\({{SO(d,1)}}\)-Invariant Yang–Baxter Operators and the dS/CFT Correspondence

Open Access


We propose a model for the dS/CFT correspondence. The model is constructed in terms of a “Yang–Baxter operator” R for unitary representations of the de Sitter group \({SO(d,1)}\). This R-operator is shown to satisfy the Yang–Baxter equation, unitarity, as well as certain analyticity relations, including in particular a crossing symmetry. With the aid of this operator we construct: (a) a chiral (light-ray) conformal quantum field theory whose internal degrees of freedom transform under the given unitary representation of \({SO(d,1)}\). By analogy with the O(N) non-linear sigma model, this chiral CFT can be viewed as propagating in a de Sitter spacetime. (b) A (non-unitary) Euclidean conformal quantum field theory on \({\mathbb{R}^{d-1}}\), where SO(d, 1) now acts by conformal transformations in (Euclidean) spacetime. These two theories can be viewed as dual to each other if we interpret \({\mathbb{R}^{d-1}}\) as conformal infinity of de Sitter spacetime. Our constructions use semi-local generator fields defined in terms of R and abstract methods from operator algebras.


  1. AF09.
    Arutyunov G., Frolov S.: Foundations of the \({AdS_5 \times S^5}\) Superstring. Part I. J. Phys. A 42, 254003 (2009)ADSMATHGoogle Scholar
  2. AAR01.
    Abdalla E., Abdalla M., Rothe K.: Non-perturbative methods in two-dimensional quantum field theory. World Scientific, Singapore (2001)CrossRefMATHGoogle Scholar
  3. AFZ79.
    Arinshtein A.E., Fateev V.A., Zamolodchikov A.B.: Quantum S-matrix of the (1 + 1)-dimensional toda chain. Phys. Lett. B 87, 389–392 (1979)ADSCrossRefGoogle Scholar
  4. Ala14.
    Alazzawi, S.: Deformations of quantum field theories and the construction of interacting models. Ph.D. Thesis, University of Vienna (2014). arXiv:1503.00897
  5. Bei12.
    Beisert N. et al.: Review of AdS/CFT integrability: an overview. Lett. Math. Phys. 99, 3 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  6. Bom16.
    Bombardelli D. et al.: An integrability primer for the gauge–gravity correspondence: an introduction. J. Phys. A 49(32), 320301 (2016)MathSciNetCrossRefMATHGoogle Scholar
  7. BBS01.
    Borchers H., Buchholz D., Schroer B.: Polarization-free generators and the S-matrix. Commun. Math. Phys. 219, 125–140 (2001) arXiv:hep-th/0003243 ADSMathSciNetCrossRefMATHGoogle Scholar
  8. BC12.
    Bostelmann H., Cadamuro D.: An operator expansion for integrable quantum field theories. J. Phys. A: Math. Theor. 46, 095401 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  9. BFK06.
    Babujian H.M., Foerster A., Karowski M.: The form factor program: a review and new results—the nested SU(N) off-shell bethe ansatz. SIGMA 2, 082 (2006) arXiv:hep-th/0609130 MathSciNetMATHGoogle Scholar
  10. BGL02.
    Brunetti R., Guido D., Longo R.: Modular localization and Wigner particles. Rev. Math. Phys. 14, 759–786 (2002) arXiv:math-ph/0203021 MathSciNetCrossRefMATHGoogle Scholar
  11. BL04.
    Buchholz D., Lechner G.: Modular nuclearity and localization. Ann. Henri Poincaré 5, 1065–1080 (2004) arXiv:math-ph/0402072 ADSMathSciNetCrossRefMATHGoogle Scholar
  12. BLM11.
    Bostelmann H., Lechner G., Morsella G.: Scaling limits of integrable quantum field theories. Rev. Math. Phys. 23, 1115–1156 (2011) arXiv:1105.2781 MathSciNetCrossRefMATHGoogle Scholar
  13. BM96.
    Bros J., Moschella U.: Two-point functions and quantum fields in de Sitter universe. Rev. Math. Phys. 8, 327–392 (1996) arXiv:gr-qc/9511019 MathSciNetCrossRefMATHGoogle Scholar
  14. Bos05.
    Bostelmann H.: Operator product expansions as a consequence of phase space properties. J. Math. Phys. 46, 082304 (2005) arXiv:math-ph/0502004v3 ADSMathSciNetCrossRefMATHGoogle Scholar
  15. BT15.
    Bischoff M., Tanimoto Y.: Integrable QFT and Longo–Witten endomorphisms. Ann. Henri Poincaré 16, 569–608 (2015) arXiv:1305.2171 ADSMathSciNetCrossRefMATHGoogle Scholar
  16. BW76.
    Bisognano J.J., Wichmann E.H.: On the duality condition for quantum fields. J. Math. Phys. 17, 303–321 (1976)ADSMathSciNetCrossRefGoogle Scholar
  17. CDI13.
    Chicherin D., Derkachov S., Isaev A.P.: Conformal algebra: R-matrix and star-triangle relation. JHEP 1304, 20 (2013) arXiv:1206.4150v2 ADSMathSciNetCrossRefMATHGoogle Scholar
  18. dMH13.
    de Medeiros P., Hollands S.: Conformal symmetry superalgebras. Class. Quant. Grav. 30, 175016 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. DKM01.
    Derkachov S.E., Korchemsky G.P., Manashov A.N.: Noncompact Heisenberg spin magnets from high-energy QCD: I. Baxter Q-operator and Separation of Variables. Nucl. Phys. B 617, 375–440 (2001) arXiv:hep-th/0107193v2 ADSMathSciNetCrossRefMATHGoogle Scholar
  20. DM06.
    Derkachov S.E., Manashov A.N.: R-matrix and baxter Q-operators for the noncompact SL(N,C) invariant spin chain. SIGMA 2, 084 (2006) arXiv:nlin/0612003v1 MathSciNetMATHGoogle Scholar
  21. DM11.
    Derkachov S.E., Manashov A.N.: Noncompact SL(N) spin chains: BGG-resolution, Q-operators and alternating sum representation for finite dimensional transfer matrices. Lett. Math. Phys. 97, 185–202 (2011) arXiv:1008.4734v2 ADSMathSciNetCrossRefMATHGoogle Scholar
  22. Dur70.
    Duren P.: Theory of H p Spaces. Dover Books on Mathematics. Dover Publications, Inc., New York (1970)Google Scholar
  23. EM14.
    Epstein H., Moschella U.: de Sitter tachyons and related topics. Commun. Math. Phys. 336(1), 381–430 (2015) arXiv:1403.3319v2 ADSMathSciNetCrossRefMATHGoogle Scholar
  24. Fad84.
    Faddeev, L.D.: Quantum completely integrable models in field theory, volume~1 of Mathematical Physics Reviews, pp. 107–155 (1984). In Novikov, S.P. (Ed.): Mathematical Physics Reviews, Vol. 1, 107–155Google Scholar
  25. FH81.
    Fredenhagen K., Hertel J.: Local algebras of observables and point-like localized fields. Commun. Math. Phys. 80, 555 (1981)ADSCrossRefMATHGoogle Scholar
  26. FOS83.
    Fröhlich J., Osterwalder K., Seiler E.: On Virtual representations of symmetric spaces and their analytic continuation. Ann. Math. 118, 461–489 (1983) http://www.jstor.org/stable/2006979 MathSciNetCrossRefMATHGoogle Scholar
  27. GHSS09.
    Guica M., Hartman T., Song W., Strominger A.: The Kerr/CFT correspondence. Phys. Rev. D 80, 124008 (2009). arXiv:0809.4266v1 ADSMathSciNetCrossRefGoogle Scholar
  28. GLW98.
    Guido D., Longo R., Wiesbrock H.: Extensions of conformal nets and superselection structures. Commun. Math. Phys. 192, 217–244 (1998) arXiv:hep-th/9703129 ADSMathSciNetCrossRefMATHGoogle Scholar
  29. GW09.
    Goodman R., Wallach N.R.: Symmetry, Representations, and Invariants. Springer, Berlin (2009)CrossRefMATHGoogle Scholar
  30. Haa96.
    Haag R.: Local Quantum Physics—Fields, Particles, Algebras, 2nd edn. Springer, Berlin (1996)MATHGoogle Scholar
  31. Hol12.
    Hollands S.: Massless interacting quantum fields in de Sitter spacetime. Ann. Henri Poincaré 13, 1039–1081 (2012)ADSMathSciNetCrossRefMATHGoogle Scholar
  32. Hol13.
    Hollands S.: Correlators, Feynman diagrams, and quantum no-hair in de Sitter spacetime. Commun. Math. Phys. 319, 1–68 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  33. H98.
    Hull C.M.: Timelike T duality, de Sitter space, large N gauge theories and topological field theory. JHEP 9807, 021 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  34. Iag93.
    Iagolnitzer D.: Scattering in Quantum Field Theories. Princeton University Press, Princeton (1993)CrossRefMATHGoogle Scholar
  35. Ket00.
    Ketov S.V.: Quantum Non-linear Sigma-Models. Springer, Berlin (2000)CrossRefMATHGoogle Scholar
  36. KR86.
    Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras II—Advanced Theory (1986)Google Scholar
  37. Lan75.
    Lang S.: SL 2(R). Springer, (1975)Google Scholar
  38. Lec03.
    Lechner G.: Polarization-free quantum fields and interaction. Lett. Math. Phys. 64, 137–154 (2003) arXiv:hep-th/0303062 MathSciNetCrossRefMATHGoogle Scholar
  39. Lec06.
    Lechner, G.: On the construction of quantum field theories with factorizing S-matrices. Ph.D. Thesis, University of Göttingen (2006). arXiv:math-ph/0611050
  40. Lec15.
    Lechner, G.: Algebraic constructive quantum field theory: integrable models and deformation techniques. In: Brunetti, R. et al. (eds.) Advances in Algebraic Quantum Field Theory, pp. 397–449. Springer, Berlin (2015)Google Scholar
  41. LM95.
    Liguori A., Mintchev M.: Fock spaces with generalized statistics. Lett. Math. Phys. 33, 283–295 (1995)ADSMathSciNetCrossRefMATHGoogle Scholar
  42. LS14.
    Lechner G., Schützenhofer C.: Towards an operator-algebraic construction of integrable global gauge theories. Ann. Henri Poincaré 15, 645–678 (2014) arXiv:1208.2366v1 ADSMathSciNetCrossRefMATHGoogle Scholar
  43. Mal98.
    Maldacena J.M.: The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998)ADSMathSciNetCrossRefMATHGoogle Scholar
  44. MM11.
    Marolf D., Morrison I.A.: The IR stability of de Sitter QFT: results at all orders. Phys. Rev. 84, 044040 (2011) arXiv:1010.5327v2 ADSGoogle Scholar
  45. NO14.
    Neeb K.-H., Olafsson G.: Reflection positivity and conformal symmetry. J. Funct. Anal. 266, 2174–2224 (2014) arXiv:1206.2039 MathSciNetCrossRefMATHGoogle Scholar
  46. Sch66.
    Schwartz L.: Theorie des Distributions. Hermann, Paris (1966)MATHGoogle Scholar
  47. Sch95.
    Schmüdgen K.: An operator-theoretic approach to a cocycle problem in the complex plane. Bull. Lond. Math. Soc. 27, 341–346 (1995)MathSciNetCrossRefMATHGoogle Scholar
  48. Smi92.
    Smirnov F.A.: Form Factors in Completely Integrable Models of Quantum Field Theory. World Scientific, Singapore (1992)CrossRefMATHGoogle Scholar
  49. Str01.
    Strominger A.: The dS/CFT Correspondence. JHEP 0110, 341–346 (2001) arXiv:hep-th/0106113v2 Google Scholar
  50. SW00.
    Schroer B., Wiesbrock H.: Modular constructions of quantum field theories with interactions. Rev. Math. Phys. 12, 301–326 (2000) arXiv:hep-th/9812251 MathSciNetCrossRefMATHGoogle Scholar
  51. VK91.
    Vilenkin N., Klimyk A.: Representations of Lie Groups and Special Functions Vol. I. Kluwer, Dordrecht (1991)CrossRefMATHGoogle Scholar
  52. Wit98.
    Witten E.: Anti De Sitter Space And Holography. Adv. Theor. Math. Phys. 2, 253–291 (1998) arXiv:hep-th/9802150v2 ADSMathSciNetCrossRefMATHGoogle Scholar
  53. Zam78.
    Zamolodchikov A.: Relativistic factorized S-matrix in two dimensions having O(N) isotopic symmetry. Nucl. Phys. B 133, 525–535 (1978)ADSMathSciNetCrossRefGoogle Scholar
  54. Zam79.
    Zamolodchikov A.B.: Factorized S-matrices in two dimensions as the exact solutions of certain relativistic quantum field models. Ann. Phys. 120, 253–291 (1979)ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institut für Theoretische PhysikUniversität LeipzigLeipzigGermany
  2. 2.School of MathematicsCardiff UniversityCardiffUK

Personalised recommendations