Journal of Mathematical Sciences

, Volume 238, Issue 6, pp 819–833 | Cite as

SOS-Representation for the SL(2,ℂ)-Invariant R-Operator and Feynman Diagrams

  • P. A. ValinevichEmail author
  • S. E. Derkachov
  • A. P. Isaev

We discuss the construction of the SL(2, ℂ)-invariant R-operator acting in the tensor product of two principal series representations of SL(2, ℂ) and satisfying the Yang–Baxter equation. We present a closed-form expression for this R-operator as a multiple two-dimensional propagator-type Feynman integral, which can be reduced to a double integral of Mellin–Barnes type. The obtained R-operator can be interpreted as a Boltzmann weight of the corresponding SOS model. All necessary formulas concerning principal series representations of SL(2, ℂ) are presented.


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  1. 1.
    I. M. Gelfand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Vol. 5, Academic Press (1966).Google Scholar
  2. 2.
    I. M. Gelfand and M. A. Naimark, “Unitary representations of the classical groups,” Trudy Mat. Inst. Steklov, 36, 3–288 (1950).Google Scholar
  3. 3.
    M. A. Naimark and A. I. Stern, Theory of Group Representations, Springer (1982).Google Scholar
  4. 4.
    I. M. Gelfand and G. E. Shilov, Generalized Functions, Vol. 1: Properties and Operations, AMS Chelsea Publishing (1964).Google Scholar
  5. 5.
    M. A. Naimark, “Decomposition of a tensor product of irreducible representations of the proper Lorentz group into irreducible representations,” Tr. Mosk. Mat. Obs., 8, 121–153 (1959).Google Scholar
  6. 6.
    A. A. Belavin, A. M. Polyakov, and A. B. Zamolodchikov, “Infinite conformal symmetry in two-dimensional quantum field theory,” Nucl. Phys. B, 241, No. 2, 333–380 (1984).Google Scholar
  7. 7.
    P. P. Kulish and E. K. Sklyanin, “Quantum spectral transform method. Recent developments,” Lect. Notes Physics, 151, 61–119 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    E. K. Sklyanin, “Quantum inverse scattering method. Selected topics,” in: Mo-Lin Ge (ed.), Quantum Group and Quantum Integrable Systems (Nankai Lectures in Mathematical Physics), World Scientific, Singapore (1992), pp. 63–97.Google Scholar
  9. 9.
    L. D. Faddeev, “How Algebraic Bethe Anstz works for integrable model,” in: A. Connes, K. Gawedzki, and J. Zinn-Justin (eds.), Quantum Symmetries/Sym´etries Qantiques, Proceedings of the Les Houches summer school, LXIV, North-Holland (1998), pp. 149–211.Google Scholar
  10. 10.
    P. P. Kulish, N. Yu. Reshetikhin, and E. K. Sklyanin, “Yang–Baxter equation and representation theory,” Lett. Math. Phys., 5, 393–403 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London (1982).zbMATHGoogle Scholar
  12. 12.
    M. Jimbo, “Introduction to the Yang–Baxter equation,” Int. J. Mod. Phys A, 4, 3759-3777 (1989); M. Jimbo (ed.), “Yang–Baxter equation in integrable systems,” Adv. Ser. Math. Phys., 10, World Scientific, Singapore (1990).Google Scholar
  13. 13.
    E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, “Exactly solvable SOS models II,” Adv. Stud. Pure Math., 16, 17–122 (1988).CrossRefzbMATHGoogle Scholar
  14. 14.
    I. B. Frenkel and V. G. Turaev, “Trigonometric solutions of the Yang–Baxter equation, nets, and hypergeometric functions,” in: Functional Analysis on the Eve of 21st Century, Vol. 1, Progress in Math., 131, Birkhäuser, Boston (1995), pp. 65–118.Google Scholar
  15. 15.
    I. Frenkel and V. Turaev, “Elliptic solutions of the Yang–Baxter equation and modular hypergeometric functions,” in: The Arnold–Gelfand Mathematical Seminars, Birkhauser Boston (1997), pp. 171–204.Google Scholar
  16. 16.
    V. Pasquier, “Etiology of IRF models,” Comm. Math. Phys., 118, 355–364 (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    S. E. Derkachov, G. P. Korchemsky, and A. N. Manashov, “Noncompact Heisenberg spin magnets from high-energy QCD. I: Baxter Q-operator and separation of variables,” Nucl. Phys. B, 617, 375–440 (2001).Google Scholar
  18. 18.
    S. E. Derkachov and A. N. Manashov, “General solution of the Yang–Baxter equation with the symmetry group SL(n,ℂ),” Algebra Analiz, 21, No. 4, 1–94 (2009).MathSciNetGoogle Scholar
  19. 19.
    S. G. Gorishnii and A. P. Isaev, “An approach to the calculation of many-loop massless Feynman integrals,” Teor. Mat. Fiz., 62, No. 3, 345–358 (1985).CrossRefGoogle Scholar
  20. 20.
    R. S. Ismagilov, “On Racah operators,” Funkt. Anal. Appl., 40, No. 3, 222–224 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    R. S. Ismagilov, “Racah operators for principal series of representations of the group SL(2, ℂ),” Sb. Math., 198, No. 3, 369–381 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    S. E. Derkachov and V. P. Spiridonov, “On the 6j-symbols for SL(2,C) group,” arXiv:1711.07073.Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • P. A. Valinevich
    • 1
    Email author
  • S. E. Derkachov
    • 1
  • A. P. Isaev
    • 2
  1. 1.St.Petersburg Department of Steklov Institute of MathematicsSt.PetersburgRussia
  2. 2.Bogoliubov Laboratory of Theoretical PhysicsJoint Institute of Nuclear ResearchDubnaRussia

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