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

SIGMA 5 (2009), 029, 17 pages      arXiv:0903.1604      https://doi.org/10.3842/SIGMA.2009.029
Contribution to the Proceedings of the XVIIth International Colloquium on Integrable Systems and Quantum Symmetries

Limits of Gaudin Systems: Classical and Quantum Cases

Alexander Chervov a, Gregorio Falqui b and Leonid Rybnikov a
a) Institute for Theoretical and Experimental Physics, 25 Bolshaya Cheremushkinskaya Str., 117218 Moscow, Russia
b) Dipartimento di Matematica e Applicazioni, Università di Milano - Bicocca, via R. Cozzi, 53, 20125 Milano, Italy

Received November 01, 2008, in final form February 25, 2009; Published online March 09, 2009

Abstract
We consider the XXX homogeneous Gaudin system with N sites, both in classical and the quantum case. In particular we show that a suitable limiting procedure for letting the poles of its Lax matrix collide can be used to define new families of Liouville integrals (in the classical case) and new ''Gaudin'' algebras (in the quantum case). We will especially treat the case of total collisions, that gives rise to (a generalization of) the so called Bending flows of Kapovich and Millson. Some aspects of multi-Poisson geometry will be addressed (in the classical case). We will make use of properties of ''Manin matrices'' to provide explicit generators of the Gaudin Algebras in the quantum case.

Key words: Gaudin models; Hamiltonian structures; Gaudin algebras.

pdf (321 kb)   ps (220 kb)   tex (44 kb)

References

  1. Babelon O., Bernard D., Talon M., Introduction to classical integrable systems, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2003.
  2. Chernyakov Yu.B., Integrable systems, obtained by point fusion from rational and elliptic Gaudin systems, Theoret. and Math. Phys. 141 (2004), 1361-1380, hep-th/0311027.
  3. Chervov A., Falqui G., Manin matrices and Talalaev's formula, J. Phys. A: Math. Theor. 41 (2008), 194006, 28 pages, arXiv:0711.2236.
  4. Chervov A., Falqui G., Rybnikov L., Limits of Gaudin algebras, quantization of bending flows, Jucys-Murphy elements and Gelfand-Tsetlin bases, arXiv:0710.4971.
  5. Chervov A., Falqui G., Rubtsov V., Algebraic properties of Manin matrices I, Adv. Appl. Math., to appear, arXiv:0901.0235.
  6. Chervov A., Rybnikov L., Talalaev D., Rational Lax operators and their quantization, hep-th/0404106.
  7. Chervov A., Talalaev D., Universal G-oper and Gaudin eigenproblem, hep-th/0409007.
  8. Chervov A., Talalaev D., Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence, hep-th/0604128.
  9. Dickey L., Soliton equations and Hamiltonian systems, 2nd ed., Advanced Series in Mathematical Physics, Vol. 26, World Scientific Publishing Co., Inc., River Edge, NJ, 2003.
  10. Enriquez B., Rubtsov V., Hitchin systems, higher Gaudin operators and R-matrices, Math. Res. Lett. 3 (1996), 343-357, alg-geom/9503010.
  11. Falqui G., Musso F., Gaudin models and bending flows: a geometrical point of view, J. Phys. A: Math. Gen. 36 (2003), 11655-11676, nlin.SI/0306005.
  12. Falqui G., Musso F., Bi-Hamiltonian geometry and separation of variables for Gaudin models: a case study, in SPT 2002: Symmetry and Perturbation Theory (Cala Gonone), Editors S. Abenda, G. Gaeta and S. Walcher, World Sci. Publ., River Edge, NJ, 2002, 42-50, nlin.SI/0306008.
  13. Falqui G., Musso F., On separation of variables for homogeneous sl(r) Gaudin systems, Math. Phys. Anal. Geom. 9 (2006), 233-262, nlin.SI/0402026.
  14. Falqui G., Musso F., Quantisation of bending flows, Czechoslovak J. Phys. 56 (2006), 1143-1148, nlin.SI/0610003.
  15. Falqui G., Pedroni M., Separation of variables for bi-Hamiltonian systems, Math. Phys. Anal. Geom. 6 (2003), 139-179, nlin.SI/0204029.
  16. Flaschka H., Millson J., Bending flows for sums of rank one matrices, Canad. J. Math. 57, (2005), 114-158, math.SG/0108191.
  17. Feigin B., Frenkel E., Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras, in Infinite Analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., Vol. 16, World Sci. Publ., River Edge, NJ, 1992, 197-215.
  18. Feigin B., Frenkel E., Reshetikhin N., Gaudin model, Bethe ansatz and critical level, Comm. Math. Phys. 166 (1994), 27-62, hep-th/9402022.
  19. Feigin B., Frenkel E., Toledano-Laredo V., Gaudin model with irregular singularities, math.QA/0612798.
  20. Frenkel E., Affine algebras, Langlands duality and Bethe ansatz, in Proceedings XIth International Congress of Mathematical Physics (Paris, 1994), Int. Press, Cambridge, MA, 1995, 606-642, q-alg/9506003.
  21. Gaudin M., La fonction d'onde de Bethe, Collection du Commissariat à l'Énergie Atomique: Série Scientifique, Masson, Paris, 1983.
  22. Gel'fand I.M., Zakharevich I., Webs, Lenard schemes, and the local geometry of bi-Hamiltonian Toda and Lax structures, Selecta Math. (N.S.) 6 (2000), 131-183, math.DG/9903080.
  23. Jurco B., Classical Yang-Baxter equations and quantum integrable systems, J. Math. Phys. 30 (1989), 1289-1293.
  24. Kapovich M., Millson J., The symplectic geometry of polygons in Euclidean space, J. Differential Geom. 44 (1996), 479-513.
  25. Kuznetsov V.B., Quadrics on Riemannian spaces of constant curvature. Separation of variables and a connection with the Gaudin magnet, Theoret. and Math. Phys. 91 (1992), 385-404.
  26. Manin Yu.I., Quantum groups and noncommutative geometry, Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1988.
  27. Musso F., Petrera M., Ragnisco O., Algebraic extensions of Gaudin models, J. Nonlinear Math. Phys. 12 (2005), suppl. 1, 482-498, nlin.SI/0410016.
  28. Pedroni M., Vanhaecke P., A Lie algebraic generalization of the Mumford system, its symmetries and its multi-Hamiltonian structure, Regul. Chaotic Dyn. 3 (1998), 132-160.
  29. Reyman A.G., Semenov-Tian-Shansky M.A., Group-theoretical methods in the theory of finite-dimensional integrable systems, in Dynamical Systems VII, Editors V.I. Arnold and S.P. Novikov, Encyclopaedia of Mathematical Sciences, Vol. 16, Berlin, Springer, 1994, 116-225.
  30. Rybnikov L.G., The shift of invariants method and the Gaudin model, Func. Anal. Appl. 40 (2006), 188-199, math.RT/0606380.
  31. Rybnikov L.G., Uniqueness of higher Gaudin Hamiltonians, math.QA/0608588.
  32. Sierra G., Integrability and conformal symmetry in BCS model, in Statistical Field Theories (Como, 2001), NATO Sci. Ser. II Math. Phys. Chem., Vol. 73, Kluwer Acad. Publ., Dordrecht, 2002, 317-328, hep-th/0111114.
  33. Sklyanin E.K., Separation of variables - new trends, in Quantum Field Theory, Integrable Models and Beyond (Kyoto, 1994), Progr. Theoret. Phys. Suppl. (1995), no. 118, 35-60, solv-int/9504001.
  34. Talalaev D., Quantization of the Gaudin system, Funct. Anal. Appl. 40 (2006), 73-77, hep-th/0404153.

Previous article   Next article   Contents of Volume 5 (2009)