The Number Theory of Superstring Amplitudes

  • Oliver SchlottererEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 314)


The following article is intended as a survey of recent results at the interface of number theory and superstring theory. We review the expansion of scattering amplitudes—central observables in field and string theory—in the inverse string tension where elegant patterns of multiple zeta values occur. More specifically, the Drinfeld associator and the Hopf algebra structure of motivic multiple zeta values are shown to govern tree-level amplitudes of the open superstring. Partial results on the quantum corrections are discussed where elliptic analogues of multiple zeta values play a central r\(\hat{\text {o}}\)le.


Scattering amplitudes Superstring theory Multiple zeta values Hopf algebras 



I am very grateful to Johannes Broedel, Carlos Mafra, Nils Matthes, Stephan Stieberger and Tomohide Terasoma for collaboration on the projects on which this article is based. Moreover, I would like to thank Johannes Broedel, Nils Matthes and Federico Zerbini for valuable comments on the draft. I am indebted to the organizers of the conference “Numbers and Physics” in Madrid in September 2014 which strongly shaped the research directions leading to [16, 53] and possibly further results. I also acknowledge financial support by the European Research Council Advanced Grant No. 247252 of Michael Green.


  1. 1.
    Di Vecchia, P.: The Birth of string theory. Lect. Notes Phys. 737, 59 (2008). arXiv:0704.0101 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Vafa, C.: Lectures on strings and dualities. [hep-th/9702201]Google Scholar
  3. 3.
    Lüst, D., Stieberger, S., Taylor, T.R.: The LHC string Hunter’s companion. Nucl. Phys. B 808, 1 (2009). arXiv:0807.3333 [hep-th]zbMATHCrossRefGoogle Scholar
  4. 4.
    Lüst, D., Schlotterer, O., Stieberger, S., Taylor, T.R.: The LHC string Hunter’s companion (II): five-particle amplitudes and universal properties. Nucl. Phys. B 828, 139 (2010). arXiv:0908.0409 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Feng, W.Z., Lüst, D., Schlotterer, O., Stieberger, S., Taylor, T.R.: Direct production of lightest regge resonances. Nucl. Phys. B 843, 570 (2011). arXiv:1007.5254 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Arkani-Hamed, N., Dimopoulos, S., Dvali, G.R.: The Hierarchy problem and new dimensions at a millimeter. Phys. Lett. B 429, 263 (1998). [hep-ph/9803315]zbMATHCrossRefGoogle Scholar
  7. 7.
    Antoniadis, I., Arkani-Hamed, N., Dimopoulos, S., Dvali, G.R.: New dimensions at a millimeter to a Fermi and superstrings at a TeV. Phys. Lett. B 436, 257 (1998). [hep-ph/9804398]zbMATHCrossRefGoogle Scholar
  8. 8.
    Green, M.B., Schwarz, J.H., Brink, L.: N = 4 Yang-Mills and N = 8 Supergravity as Limits of String Theories. Nucl. Phys. B 198, 474 (1982)CrossRefGoogle Scholar
  9. 9.
    Kawai, H., Lewellen, D.C., Tye, S.H.H.: A relation between tree amplitudes of closed and open strings. Nucl. Phys. B 269, 1 (1986)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bjerrum-Bohr, N.E.J., Damgaard, P.H., Vanhove, P.: Minimal basis for gauge theory amplitudes. Phys. Rev. Lett. 103, 161602 (2009). arXiv:0907.1425 [hep-th]MathSciNetCrossRefGoogle Scholar
  11. 11.
    Stieberger, S.: Open & Closed vs. Pure Open String Disk Amplitudes. arXiv:0907.2211 [hep-th]
  12. 12.
    Drummond, J.M., Ragoucy, E.: Superstring amplitudes and the associator. JHEP 1308, 135 (2013). arXiv:1301.0794 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Broedel, J., Schlotterer, O., Stieberger, S., Terasoma, T.: All order \(\alpha ^{\prime }\)-expansion of superstring trees from the Drinfeld associator. Phys. Rev. D 89(6), 066014 (2014) arXiv:1304.7304 [hep-th]
  14. 14.
    Schlotterer, O., Stieberger, S.: Motivic multiple zeta values and superstring amplitudes. J. Phys. A 46, 475401 (2013). arXiv:1205.1516 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Stieberger, S.: Closed superstring amplitudes, single-valued multiple zeta values and the Deligne associator. J. Phys. A 47, 155401 (2014). arXiv:1310.3259 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Broedel, J., Mafra, C.R., Matthes, N., Schlotterer, O.: Elliptic multiple zeta values and one-loop superstring amplitudes. JHEP 1507, 112 (2015). arXiv:1412.5535 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Enriquez, B.: Analogues elliptiques des nombres multizétas. Bull. Soc. Math. France 144, 395–427 (2016). arxiv:1301.3042MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Enriquez, B.: Elliptic associators. Selecta Math. (N.S.) 20, 491 (2014)Google Scholar
  19. 19.
    Berkovits, N.: Super Poincare covariant quantization of the superstring. JHEP 0004, 018 (2000). [hep-th/0001035]MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Mafra, C.R., Schlotterer, O., Stieberger, S.: Complete N-point superstring disk amplitude I. Pure Spinor computation. Nucl. Phys. B 873, 419 (2013) arXiv:1106.2645 [hep-th]
  21. 21.
    Drinfeld, V.G.: Quasi Hopf algebras. Leningrad Math. J 1, 1419 (1989)MathSciNetGoogle Scholar
  22. 22.
    Drinfeld, V.G.: On quasitriangular quasi-Hopf algebras and a group that is closely connected with Gal(\(\bar{\mathbb{Q}}/{\mathbb{Q}}\)). Leningrad Math. J 2(4), 829(1991)Google Scholar
  23. 23.
    Racinet, G.: Doubles mélanges des polylogarithmes multiples aux racines de l?unité. Publ. Math. Inst. Hautes Etudes Sci. 185 (2002)Google Scholar
  24. 24.
    Le, T., Murakami, J.: Kontsevich’s integral for the Kauffman polynomial. Nagoya Math J. 142, 93 (1996)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Terasoma, T.: Selberg Integrals and Multiple Zeta Values. Compositio Mathematica 133, 1 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Barreiro, L.A., Medina, R.: 5-field terms in the open superstring effective action. JHEP 0503, 055 (2005). [hep-th/0503182]MathSciNetCrossRefGoogle Scholar
  27. 27.
    Oprisa, D., Stieberger, S.: Six gluon open superstring disk amplitude, multiple hypergeometric series and Euler-Zagier sums. hep-th/0509042Google Scholar
  28. 28.
    Stieberger, S., Taylor, T.R.: Multi-gluon scattering in open superstring theory. Phys. Rev. D 74, 126007 (2006). [hep-th/0609175]MathSciNetCrossRefGoogle Scholar
  29. 29.
    Boels, R.H.: On the field theory expansion of superstring five point amplitudes. Nucl. Phys. B 876, 215 (2013). arXiv:1304.7918 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Puhlfuerst, G., Stieberger, S.: Differential equations, associators, and recurrences for amplitudes. Nucl. Phys. B 902, 186 (2016). arXiv:1507.01582 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Broedel, J., Schlotterer, O., Stieberger, S.
  32. 32.
    Mafra, C.R., Schlotterer, O., Stieberger, S.: Complete N-point superstring disk amplitude II. Amplitude and Hypergeometric Function Structure. Nucl. Phys. B 873, 461 (2013) arXiv:1106.2646 [hep-th]
  33. 33.
    Broedel, J., Schlotterer, O., Stieberger, S.: Polylogarithms, multiple zeta values and superstring amplitudes. Fortsch. Phys. 61, 812 (2013). arXiv:1304.7267 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Brown, F.: Mixed Tate motives over \(\mathbb{Z}\). Ann. Math. 175, 949 (2012)Google Scholar
  35. 35.
    Bern, Z., Carrasco, J.J.M., Johansson, H.: New relations for gauge-theory amplitudes. Phys. Rev. D 78, 085011 (2008). arXiv:0805.3993 [hep-ph]MathSciNetCrossRefGoogle Scholar
  36. 36.
    Apéry, R.: Irrationalité de \(\zeta (2)\) et \(\zeta (3)\). Astérisque 61, 11 (1979)zbMATHGoogle Scholar
  37. 37.
    Ball, K., Rivoal, T.: Irrationalité d’une infinité de valeurs de la fonction zeta aux entiers impairs. Invent. Math. 146, 193 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Blumlein, J., Broadhurst, D.J., Vermaseren, J.A.M.: The multiple zeta value data mine. Comput. Phys. Commun. 181, 582 (2010). arXiv:0907.2557 [math-ph]MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Zagier, D.: Values of zeta functions and their applications. In: First European Congress of Mathematics, Vol. II (Paris, 1992) Birkhaeuser, Basel, p. 497 (1994)Google Scholar
  40. 40.
    Goncharov, A.B.: Galois symmetries of fundamental groupoids and noncommutative geometry. Duke Math. J. 128, 209 (2005). [math/0208144]MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Brown, F.: Motivic Periods and the Projective Line minus Three Points. ICM14 arXiv:1304.7267
  42. 42.
    Brown, F.: Single-valued motivic periods and multiple zeta values. SIGMA 2, e25 (2014). arXiv:1309.5309 [math.NT]MathSciNetzbMATHGoogle Scholar
  43. 43.
    Brown, F.: On the decomposition of motivic multiple zeta values. In: Galois-Teichmüller theory and arithmetic geometry, Adv. Stud. Pure Math. 63, 31–58 (2012) arXiv:1102.1310
  44. 44.
    Bern, Z., Dixon, L.J., Perelstein, M., Rozowsky, J.S.: Multileg one loop gravity amplitudes from gauge theory. Nucl. Phys. B 546, 423 (1999). [hep-th/9811140]zbMATHCrossRefGoogle Scholar
  45. 45.
    Bjerrum-Bohr, N.E.J., Damgaard, P.H., Sondergaard, T., Vanhove, P.: The momentum kernel of gauge and gravity theories. JHEP 1101, 001 (2011). arXiv:1010.3933 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Schnetz, O.: Graphical functions and single-valued multiple polylogarithms. Commun. Num. Theor. Phys. 08, 589 (2014). arXiv:1302.6445 [math.NT]MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Stieberger, S., Taylor, T.R.: Closed string amplitudes as single-valued open string amplitudes. Nucl. Phys. B 881, 269 (2014). arXiv:1401.1218 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Green, M.B., Schwarz, J.H.: Infinity cancellations in \(SO(32)\) superstring theory. Phys. Lett. B 151, 21 (1985)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Green, M.B., Schwarz, J.H.: Anomaly cancellation in supersymmetric \(D=10\) gauge theory and superstring theory. Phys. Lett. B 149, 117 (1984)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Green, M.B., Schwarz, J.H.: The Hexagon Gauge anomaly in Type I superstring theory. Nucl. Phys. B 255, 93 (1985)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Levin, A., Racinet, G.: Towards multiple elliptic polylogarithms (2007) [math/0703237]Google Scholar
  52. 52.
    Brown, F., Levin, A.: Multiple elliptic polylogarithms (2011)Google Scholar
  53. 53.
    Broedel, J., Matthes, N., Schlotterer, O.: Relations between elliptic multiple zeta values and a special derivation algebra. J. Phys. A 49(15), 155203 (2016) arXiv:1507.02254 [hep-th]
  54. 54.
    Green, M.B., Schwarz, J.H.: Supersymmetrical dual string theory. 3. Loops and renormalization. Nucl. Phys. B 198, 441 (1982)Google Scholar
  55. 55.
    Mafra, C.R., Schlotterer, O.: The structure of n-point one-loop open superstring amplitudes. JHEP 1408, 099 (2014). arXiv:1203.6215 [hep-th]CrossRefGoogle Scholar
  56. 56.
    Mafra, C.R., Schlotterer, O.: Cohomology foundations of one-loop amplitudes in pure spinor superspace. arXiv:1408.3605 [hep-th]
  57. 57.
    Mafra, C.R., Schlotterer, O.: Towards one-loop SYM amplitudes from the pure spinor BRST cohomology. Fortsch. Phys. 63(2), 105 (2015) arXiv:1410.0668 [hep-th]
  58. 58.
    Matthes, N.: Elliptic double zeta values. J. Number Theory 171, 227 (2017). arXiv:1509.08760 [math-NT]MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Manin, Y.I.: Iterated integrals of modular forms and noncommutative modular symbols. Algebraic Geom. Number Theory, 565 (2006)Google Scholar
  60. 60.
    Brown, F.: Multiple modular values for SL\(_2(\mathbb{Z})\) (2014)Google Scholar
  61. 61.
    Tsunogai, H.: On some derivations of Lie algebras related to Galois representations. Publ. Res. Inst. Math. Sci. 31, 113 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Calaque, D., Enriquez, B., Etingof, P.: Universal KZB equations: the elliptic case. Progr. Math. 269, 165 (2009)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Hain, R.: Notes on the universal elliptic KZB equation. arXiv:1309.0580 [math-NT]
  64. 64.
    Pollack, A.: Relations between derivations arising from modular formsGoogle Scholar
  65. 65.
    Broedel, J., Matthes, N., Schlotterer, O.
  66. 66.
    Green, M.B., Gutperle, M.: Effects of D instantons. Nucl. Phys. B 498, 195 (1997). [hep-th/9701093]MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Green, M.B., Kwon, H.H., Vanhove, P.: Two loops in eleven-dimensions. Phys. Rev. D 61, 104010 (2000) [hep-th/9910055]Google Scholar
  68. 68.
    Green, M.B., Vanhove, P.: Duality and higher derivative terms in M theory. JHEP 0601, 093 (2006). [hep-th/0510027]MathSciNetCrossRefGoogle Scholar
  69. 69.
    Green, M.B., Russo, J.G., Vanhove, P.: Low energy expansion of the four-particle genus-one amplitude in type II superstring theory. JHEP 0802, 020 (2008). arXiv:0801.0322 [hep-th]MathSciNetCrossRefGoogle Scholar
  70. 70.
    D’Hoker, E., Green, M.B., Vanhove, P.: On the modular structure of the genus-one Type II superstring low energy expansion. JHEP 1508, 041 (2015). arXiv:1502.06698 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Green, M.B., Vanhove, P.: The Low-energy expansion of the one loop type II superstring amplitude. Phys. Rev. D 61, 104011 (2000). [hep-th/9910056]MathSciNetCrossRefGoogle Scholar
  72. 72.
    Richards, D.M.: The One-loop five-graviton amplitude and the effective action. JHEP 0810, 042 (2008). arXiv:0807.2421 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Green, M.B., Mafra, C.R., Schlotterer, O.: Multiparticle one-loop amplitudes and S-duality in closed superstring theory. JHEP 1310, 188 (2013). arXiv:1307.3534 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    D’Hoker, E., Green, M.B., Vanhove, P.: Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus. J. Number Theory (2018) arXiv:1509.00363 [hep-th]
  75. 75.
    D’Hoker, E., Green, M.B.: Zhang-Kawazumi invariants and superstring amplitudes. J. Number Theory 144, 111 (2014). arXiv:1308.4597 [hep-th]MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    D’Hoker, E., Green, M.B., Pioline, B., Russo, R.: Matching the \(D^{6}R^{4}\) interaction at two-loops. JHEP 1501, 031 (2015). arXiv:1405.6226 [hep-th]CrossRefGoogle Scholar
  77. 77.
    Zhang, S.W.: Gross—Schoen cycles and dualising sheaves. Inventiones Mathematicae 179, 1 (2010). arXiv:0812.0371MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    Kawazumi, N.: Johnson’s homomorphisms and the Arakelov Green function. arXiv:0801.4218 [math.GT]

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-InstitutPotsdamGermany
  2. 2.Department of Physics and AstronomyUppsala UniversityUppsalaSweden

Personalised recommendations