Advertisement

Polylogarithm Identities, Cluster Algebras and the \(\mathcal {N} = 4\)  Supersymmetric Theory

  • Cristian VerguEmail author
Conference paper
  • 40 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 314)

Abstract

Scattering amplitudes in \(\mathcal {N} = 4\) super-Yang Mills theory can be computed to higher perturbative orders than in any other four-dimensional quantum field theory. The results are interesting transcendental functions. By a hidden symmetry (dual conformal symmetry) the arguments of these functions have a geometric interpretation in terms of configurations of points in \(\mathbb {CP}^3\) and they turn out to be cluster coordinates. We briefly introduce cluster algebras and discuss their Poisson structure and the Sklyanin bracket. Finally, we present a \(40\)-term trilogarithm identity which was discovered by accident while studying the physical results.

Keywords

Scattering amplitudes Polylogarithms Cluster algebras Twistors 

Notes

Acknowledgements

First, I would like to thank the organizers of the Opening Workshop of the Research Trimester on Multiple Zeta Values, Multiple Polylogarithms, and Quantum Field Theory: José I. Burgos Gil, Kurusch Ebrahimi-Fard, D. Ellwood, Ulf Kühn, Dominique Manchon and P. Tempesta.

I would also like to thank the participants and particularly Frédéric Chapoton and Herbert Gangl for discussions during the opening workshop Numbers and Physics (NAP2014). Finally, I am grateful to my coauthors in Refs. [41, 46] for collaboration.

References

  1. 1.
    Alday, L.F., Maldacena, J.M.: Gluon scattering amplitudes at strong coupling. JHEP 0706, 064 (2007)Google Scholar
  2. 2.
    Anastasiou, C., Dixon, L., Bern, Z., Kosower, D.A.: Planar amplitudes in maximally supersymmetric yang-mills theory. Phys. Rev. Lett. 91(25) (2003)Google Scholar
  3. 3.
    Anastasiou, C., Brandhuber, A., Heslop, P., Khoze, V.V., Spence, B., et al.: Two-loop polygon wilson loops in n \(=\) 4 SYM. JHEP 0905, 115 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Arkani-Hamed, N., Bourjaily, J.L., Cachazo, F., Caron-Huot, S., Trnka, J.: The all-loop integrand for scattering amplitudes in planar n \(=\) 4 SYM. JHEP 1101, 041 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Arkani-Hamed, N., Bourjaily, J.L., Cachazo, F., Goncharov, A.B., Postnikov, A., et al.: Scattering Amplitudes and the Positive Grassmannian (2012)Google Scholar
  6. 6.
    Arkani-Hamed, N., Cachazo, F., Cheung, C., Kaplan, J.: A duality for the s matrix. JHEP 1003, 020 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Atiyah, M.F., Singer, I.M.: The index of elliptic operators on compact manifolds. Bull. Am. Math. Soc. 69(3), 422–434 (1963)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Atiyah, M.F., Hitchin, N.J., Drinfeld, V.G., Manin, Y.I.: Construction of instantons. Phys. Lett. A 65(3), 185–187 (1978)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Berenstein, A., Fomin, S., Zelevinsky, A.: Cluster algebras. iii: Upper bounds and double bruhat cells. Duke Math. J. 126(1), 1–52 (2005)Google Scholar
  10. 10.
    Bern, Z., Carrasco, J.J.M., Johansson, H., Kosower, D.A.: Maximally supersymmetric planar yang-mills amplitudes at five loops. Phys. Rev. D 76(12) (2007)Google Scholar
  11. 11.
    Bern, Z., Dixon, L.J., Kosower, D.A., Roiban, R., Spradlin, M., Vergu, C., Volovich, A.: Two-loop six-gluon maximally helicity violating amplitude in maximally supersymmetric yang-mills theory. Phys. Rev. D 78(4) (2008)Google Scholar
  12. 12.
    Bern, Z., Dixon, L.J., Kosower, D.A., Roiban, R., Spradlin, M., et al.: The two-loop six-gluon mhv amplitude in maximally supersymmetric yang-mills theory. Phys. Rev. D 78, 045007 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bern, Z., Chalmers, G.: Factorization in one-loop gauge theory. Nucl. Phys. B 447(2–3), 465–518 (1995)CrossRefGoogle Scholar
  14. 14.
    Bern, Z., Czakon, M., Dixon, L.J., Kosower, D.A., Smirnov, V.A.: Four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric yang-mills theory. Phys. Rev. D 75(8) (2007)Google Scholar
  15. 15.
    Bern, Z., Dixon, L.J., Smirnov, V.A.: Iteration of planar amplitudes in maximally supersymmetric yang-mills theory at three loops and beyond. Phys. Rev. D 72, 085001 (2005)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bloch, S.J.: Higher regulators, algebraic \(K\)-theory, and zeta functions of elliptic curves. CRM Monograph Series, vol. 11. American Mathematical Society, Providence, RI (2000)Google Scholar
  17. 17.
    Brandhuber, A., Heslop, P., Travaglini, G.: Mhv amplitudes in super-yang-mills and wilson loops. Nucl. Phys. B 794(1–2), 231–243 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Brink, L., Schwarz, J.H., Scherk, J.: Supersymmetric yang-mills theories. Nucl. Phys. B 121(1), 77–92 (1977)Google Scholar
  19. 19.
    Britto, R., Cachazo, F., Feng, B., Witten, E.: Direct proof of the tree-level scattering amplitude recursion relation in yang-mills theory. Phys. Rev. Lett. 94(18) (2005)Google Scholar
  20. 20.
    Broadhurst, D.J., Kreimer, D.: Knots and numbers in \({\phi }^{4}\) theory to 7 loops and beyond. Int. J. Mod. Phys. C 06(04), 519–524 (1995)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Caron-Huot, S.: Superconformal symmetry and two-loop amplitudes in planar n \(=\) 4 super yang-mills. JHEP 1112, 066 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Caron-Huot, S., He, S.: Jumpstarting the all-loop s-matrix of planar \( \cal{N}= {4} \) super yang-mills. J. High Energy Phys. 2012(7) (2012)Google Scholar
  23. 23.
    Caron-Huot, S.: Notes on the scattering amplitude/wilson loop duality. JHEP 1107, 058 (2011)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Coleman, S.R., Mandula, J.: All possible symmetries of the S matrix. Phys. Rev. 159(5), 1251–1256 (1967)Google Scholar
  25. 25.
    Dixon, L.J., Drummond, J.M., Duhr, C., Pennington, J.: The four-loop remainder function and multi-regge behavior at nnlla in planar \( \cal{N} =\) 4 super-yang-mills theory. J. High Energy Phys. 2014(6) (2014)Google Scholar
  26. 26.
    Dixon, L.J., Drummond, J.M., Henn, J.M.: Bootstrapping the three-loop hexagon. JHEP 1111, 023 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Dixon, L.J., Drummond, J.M., Henn, J.M.: Analytic result for the two-loop six-point NMHV amplitude in \( \cal{N}= {4} \) super yang-mills theory. J. High Energy Phys. 2012(1) (2012)Google Scholar
  28. 28.
    Dixon, L.J., Drummond, J.M., von Hippel, M., Pennington, J.: Hexagon functions and the three-loop remainder function. J. High Energy Phys. 2013(12) (2013)Google Scholar
  29. 29.
    Dixon, L.J., von Hippel, M.: Bootstrapping an NMHV amplitude through three loops. J. High Energy Phys. 2014(10) (2014)Google Scholar
  30. 30.
    Drummond, J.M., Papathanasiou, G., Spradlin, M.: A symbol of uniqueness: the cluster bootstrap for the 3-loop mhv heptagon. J. High Energy Phys. 2015(3) (2015)Google Scholar
  31. 31.
    Drummond, J.M., Henn, J., Korchemsky, G.P., Sokatchev, E.: The hexagon wilson loop and the bds ansatz for the six-gluon amplitude. Phys. Lett. B 662, 456–460 (2008)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Drummond, J.M., Henn, J., Korchemsky, G.P., Sokatchev, E.: Hexagon wilson loop \(=\) six-gluon mhv amplitude. Nucl. Phys. B 815, 142–173 (2009)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Drummond, J.M., Henn, J., Korchemsky, G.P., Sokatchev, E.: Dual superconformal symmetry of scattering amplitudes in n \(=\) 4 super-yang-mills theory. Nucl. Phys. B 828, 317–374 (2010)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Drummond, J.M., Henn, J., Smirnov, V.A., Sokatchev, E.: Magic identities for conformal four-point integrals. JHEP 0701, 064 (2007)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Drummond, J.M., Korchemsky, G.P., Sokatchev, E.: Conformal properties of four-gluon planar amplitudes and wilson loops. Nucl. Phys. B 795(1–2), 385–408 (2008)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Fock, V.V., Goncharov, A.B.: Cluster ensembles, quantization and the dilogarithm. Ann. Sci. Éc. Norm. Supér. (4) 42(6), 865–930 (2009)Google Scholar
  37. 37.
    Fomin, S., Zelevinsky, A.: Cluster algebras. i: Foundations. J. Am. Math. Soc. 15(2), 497–529 (2002)Google Scholar
  38. 38.
    Fomin, S., Zelevinsky, A.: Cluster algebras. ii: Finite type classification. Invent. Math. 154(1), 63–121 (2003)Google Scholar
  39. 39.
    Fomin, S., Zelevinsky, A.: Cluster algebras. iv: coefficients. Compos. Math. 143(1), 112–164 (2007)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster algebras and Poisson geometry. Mosc. Math. J. 3(3), 899–934, 1199 (2003). Dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthdayGoogle Scholar
  41. 41.
    Golden, J., Goncharov, A.B., Spradlin, M., Vergu, C., Volovich, A.: Motivic amplitudes and cluster coordinates. JHEP 1401, 091 (2014)CrossRefGoogle Scholar
  42. 42.
    Golden, J., Paulos, M.F., Spradlin, M., Volovich, A.: Cluster polylogarithms for scattering amplitudes. J. Phys. A: Math. Theor. 47(47), 474005 (2014)Google Scholar
  43. 43.
    Golden, J., Spradlin, M.: An analytic result for the two-loop seven-point mhv amplitude in n \(=\) 4 SYM. J. High Energy Phys. 8, 2014 (2014)Google Scholar
  44. 44.
    Golden, J., Spradlin, M.: A cluster bootstrap for two-loop MHV amplitudes. J. High Energy Phys. 2015(2) (2015)Google Scholar
  45. 45.
    Goncharov, A.B.: Geometry of configurations, polylogarithms, and motivic cohomology. Adv. Math. 114(2), 197–318 (1995)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Goncharov, A.B., Spradlin, M., Vergu, C., Volovich, A.: Classical polylogarithms for amplitudes and wilson loops. Phys. Rev. Lett. 105(15), 11 (2010)Google Scholar
  47. 47.
    Haag, R., Lopuszanski, J.T., Sohnius, M.: All possible generators of supersymmetries of the s matrix. Nucl. Phys. B 88(2), 257 (1975)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Hodges, A.: Eliminating spurious poles from gauge-theoretic amplitudes (2009)Google Scholar
  49. 49.
    ’t Hooft, G.: A planar diagram theory for strong interactions. Nucl. Phys. B 72(3), 461—473 (1974)Google Scholar
  50. 50.
    Huang, Y.-T., Wen, C., Xie, D.: The positive orthogonal grassmannian and loop amplitudes of abjm. J. Phys. A: Math. Theor. 47(47), 474008 (2014)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Keller, B.: Cluster algebras, quiver representations and triangulated categories. In: Triangulated Categories. Cambridge University Press, Cambridge (2010)Google Scholar
  52. 52.
    Kotikov, A.V., Lipatov, L.N., Onishchenko, A.I., Velizhanin, V.N.: Three-loop universal anomalous dimension of the wilson operators in n \(=\) 4 susy yang-mills model. Phys. Lett. B 595(1–4), 521–529 (2004)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Maldacena, J.: The large-n limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 38(4), 1113–1133 (1999)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Mason, L., Skinner, D.: Dual superconformal invariance, momentum twistors and grassmannians. J. High Energy Phys. 2009(11), 045 (2009)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Mason, L.J., Skinner, D.: The complete planar s-matrix of n \(=\) 4 SYM as a wilson loop in twistor space. JHEP 1012, 018 (2010)Google Scholar
  56. 56.
    Ovsienko, V.: Cluster superalgebras (2015)Google Scholar
  57. 57.
    Parke, S.J., Taylor, T.R.: Amplitude for n-gluon scattering. Phys. Rev. Lett. 56, 2459–2460 (1986)Google Scholar
  58. 58.
    Penrose, R.: Twistor algebra. J. Math. Phys. 8, 345 (1967)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Schnetz, O.: Quantum periods: a census of \(\phi ^4\)-transcendentals. Commun. Number Theory Phys. 4(1), 1–47 (2010)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Scott, J.S.: Grassmannians and cluster algebras. Proc. Lond. Math. Soc. III. Ser. 92(2), 345–380 (2006)Google Scholar
  61. 61.
    Witten, E.: Quantum field theory and the jones polynomial. Commun. Math. Phys. 121(3), 351–399 (1989)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Witten, E.: Monopoles and four-manifolds. Math. Res. Lett. 1(6), 769–796 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of MathematicsKing’s College London The StrandLondonUK

Personalised recommendations