Communications in Mathematical Physics

, Volume 354, Issue 1, pp 1–30 | Cite as

The Tetrahedral Zamolodchikov Algebra and the \({AdS_5\times S^5}\) S-matrix

  • Vladimir MitevEmail author
  • Matthias Staudacher
  • Zengo Tsuboi


The S-matrix of the \({AdS_5\times S^5}\) string theory is a tensor product of two centrally extended su\({(2|2)\ltimes \mathbb{R}^2}\) S-matrices, each of which is related to the R-matrix of the Hubbard model. The R-matrix of the Hubbard model was first found by Shastry, who ingeniously exploited the fact that, for zero coupling, the Hubbard model can be decomposed into two XX models. In this article, we review and clarify this construction from the AdS/CFT perspective and investigate the implications this has for the \({AdS_5\times S^5}\) S-matrix.


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  1. 1.
    Shastry B.S.: Exact integrability of the one-dimensional Hubbard model. Phys. Rev. Lett. 56, 2453 (1986)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Essler F.H.L., Frahm H., Göhmann F., Klümper A., Korepin V.E.: The One-Dimensional Hubbard Model. Cambridge University Press, Cambridge (2005)CrossRefzbMATHGoogle Scholar
  3. 3.
    Beisert N.: The analytic Bethe ansatz for a chain with centrally extended su(2|2) symmetry. J. Stat. Mech. 0701, P01017 (2007) arXiv:nlin/0610017 MathSciNetGoogle Scholar
  4. 4.
    Beisert N., Eden B., Staudacher M.: Transcendentality and crossing. J. Stat. Mech. 0701, P01021 (2007) arXiv:hep-th/0610251 Google Scholar
  5. 5.
    Martins M.J., Melo C.S.: The Bethe ansatz approach for factorizable centrally extended S-matrices. Nucl. Phys. B 785, 246 (2007) arXiv:hep-th/0703086 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Beisert N.: The \({SU(2|2)}\) dynamic S-matrix. Adv. Theor. Math. Phys. 12, 945 (2008) arXiv:hep-th/0511082 CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Plefka J., Spill F., Torrielli A.: On the Hopf algebra structure of the AdS/CFT S-matrix. Phys. Rev. D 74, 066008 (2006) arXiv:hep-th/0608038 ADSCrossRefMathSciNetGoogle Scholar
  8. 8.
    Beisert, N.: The S-matrix of AdS/CFT and Yangian symmetry. PoS SOLVAY: 002 (2006). arXiv:0704.0400 [nlin.SI]
  9. 9.
    de Leeuw, M.: The S-matrix of the \({AdS_5 \times S^5}\) superstring. arXiv:1007.4931[hep-th].
  10. 10.
    Ahn C., Nepomechie R.I: Review of AdS/CFT integrability, chapter III. 2: exact world-sheet S-matrix. Lett. Math. Phys. 99, 209 (2012) arXiv:1012.3991 [hep-th]ADSCrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Torrielli A.: Yangians, S-matrices and AdS/CFT. J. Phys. A 44, 263001 (2011) arXiv:1104.2474 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Gomez C., Hernandez R.: The magnon kinematics of the AdS/CFT correspondence. JHEP 0611, 021 (2006) arXiv:hep-th/0608029 [hep-th]ADSCrossRefMathSciNetGoogle Scholar
  13. 13.
    Rej A., Serban D., Staudacher M.: Planar \({\mathcal {N}=4}\) gauge theory and the Hubbard model. JHEP 0603, 018 (2006) arXiv:hep-th/0512077 [hep-th]ADSCrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Shiroishi M., Wadati M.: Yang–Baxter equation for the R-matrix of the one-dimensional Hubbard model. J. Phys. Soc. Jpn 64, 57 (1995)ADSCrossRefGoogle Scholar
  15. 15.
    Korepanov, I.G.: Vacuum curves, classical integrable systems in discrete space-time and statistical physics. Zap. Nauchn. Semin. 235, 272 Talk made at the Lobachevsky Semester in Euler Int. Math. Inst., St. Petersburg, Nov 1992. (1996)Google Scholar
  16. 16.
    Korepanov I.G.: Tetrahedral Zamolodchikov algebras corresponding to Baxter’s L-operators. Commun. Math. Phys. 154, 85 (1993)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Zamolodchikov A.B.: Tetrahedra equations and integrable systems in three dimensional space. Zh. Eksp. Teor. Fiz. 79, 641 (1980)MathSciNetGoogle Scholar
  18. 18.
    Arutyunov G., Frolov S., Zamaklar M.: The Zamolodchikov–Faddeev algebra for \({AdS_5 \times S^5}\) superstring. JHEP 0704, 002 (2007) arXiv:0612229 [hep-th]ADSCrossRefGoogle Scholar
  19. 19.
    Shiroishi M., Wadati M.: Tetrahedral Zamolodchikov algebra related to the six-vertex free-fermion model and a new solution of the Yang–Baxter equation. J. Phys. Soc. Jpn 64, 4598 (1995)ADSCrossRefGoogle Scholar
  20. 20.
    Bazhanov V.V., Stroganov Y.G.: Hidden symmetry of free fermion model. 1. Triangle equations and symmetric parametrization. Theor. Math. Phys. 62, 253 (1985)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Gomez C., Ruiz-Altaba M., Sierra G.: Quantum Groups in Two-Dimensional Physics. Cambridge Monographs on Mathematical Physics, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  22. 22.
    Reshetikhin N.: Multiparameter quantum groups and twisted quasitriangular Hopf algebras. Lett. Math. Phys. 20, 331 (1990)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Beisert N., Galleas W., Matsumoto T.: A quantum affine algebra for the deformed Hubbard chain. J. Phys. A 45, 365206 (2012) arXiv:math-ph/1102.5700 [math-ph]CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Horibe M., Shigemoto K.: On solutions of tetrahedron equations based on Korepanov mechanism. Prog. Theor. Phys. 93, 871 (1995) arXiv:hep-th/9410171 [hep-th]ADSCrossRefMathSciNetGoogle Scholar
  25. 25.
    Janik R.A.: The \({AdS_5 \times S^5}\) superstring worldsheet S-matrix and crossing symmetry. Phys. Rev. D 73, 086006 (2006) arXiv:hep-th/0603038 [hep-th]ADSCrossRefMathSciNetGoogle Scholar
  26. 26.
    Umeno Y., Shiroishi M., Wadati M.: Fermionic R-operator and integrability of the one-dimensional Hubbard model. J. Phys. Soc. Jpn 67, 2242 (1998)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Beisert N., Koroteev P.: Quantum deformations of the one-dimensional Hubbard model. J. Phys. A 41, 255204 (2008) arXiv:0802.0777 [hep-th]ADSCrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Beisert N., Staudacher M.: Long-range psu(2,2|4) Bethe ansätze for gauge theory and strings. Nucl. Phys. B 727, 1 (2005) arXiv:hep-th/0504190 [hep-th]ADSCrossRefzbMATHGoogle Scholar
  29. 29.
    Gromov, N., Kazakov, V., Vieira, P.: Integrability for the full spectrum of planar AdS/CFT. arXiv:0901.3753
  30. 30.
    Gromov N., Kazakov V., Vieira P.: Exact spectrum of anomalous dimensions of planar \({\mathcal{N}=4}\) supersymmetric Yang–Mills theory. Phys. Rev. Lett. 103, 131601 (2009)ADSCrossRefMathSciNetGoogle Scholar
  31. 31.
    Bombardelli D., Fioravanti D., Tateo R.: Thermodynamic Bethe ansatz for planar AdS/CFT: a proposal. J. Phys. A 42, 375401 (2009) arXiv:0902.3930 [hep-th]ADSCrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Gromov, N., Kazakov, V., Kozak, A., Vieira, P.: Integrability for the full spectrum of planar AdS/CFT II. arXiv:0902.4458 [hep-th]
  33. 33.
    Gromov N., Kazakov V., Kozak A., Vieira P.: Exact spectrum of anomalous dimensions of planar \({N = 4}\) supersymmetric Yang–Mills theory: TBA and excited states. Lett. Math. Phys. 91, 265 (2010)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Arutyunov G., Frolov S.: Thermodynamic Bethe ansatz for the \({{\rm AdS_5 \times S^5}}\) mirror model. JHEP 0905, 068 (2009) arXiv:0903.0141 [hep-th]ADSCrossRefGoogle Scholar
  35. 35.
    Gromov N., Kazakov V., Leurent S., Volin D.: Solving the AdS/CFT Y-system. JHEP 1207, 023 (2012) arXiv:1110.0562 [hep-th]ADSCrossRefMathSciNetGoogle Scholar
  36. 36.
    Balog J., Hegedus A.: Hybrid-NLIE for the AdS/CFT spectral problem. JHEP 1208, 022 (2012) arXiv:1202.3244 [hep-th]ADSCrossRefGoogle Scholar
  37. 37.
    Suzuki R.: Hybrid NLIE for the mirror \({AdS_5 \times S^5}\). J. Phys. A 44, 235401 (2011) arXiv:1101.5165 [hep-th]ADSCrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    Gromov N., Kazakov V., Tsuboi Z.: \({PSU(2,2|4)}\) character of quasiclassical AdS/CFT. JHEP 1007, 097 (2010) arXiv:1002.3981 [hep-th]ADSCrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Bazhanov V.V., Sergeev S.M.: Zamolodchikov’s tetrahedron equation and hidden structure of quantum groups. J. Phys. A 39, 3295 (2006) arXiv:hep-th/0509181 ADSCrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Kulish P.P., Sklyanin E.K.: Solutions of the Yang–Baxter equation. J. Soviet Math. 19, 1596 (1982)CrossRefzbMATHGoogle Scholar
  41. 41.
    Shastry B.: Decorated star-triangle relations and exact integrability of the one-dimensional Hubbard model. J. Stat. Phys. 50, 57 (1986)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Akutsu Y., Olmedilla E., Wadati M.: Yang–Baxter relations for spin models and fermion models. JPSJ 56, 2298 (1987)ADSCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Vladimir Mitev
    • 1
    • 2
    Email author
  • Matthias Staudacher
    • 1
    • 3
  • Zengo Tsuboi
    • 1
    • 4
  1. 1.Institut für Mathematik und Institut für PhysikHumboldt-Universität zu Berlin IRIS HausBerlinGermany
  2. 2.PRISMA Cluster of Excellence, Institut für Physik, WA THEPJohannes Gutenberg-Universität MainzMainzGermany
  3. 3.Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-InstitutPotsdamGermany
  4. 4.Laboratoire de Physique Théorique (LPT ENS), Département de physique de l’ENS, École normale supérieure, PSL Research University, Sorbonne Universités, UPMC Univ. Paris 06, CNRSParisFrance

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