, G. Richter and O. Schlotterer, Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes, J. Phys. A 51 (2018) 285401 [arXiv:1704.03449] [INSPIRE].MathSciNetzbMATHGoogle Scholar
  26. [26]
    M.B. Green and J.H. Schwarz, Infinity Cancellations in SO(32) Superstring Theory, Phys. Lett. 151B (1985) 21 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    F. Brown, Notes on motivic periods, Commun. Num. Theor. Phys. 11 (2015) 557 [arXiv:1512.06410].MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    F. Brown, A class of non-holomorphic modular forms I, Res. Math. Sci. 5 (2018) arXiv:1707.01230 [INSPIRE].
  29. [29]
    F. Brown, A class of non-holomorphic modular forms II: equivariant iterated Eisenstein integrals, arXiv:1708.03354.
  30. [30]
    C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N-Point Superstring Disk Amplitude I. Pure Spinor Computation, Nucl. Phys. B 873 (2013) 419 [arXiv:1106.2645] [INSPIRE].
  31. [31]
    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
  32. [32]
    J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string. Cambridge University Press, (2007).Google Scholar
  33. [33]
    T. Terasoma, Selberg Integrals and Multiple Zeta Values, Compos. Math. 133 (2002) 1.MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    F. Brown, Multiple zeta values and periods of moduli spaces \( {\overline{\mathfrak{m}}}_{0,n} \), Ann. Sci. Éc. Norm. Supér. 42 (2009) 371 [math/0606419].
  35. [35]
    S. Stieberger, Constraints on Tree-Level Higher Order Gravitational Couplings in Superstring Theory, Phys. Rev. Lett. 106 (2011) 111601 [arXiv:0910.0180] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    L.A. Barreiro and R. Medina, 5-field terms in the open superstring effective action, JHEP 03 (2005) 055 [hep-th/0503182] [INSPIRE].
  37. [37]
    D. Oprisa and S. Stieberger, Six gluon open superstring disk amplitude, multiple hypergeometric series and Euler-Zagier sums, hep-th/0509042 [INSPIRE].
  38. [38]
    S. Stieberger and T.R. Taylor, Multi-Gluon Scattering in Open Superstring Theory, Phys. Rev. D 74 (2006) 126007 [hep-th/0609175] [INSPIRE].ADSGoogle Scholar
  39. [39]
    S. Stieberger and T.R. Taylor, Supersymmetry Relations and MHV Amplitudes in Superstring Theory, Nucl. Phys. B 793 (2008) 83 [arXiv:0708.0574] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    R.H. Boels, On the field theory expansion of superstring five point amplitudes, Nucl. Phys. B 876 (2013) 215 [arXiv:1304.7918] [INSPIRE].
  41. [41]
    G. Puhlfürst and S. Stieberger, Differential Equations, Associators and Recurrences for Amplitudes, Nucl. Phys. B 902 (2016) 186 [arXiv:1507.01582] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    J. Broedel, O. Schlotterer, S. Stieberger and T. Terasoma, All order α-expansion of superstring trees from the Drinfeld associator, Phys. Rev. D 89 (2014) 066014 [arXiv:1304.7304] [INSPIRE].ADSGoogle Scholar
  43. [43]
    C.R. Mafra and O. Schlotterer, Non-abelian Z-theory: Berends-Giele recursion for the α-expansion of disk integrals, JHEP 01 (2017) 031 [arXiv:1609.07078] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    J. Broedel, O. Schlotterer and S. Stieberger, α-expansion of open superstring amplitudes, http://mzv.mpp.mpg.de.
  45. [45]
    H. Kawai, D.C. Lewellen and S.H.H. Tye, A Relation Between Tree Amplitudes of Closed and Open Strings, Nucl. Phys. B 269 (1986) 1 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  46. [46]
    S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theor. 148 (2015) 328 [arXiv:1309.5865] [INSPIRE].
  47. [47]
    S. Bloch, M. Kerr and P. Vanhove, A Feynman integral via higher normal functions, Compos. Math. 151 (2015) 2329 [arXiv:1406.2664] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms, J. Math. Phys. 55 (2014) 102301 [arXiv:1405.5640] [INSPIRE].
  49. [49]
    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case, J. Math. Phys. 56 (2015) 072303 [arXiv:1504.03255] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    L. Adams and S. Weinzierl, Feynman integrals and iterated integrals of modular forms, Commun. Num. Theor. Phys. 12 (2018) 193 [arXiv:1704.08895] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    E. Remiddi and L. Tancredi, Schouten identities for Feynman graph amplitudes; The Master Integrals for the two-loop massive sunrise graph, Nucl. Phys. B 880 (2014) 343 [arXiv:1311.3342] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  52. [52]
    E. Remiddi and L. Tancredi, Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral, Nucl. Phys. B 907 (2016) 400 [arXiv:1602.01481] [INSPIRE].
  53. [53]
    E. Remiddi and L. Tancredi, An Elliptic Generalization of Multiple Polylogarithms, Nucl. Phys. B 925 (2017) 212 [arXiv:1709.03622] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism, JHEP 05 (2018) 093 [arXiv:1712.07089] [INSPIRE].
  55. [55]
    J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral, Phys. Rev. D 97 (2018) 116009 [arXiv:1712.07095] [INSPIRE].ADSGoogle Scholar
  56. [56]
    B. Enriquez, Elliptic associators, Selecta Math. (N.S.) 20 (2014) 491.Google Scholar
  57. [57]
    N. Matthes, Elliptic multiple zeta values, Ph.D. thesis, Universität Hamburg, Germany, (2016).Google Scholar
  58. [58]
    F. Zerbini, Elliptic multiple zeta values, modular graph functions and genus 1 superstring scattering amplitudes, Ph.D. Thesis, Bonn University, Germany, (2017), arXiv:1804.07989 [INSPIRE].
  59. [59]
    F. Brown, Multiple modular values and the relative completion of the fundamental group of m 1,1, arXiv:1407.5167.
  60. [60]
    N. Matthes, On the algebraic structure of iterated integrals of quasimodular forms, Alg. Numb. Theor. 11 (2017) 2113 [arXiv:1708.04561].MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    M.B. Green, J.H. Schwarz and L. Brink, N = 4 Yang-Mills and N = 8 Supergravity as Limits of String Theories, Nucl. Phys. B 198 (1982) 474 [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    D.M. Richards, The One-Loop Five-Graviton Amplitude and the Effective Action, JHEP 10 (2008) 042 [arXiv:0807.2421] [INSPIRE].
  63. [63]
    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
  64. [64]
    C.R. Mafra and O. Schlotterer, One-loop superstring six-point amplitudes and anomalies in pure spinor superspace, JHEP 04 (2016) 148 [arXiv:1603.04790] [INSPIRE].ADSzbMATHGoogle Scholar
  65. [65]
    W. Lerche, B.E.W. Nilsson, A.N. Schellekens and N.P. Warner, Anomaly Cancelling Terms From the Elliptic Genus, Nucl. Phys. B 299 (1988) 91 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  66. [66]
    S. Stieberger and T.R. Taylor, NonAbelian Born-Infeld action and type I-heterotic duality 2: Nonrenormalization theorems, Nucl. Phys. B 648 (2003) 3 [hep-th/0209064] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  67. [67]
    L. Dolan and P. Goddard, Current Algebra on the Torus, Commun. Math. Phys. 285 (2009) 219 [arXiv:0710.3743] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. [68]
    A. Basu, Low momentum expansion of one loop amplitudes in heterotic string theory, JHEP 11 (2017) 139 [arXiv:1708.08409] [INSPIRE].
  69. [69]
    F. Zerbini, Single-valued multiple zeta values in genus 1 superstring amplitudes, Commun. Num. Theor. Phys. 10 (2016) 703 [arXiv:1512.05689] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  70. [70]
    D. Zagier, Genus 0 and genus 1 string amplitudes and multiple zeta values, in preparation.Google Scholar
  71. [71]
    A. Tsuchiya, More on One Loop Massless Amplitudes of Superstring Theories, Phys. Rev. D 39 (1989) 1626 [INSPIRE].
  72. [72]
    C.R. Mafra and O. Schlotterer, Double-Copy Structure of One-Loop Open-String Amplitudes, Phys. Rev. Lett. 121 (2018) 011601 [arXiv:1711.09104] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    J. Broedel, N. Matthes and O. Schlotterer, Elliptic multiple zeta values, https://tools.aei.mpg.de/emzv.
  74. [74]
    D. Zagier, Periods of modular forms and Jacobi theta functions, Invent. Math. 104 (1991) 449.ADSMathSciNetCrossRefzbMATHGoogle Scholar
  75. [75]
    K. Haberland, Perioden von Modulformen einer Variabler und Gruppencohomologie, I, Math. Nachr. 112 (1983) 245.MathSciNetCrossRefzbMATHGoogle Scholar
  76. [76]
    D. Calaque, B. Enriquez and P. Etingof, Universal KZB equations: the elliptic case, in Algebra, arithmetic, and geometry: in honor of Yu. I. Manin, Vol. I, volume 269 of Progr. Math., Birkhäuser Boston, Inc., Boston, MA, U.S.A., (2009), p. 165.Google Scholar
  77. [77]
    R. Hain, Notes on the universal elliptic KZB equation, arXiv:1309.0580.
  78. [78]
    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
  79. [79]
    F. Brown, Single-valued multiple polylogarithms in one variable, C. R. Acad. Sci. Paris 338 (2004) 527.Google Scholar
  80. [80]
    M.B. Green, J. Schwarz and E. Witten, Superstring Theory. Vol. 2: Loop amplitudes, anomalies and phenomenology, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, U.K., (1987).Google Scholar
  81. [81]
    S. Hohenegger and S. Stieberger, Monodromy Relations in Higher-Loop String Amplitudes, Nucl. Phys. B 925 (2017) 63 [arXiv:1702.04963] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  82. [82]
    M.B. Green and J.H. Schwarz, Supersymmetrical Dual String Theory. 3. Loops and Renormalization, Nucl. Phys. B 198 (1982) 441 [INSPIRE].
  83. [83]
    P. Tourkine and P. Vanhove, Higher-loop amplitude monodromy relations in string and gauge theory, Phys. Rev. Lett. 117 (2016) 211601 [arXiv:1608.01665] [INSPIRE].ADSCrossRefGoogle Scholar
  84. [84]
    E. D’Hoker and M.B. Green, Zhang-Kawazumi Invariants and Superstring Amplitudes, J. Number Theory 144 (2014) 111 arXiv:1308.4597 [INSPIRE].
  85. [85]
    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
  86. [86]
    E. D’Hoker, M.B. Green and B. Pioline, Higher genus modular graph functions, string invariants and their exact asymptotics, arXiv:1712.06135 [INSPIRE].
  87. [87]
    C.M. Hull and P.K. Townsend, Unity of superstring dualities, Nucl. Phys. B 438 (1995) 109 [hep-th/9410167] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  88. [88]
    M.B. Green and M. Gutperle, Effects of D instantons, Nucl. Phys. B 498 (1997) 195 [hep-th/9701093] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  89. [89]
    M.B. Green, H.-h. Kwon and P. Vanhove, Two loops in eleven-dimensions, Phys. Rev. D 61 (2000) 104010 [hep-th/9910055] [INSPIRE].
  90. [90]
    M.B. Green and P. Vanhove, Duality and higher derivative terms in M-theory, JHEP 01 (2006) 093 [hep-th/0510027] [INSPIRE].
  91. [91]
    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].CrossRefzbMATHGoogle Scholar
  92. [92]
    V.G. Drinfeld, Quasi-Hopf algebras, Alg. Anal. 1 (1989) 114.MathSciNetGoogle Scholar
  93. [93]
    V. Drinfeld, On quasitriangular quasi-Hopf algebras and on a group that is closely connected with \( Gal\left(\overline{\mathbb{Q}}/\mathbb{Q}\right) \), Leningrad Math. J. 2 (1991) 829.MathSciNetGoogle Scholar
  94. [94]
    T. Le and J. Murakami, Kontsevichs integral for the Kauffman polynomial, Nagoya Math. J. 142 (1996) 39.Google Scholar
  95. [95]
    G. Racinet, Doubles mélanges des polylogarithmes multiples aux racines de lunité, Publ. Math. Inst. Hautes Études Sci. (2002) 185.Google Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.Institut für Mathematik und Institut für PhysikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.KAVLI Institute for Theoretical Physics, Kohn HallUniversity of CaliforniaSanta BarbaraU.S.A.
  3. 3.Max-Planck-Institut für GravitationsphysikAlbert-Einstein-InstitutPotsdamGermany
  4. 4.Perimeter Institute for Theoretical PhysicsWaterlooCanada
  5. 5.Institut de Physique Théorique (IPhT), CEA-SaclayGif-sur-YvetteFrance
  6. 6.Max-Planck-Institut für MathematikBonnGermany

Personalised recommendations