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Classical Lie Bialgebras for AdS/CFT Integrability by Contraction and Reduction (2210.11150v2)

Published 20 Oct 2022 in hep-th, math-ph, math.MP, and math.QA

Abstract: Integrability of the one-dimensional Hubbard model and of the factorised scattering problem encountered on the worldsheet of AdS strings can be expressed in terms of a peculiar quantum algebra. In this article, we derive the classical limit of these algebraic integrable structures based on established results for the exceptional simple Lie superalgebra d(2,1;epsilon) along with standard sl(2) which form supersymmetric isometries on 3D AdS space. The two major steps in this construction consist in the contraction to a 3D Poincar\'e superalgebra and a certain reduction to a deformation of the u(2|2) superalgebra. We apply these steps to the integrable structure and obtain the desired Lie bialgebras with suitable classical r-matrices of rational and trigonometric kind. We illustrate our findings in terms of representations for on-shell fields on AdS and flat space.

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References (55)
  1. V. G. Drinfel’d, “Hopf algebras and the quantum Yang–Baxter equation”, Sov. Math. Dokl. 32, 254 (1985).
  2. V. G. Drinfel’d, “Quantum groups”, J. Sov. Math. 41, 898 (1988).
  3. J. Hubbard, “Electron Correlations in Narrow Energy Bands”, Proc. R. Soc. London A 276, 238 (1963), http://www.jstor.org/stable/2414761.
  4. F. H. L. Essler, H. Frahm, F. Göhmann, A. Klümper and V. E. Korepin, “The one-dimensional Hubbard model”, Cambridge University Press (2005), Cambridge, UK.
  5. B. S. Shastry, “Exact Integrability of the One-Dimensional Hubbard Model”, Phys. Rev. Lett. 56, 2453 (1986).
  6. N. Beisert, “The Analytic Bethe Ansatz for a Chain with Centrally Extended su(2///2) Symmetry”, J. Stat. Mech. 0701, P01017 (2007), nlin/0610017.
  7. N. Beisert, “The SU(2///2) dynamic S-matrix”, Adv. Theor. Math. Phys. 12, 945 (2008), hep-th/0511082.
  8. N. Beisert et al., “Review of AdS/CFT Integrability: An Overview”, Lett. Math. Phys. 99, 3 (2012), arxiv:1012.3982.
  9. C. Gómez and R. Hernández, “The Magnon kinematics of the AdS/CFT correspondence”, JHEP 0611, 021 (2006), hep-th/0608029.
  10. J. Plefka, F. Spill and A. Torrielli, “On the Hopf algebra structure of the AdS/CFT S-matrix”, Phys. Rev. D 74, 066008 (2006), hep-th/0608038.
  11. N. Beisert, “The S-matrix of AdS/CFT and Yangian symmetry”, PoS SOLVAY, 002 (2006), arxiv:0704.0400.
  12. N. Dorey, “Magnon Bound States and the AdS/CFT Correspondence”, J. Phys. A 39, 13119 (2006), hep-th/0604175.
  13. H.-Y. Chen, N. Dorey and K. Okamura, “The Asymptotic spectrum of the N = 4 super Yang-Mills spin chain”, JHEP 0703, 005 (2007), hep-th/0610295.
  14. T. Matsumoto and A. Molev, “Representations of centrally extended Lie superalgebra psl(2///2)”, J. Math. Phys. 55, 091704 (2014), arxiv:1405.3420.
  15. G. Arutyunov and S. Frolov, “The S-matrix of String Bound States”, Nucl. Phys. B 804, 90 (2008), arxiv:0803.4323.
  16. M. de Leeuw, “Bound States, Yangian Symmetry and Classical r-matrix for the AdS55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT ×\times× S55{}^{5}start_FLOATSUPERSCRIPT 5 end_FLOATSUPERSCRIPT Superstring”, JHEP 0806, 085 (2008), arxiv:0804.1047.
  17. G. Arutyunov, M. de Leeuw and A. Torrielli, “The Bound State S-Matrix for AdS55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT ×\times× S55{}^{5}start_FLOATSUPERSCRIPT 5 end_FLOATSUPERSCRIPT Superstring”, Nucl. Phys. B 819, 319 (2009), arxiv:0902.0183.
  18. R. A. Janik, “The AdS55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT ×\times× S55{}^{5}start_FLOATSUPERSCRIPT 5 end_FLOATSUPERSCRIPT superstring worldsheet S-matrix and crossing symmetry”, Phys. Rev. D 73, 086006 (2006), hep-th/0603038.
  19. R. Hernández and E. López, “Quantum corrections to the string Bethe ansatz”, JHEP 0607, 004 (2006), hep-th/0603204.
  20. G. Arutyunov and S. Frolov, “On AdS55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT ×\times× S55{}^{5}start_FLOATSUPERSCRIPT 5 end_FLOATSUPERSCRIPT String S-matrix”, Phys. Lett. B 639, 378 (2006), hep-th/0604043.
  21. N. Beisert, R. Hernández and E. López, “A Crossing-symmetric phase for AdS55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT ×\times× S55{}^{5}start_FLOATSUPERSCRIPT 5 end_FLOATSUPERSCRIPT strings”, JHEP 0611, 070 (2006), hep-th/0609044.
  22. N. Beisert, B. Eden and M. Staudacher, “Transcendentality and Crossing”, J. Stat. Mech. 0701, P01021 (2007), hep-th/0610251.
  23. N. Dorey, D. M. Hofman and J. M. Maldacena, “On the Singularities of the Magnon S-matrix”, Phys. Rev. D 76, 025011 (2007), hep-th/0703104.
  24. F. Spill and A. Torrielli, “On Drinfeld’s second realization of the AdS/CFT su(2///2) Yangian”, J. Geom. Phys. 59, 489 (2009), arxiv:0803.3194.
  25. N. Beisert and M. de Leeuw, “The RTT realization for the deformed gl(2///2) Yangian”, J. Phys. A 47, 305201 (2014), arxiv:1401.7691.
  26. N. Beisert, M. de Leeuw and R. Hecht, “Maximally extended sl(2///2) as a quantum double”, J. Phys. A 49, 434005 (2016), arxiv:1602.04988.
  27. T. Matsumoto, “Drinfeld realization of the centrally extended psl(2///2) Yangian algebra with the manifest coproducts”, arxiv:2208.11889.
  28. T. Klose, T. McLoughlin, R. Roiban and K. Zarembo, “Worldsheet scattering in AdS55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT ×\times× S55{}^{5}start_FLOATSUPERSCRIPT 5 end_FLOATSUPERSCRIPT”, JHEP 0703, 094 (2007), hep-th/0611169.
  29. A. Torrielli, “Classical r-matrix of the su(2///2) SYM spin-chain”, Phys. Rev. D 75, 105020 (2007), hep-th/0701281.
  30. S. Moriyama and A. Torrielli, “A Yangian double for the AdS/CFT classical r-matrix”, JHEP 0706, 083 (2007), arxiv:0706.0884.
  31. N. Beisert and F. Spill, “The Classical r-matrix of AdS/CFT and its Lie Bialgebra Structure”, Commun. Math. Phys. 285, 537 (2009), arxiv:0708.1762.
  32. T. Matsumoto, S. Moriyama and A. Torrielli, “A Secret Symmetry of the AdS/CFT S-matrix”, JHEP 0709, 099 (2007), arxiv:0708.1285.
  33. N. Beisert, R. Hecht and B. Hoare, “Maximally extended sl(2///2), q-deformed d(2,1;ϵitalic-ϵ\epsilonitalic_ϵ) and 3D kappa-Poincaré”, J. Phys. A 50, 314003 (2017), arxiv:1704.05093.
  34. T. Matsumoto and S. Moriyama, “An Exceptional Algebraic Origin of the AdS/CFT Yangian Symmetry”, JHEP 0804, 022 (2008), arxiv:0803.1212.
  35. V. Chari and A. Pressley, “A guide to quantum groups”, Cambridge University Press (1994), Cambridge, UK.
  36. V. G. Drinfel’d and A. A. Belavin, “Solutions of the classical Yang-Baxter equation for simple Lie algebras”, Func. Anal. Appl. 16, 159 (1982).
  37. N. Reshetikhin, “Multiparameter quantum groups and twisted quasitriangular Hopf algebras”, Lett. Math. Phys. 20, 331 (1990).
  38. G. Arutyunov, S. Frolov, J. Plefka and M. Zamaklar, “The Off-shell Symmetry Algebra of the Light-cone AdS55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT ×\times× S55{}^{5}start_FLOATSUPERSCRIPT 5 end_FLOATSUPERSCRIPT Superstring”, J. Phys. A 40, 3583 (2007), hep-th/0609157.
  39. D. M. Hofman and J. M. Maldacena, “Giant Magnons”, J. Phys. A 39, 13095 (2006), hep-th/0604135.
  40. N. Beisert, “The Classical Trigonometric r-Matrix for the Quantum-Deformed Hubbard Chain”, J. Phys. A 44, 265202 (2011), arxiv:1002.1097.
  41. N. Beisert and P. Koroteev, “Quantum Deformations of the One-Dimensional Hubbard Model”, J. Phys. A 41, 255204 (2008), arxiv:0802.0777.
  42. N. Beisert, W. Galleas and T. Matsumoto, “A Quantum Affine Algebra for the Deformed Hubbard Chain”, J. Phys. A 45, 365206 (2012), arxiv:1102.5700.
  43. F. Delduc, M. Magro and B. Vicedo, “An integrable deformation of the AdS55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT ×\times× S55{}^{5}start_FLOATSUPERSCRIPT 5 end_FLOATSUPERSCRIPT superstring action”, Phys. Rev. Lett. 112, 051601 (2014), arxiv:1309.5850.
  44. G. Arutyunov, R. Borsato and S. Frolov, “S-matrix for strings on η𝜂\etaitalic_η-deformed AdS55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT ×\times× S55{}^{5}start_FLOATSUPERSCRIPT 5 end_FLOATSUPERSCRIPT”, JHEP 1404, 002 (2014), arxiv:1312.3542.
  45. F. Delduc, M. Magro and B. Vicedo, “Derivation of the action and symmetries of the q-deformed AdS55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT ×\times× S55{}^{5}start_FLOATSUPERSCRIPT 5 end_FLOATSUPERSCRIPT superstring”, JHEP 1410, 132 (2014), arxiv:1406.6286.
  46. W. Nahm, “Supersymmetries and their Representations”, Nucl. Phys. B 135, 149 (1978).
  47. J. Van der Jeugt, “Irreducible representations of the exceptional Lie superalgebras D(2,1;α𝛼\alphaitalic_α)”, J. Math. Phys. 26, 913 (1985).
  48. O. Ohlsson Sax and B. Stefanski, Jr., “Integrability, spin-chains and the AdS33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT/CFT22{}_{2}start_FLOATSUBSCRIPT 2 end_FLOATSUBSCRIPT correspondence”, JHEP 1108, 029 (2011), arxiv:1106.2558.
  49. L. Eberhardt and M. R. Gaberdiel, “Strings on AdS33{}_{3}start_FLOATSUBSCRIPT 3 end_FLOATSUBSCRIPT ×\times× S33{}^{3}start_FLOATSUPERSCRIPT 3 end_FLOATSUPERSCRIPT ×\times× S33{}^{3}start_FLOATSUPERSCRIPT 3 end_FLOATSUPERSCRIPT ×\times× S11{}^{1}start_FLOATSUPERSCRIPT 1 end_FLOATSUPERSCRIPT”, JHEP 1906, 035 (2019), arxiv:1904.01585.
  50. G. Arutyunov, S. Frolov and M. Staudacher, “Bethe ansatz for quantum strings”, JHEP 0410, 016 (2004), hep-th/0406256.
  51. N. Beisert and E. Im, “Affine Classical Lie Bialgebras for AdS/CFT Integrability”, in preparation.
  52. N. Beisert, “The Dilatation operator of N = 4 super Yang-Mills theory and integrability”, Phys. Rept. 405, 1 (2004), hep-th/0407277.
  53. N. Beisert and E. Im, work in progress.
  54. C. Gómez and R. Hernández, “Quantum deformed magnon kinematics”, JHEP 0703, 108 (2007), hep-th/0701200.
  55. R. Borsato and A. Torrielli, “q𝑞qitalic_q-Poincaré supersymmetry in AdS55{}_{5}start_FLOATSUBSCRIPT 5 end_FLOATSUBSCRIPT/CFT44{}_{4}start_FLOATSUBSCRIPT 4 end_FLOATSUBSCRIPT”, Nucl. Phys. B 928, 321 (2018), arxiv:1706.10265.
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Authors (2)

  1. Niklas Beisert
  2. Egor Im

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