Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 067, 26 pages      arXiv:1012.1072      https://doi.org/10.3842/SIGMA.2011.067

1+1 Gaudin Model

Andrei V. Zotov
Institute of Theoretical and Experimental Physics, Moscow, Russia

Received January 29, 2011, in final form July 03, 2011; Published online July 13, 2011

Abstract
We study 1+1 field-generalizations of the rational and elliptic Gaudin models. For sl(N) case we introduce equations of motion and L-A pair with spectral parameter on the Riemann sphere and elliptic curve. In sl(2) case we study the equations in detail and find the corresponding Hamiltonian densities. The n-site model describes n interacting Landau-Lifshitz models of magnets. The interaction depends on position of the sites (marked points on the curve). We also analyze the 2-site case in its own right and describe its relation to the principal chiral model. We emphasize that 1+1 version impose a restriction on a choice of flows on the level of the corresponding 0+1 classical mechanics.

Key words: integrable systems; field theory; Gaudin models.

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References

  1. Gaudin M., Diagonalisation dùne classe d'Hamiltoniens de spin, J. Physique 37 (1976), 1087-1098.
    Gaudin M., La fonction d'onde de Bethe, Masson, Paris, 1983 (in French); Mir, Moscow, 1987 (in Russian).
  2. Baxter R.J., One-dimensional anisotropic Heisenberg chain, Ann. Physics 70 (1972), 323-337.
  3. Sklyanin E.K., Takebe T., Algebraic bethe ansatz for the XYZ Gaudin model, Phys. Lett. A 219 (1996), 217-225, q-alg/9601028.
  4. Zotov A., Elliptic linear problem for Calogero-Inozemtsev model and Painlevé VI equation, Lett. Math. Phys. 67 (2004), 153-165, hep-th/0310260.
    Levin A., Olshanetsky M., Zotov A., Painlevé VI, rigid tops and reflection equation, Comm. Math. Phys. 268 (2006), 67-103, math.QA/0508058.
  5. Levin A., Zotov A., On rational and elliptic forms of Painlevé VI equation, Amer. Math. Soc. Transl. Ser. 2, Vol. 221, Amer. Math. Soc., Providence, RI, 173-184.
  6. Fuchs R., Über lineare homogene Differentialgleichungen zweiterordnung mit im endlich gelegne wesentlich singulären Stellen, Math. Ann. 63 (1907), 301-323.
    Schlesinger L., Über eine Klasse von Differentialsystemen beliebiger Ordnung mit festen kritischen Punkten, J. Reine Angew. Math. 141 (1912), 96-145.
  7. Reyman A.G., Semenov-Tian-Shansky M.A., Lie algebras and Lax equations with spectral parameter on an elliptic curve, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 150 (1986), 104-118 (in Russian).
  8. Belavin A.A., Drinfel'd V.G., Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funktsional. Anal. i Prilozhen. 16 (1982), no. 3, 1-29 (English transl.: Funct. Anal. Appl. 16 (1982), 159-180).
  9. Zotov A.V., Levin A.M.; Olshanetsky M.A., Chernyakov Yu.B., Quadratic algebras related to elliptic curves, Theoret. and Math. Phys. 156 (2008), 1103-1122, arXiv:0710.1072.
  10. Sklyanin E.K., Separation of variables in the Gaudin model, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 164 (1987), 151-169 (English transl.: J. Soviet Math. 47 (1989), 2473-2488).
    Sklyanin E.K., Takebe T., Separation of variables in the elliptic Gaudin model, Comm. Math. Phys. 204 (1999), 17-38, solv-int/9807008.
  11. Sklyanin E.K., Generating function of correlators in the sl2 Gaudin model, Lett. Math. Phys. 47 (1999), 275-292, solv-int/9708007.
    Takasaki K., Gaudin model, KZ equation and an isomonodromic problem on the tours, Lett. Math. Phys. 44 (1998), 143-156, hep-th/9711058.
    Feigin B., Frenkel E., Reshetikhin N., Gaudin model, Bethe ansatz and critical level, Comm. Math. Phys. 166 (1994), 27-62, hep-th/9402022.
    Gould M.D., Zhang Y.-Z., Zhao S.-Y., Elliptic Gaudin models and elliptic KZ equations, Nuclear Phys. B 630 (2002), 492-508, nlin.SI/0110038.
    Chernyakov Yu., Levin A., Olshanetsky M., Zotov A., Elliptic Schlesinger system and Painlevé VI, J. Phys. A: Math. Gen. 39 (2006), 12083-12101, nlin.SI/0602043.
  12. Khesin B., Levin A., Olshanetsky M., Bihamiltonian structures and quadratic algebras in hydrodynamics and on non-commutative torus, Comm. Math. Phys. 250 (2004), 581-612, nlin.SI/0309017.
  13. Petrera M., Suris Yu.B., An integrable discretization of the rational su(2) Gaudin model and related systems, Comm. Math. Phys. 283 (2008), 227-253, arXiv:0707.4088.
    Ragnisco O., Zullo F., Bäcklund transformations for the trigonometric Gaudin magnet, SIGMA 6 (2010), 012, 6 pages, arXiv:0912.2456.
    Veselov A.P., What is an integrable mapping?, in What is Integrability?, Editor V.E. Zakharov, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991, 251-272.
    Hone A.N.W., Kuznetsov V.B., Ragnisco O., Bäcklund transformations for the sl(2) Gaudin magnet, J. Phys. A: Math. Gen. 34 (2001), 2477-2490, nlin.SI/0007041.
  14. Talalaev D.V., The quantum Gaudin system, Funktsional. Anal. i Prilozhen. 40 (2006), no. 1, 86-91 (English transl.: Funct. Anal. Appl. 40 (2006), 73-77).
    Rubtsov V., Silantyev A., Talalaev D., Manin matrices, quantum elliptic commutative families and characteristic polynomial of elliptic Gaudin model, SIGMA 5 (2009), 110, 22 pages, arXiv:0908.4064.
  15. Frenkel E., Affine algebras, Langlands duality and Bethe ansatz, q-alg/9506003.
    Chervov A., Talalaev D., Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence, hep-th/0604128.
    Teschner J., Quantization of the Hitchin moduli spaces, Liouville theory, and the geometric Langlands correspondence I, arXiv:1005.2846.
  16. Enriquez B., Rubtsov V., Hitchin systems, higher Gaudin operators and R-matrices, Math. Res. Lett. 3 (1996), 343-357, alg-geom/9503010.
  17. Hitchin N., Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91-114.
  18. Nekrasov N., Holomorphic bundles and many-body systems, Comm. Math. Phys. 180 (1996), 587-603, hep-th/9503157.
    Gorsky A., Nekrasov N., Elliptic Calogero-Moser system from two dimensional current algebra, hep-th/9401021.
  19. Levin A., Olshanetsky M., Zotov A., Hitchin systems - symplectic Hecke correspondence and two-dimensional version, Comm. Math. Phys. 236 (2003), 93-133, nlin.SI/0110045.
  20. Petrera M., Ragnisco O., From su(2) gaudin models to integrable tops, SIGMA 3 (2007), 058, 14 pages, math-ph/0703044.
    Levin A., Zotov A., An integrable system of interacting elliptic tops, Teoret. Mat. Fiz. 146 (2006), 55-64 (English transl.: Theoret. and Math. Phys. 146 (2006), 45-52).
  21. Kuznetsov V.B., Isomorphism of the n-dimensional Neumann system and the n-site Gaudin magnet, Funktsional. Anal. i Prilozhen. 26 (1992), no. 4, 88-90 (English transl.: Funct. Anal. Appl. 26 (1992), 302-304).
  22. Zakharov V.E., Shabat A.B., A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I, Funktsional. Anal. i Prilozhen. 8 (1974), no. 3, 43-53 (English transl.: Funct. Anal. Appl. 8 (1974), 226-235).
  23. Faddeev L., Takhtajan L., Hamiltonian approach to solitons theory, Nauka, Moscow, 1986 (in Russian).
  24. Dubrovin B.A., Krichever I.M., Novikov S.P., Integrable systems. I, Current Problems in Mathematics. Fundamental Directions, Vol. 4, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985, 179-284 (in Russian).
  25. Sokolov V.V., Schabat A.B., Classification of integrable evolution equations, Soviet Sci. Rev. Sect. C Math. Phys. Rev., Vol. 4, Harwood Academic Publ., Chur, 1984, 221-280.
    Mikhailov A.V., Schabat A.B., Yamilov R.I., The symmetry approach to classification of nonlinear equations. Complete list of integrable systems, Uspekhi Mat. Nauk 42 (1987), no. 4, 3-53 (English transl.: Russian Math. Surveys 42 (1987), no. 4, 1-63).
    Fokas A.S., Symmetries and integrability, Stud. Appl. Math. 77 (1987), 253-299.
  26. Landau L., Lifshitz E., On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Zeitsch. der Sow. 8 (1935), 153-169.
  27. Sklyanin E., On complete integrability of the Landau-Lifshitz equation, Preprint LOMI E-3-79, 1979.
    Borovik A.E., Robuk V.N., Linear pseudopotentials and conservation laws for the Landau-Lifshits equation describing the nonlinear dynamics of a ferromagnet with uniaxial anisotropy, Teoret. Mat. Fiz. 46 (1981), 371-381 (English transl.: Theoret. and Math. Phys. 46 (1981), 242-248).
  28. Baxter R.J., Eight-vertex model in lattice statistics and one-dimensional anisotropic Heisenberg chain. I. Some fundamental eigenvectors, Ann. Physics 76 (1973), 1-24.
    Takhtajan L.A., Faddeev L.D., The quantum method of the inverse problem and the Heisenberg XYZ model, Uspekhi Mat. Nauk 34 (1979), no. 5, 13-63 (English transl.: Russian Math. Surveys 34 (1979), no. 5, 11-68).
  29. Zakharov V.E., Mikhailov A.V., Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method, Soviet Phys. JETP 74 (1978), 1953-1973.
    Pohlmeyer K., Integrable Hamiltonian systems and interactions through quadratic constraints, Comm. Math. Phys. 46 (1976), 207-221.
  30. Cherednik I.V., Local conservation laws of principal chiral fields (d=1), Teoret. Mat. Fiz. 38 (1979), 179-185 (English transl.: Theoret. and Math. Phys. 38 (1979), 120-124).
    Cherednik I.V., Relativistically invariant quasiclassical limits of integrable two-dimensional quantum models, Teoret. Mat. Fiz. 47 (1981), 225-229 (English transl.: Theoret. and Math. Phys. 47 (1981), 422-425).
  31. Krichever I., Vector bundles and Lax equations on algebraic curves, Comm. Math. Phys. 229 (2002), 229-269, hep-th/0108110.
  32. Dubrovin B.A., Matveev V.B., Novikov S.P., Non-linear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian varieties, Uspekhi Mat. Nauk 31 (1976), no. 1, 55-136 (English transl.: Russian Math. Surveys 31 (1976), no. 1, 59-146).
  33. Golubchik I.Z., Sokolov V.V., Multicomponent generalization of the hierarchy of the Landau-Lifshitz equation, Teoret. Mat. Fiz. 124 (2000), 62-71 (English transl.: Theoret. and Math. Phys. 124 (2000), 909-917).
  34. Akhmetshin A.A., Krichever I.M., Volvovski Yu.S., Elliptic families of solutions of the Kadomtsev-Petviashvili equation, and the field analogue of the elliptic Calogero-Moser system, Funktsional. Anal. i Prilozhen. 36 (2002), no. 4, 1-17 (English transl.: Funct. Anal. Appl. 36 (2002), 253-266), hep-th/0203192.
  35. Skrypnik T., 'Doubled' generalized Landau-Lifshiz hierarchies and special quasigraded Lie algebras, J. Phys. A: Math. Gen. 37 (2004), 7755-7768.
    Skrypnik T., Quasigraded Lie algebras and matrix generalization of Landau-Lifshitz equation, in Proceedinds of Fifth International Conference "Symmetry in Nonlinear Mathematical Physics" (June 23-29, 2003, Kyiv), Editors A.G. Nikitin, V.M. Boyko, R.O. Popovych and I.A. Yehorchenko, Proceedings of Institute of Mathematics, Kyiv 50 (2004), Part 1, 462-469.
  36. Holod P.I., The hidden symmetry of the Landau-Lifshits equation, the hierarchy of higher equations and a dual equation for an asymmetric chiral field, Teoret. Mat. Fiz. 70 (1987), 18-29 (English transl.: Theoret. and Math. Phys. 70 (1987), 11-19).
  37. Orlov A.Yu., N-soliton solution of chiral fields on Grassmann manifolds (σ model), Teoret. Mat. Fiz. 61 (1984), 214-225 (English transl.: Theoret. and Math. Phys. 61 (1984), 1099-1107).
  38. Weyl A., Elliptic functions according to Eisenstein and Kronecker, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 88, Springer-Verlag, Berlin - New York, 1976.
  39. Mumford D., Tata lectures on theta. I, Progress in Mathematics, Vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983.
    Mumford D., Tata lectures on theta. II. Jacobian theta functions and differential equations, Progress in Mathematics, Vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984.

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