International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2006 / Article

Open Access

Volume 2006 |Article ID 92064 | 31 pages | https://doi.org/10.1155/IJMMS/2006/92064

Quantum curve in q-oscillator model

Received16 Feb 2006
Accepted09 May 2006
Published13 Aug 2006

Abstract

A lattice model of interacting q-oscillators, proposed by V. Bazhanov and S. Sergeev in 2005 is the quantum-mechanical integrable model in 2+1 dimensional space-time. Its layer-to-layer transfer matrix is a polynomial of two spectral parameters, it may be regarded in terms of quantum groups both as a sum of sl(N) transfer matrices of a chain of length M and as a sum of sl(M) transfer matrices of a chain of length N for reducible representations. The aim of this paper is to derive the Bethe ansatz equations for the q-oscillator model entirely in the framework of 2+1 integrability providing the evident rank-size duality.

References

  1. R. J. Baxter, “On Zamolodchikov's solution of the tetrahedron equations,” Communications in Mathematical Physics, vol. 88, no. 2, pp. 185–205, 1983. View at: Publisher Site | Google Scholar | MathSciNet
  2. R. J. Baxter, “The Yang-Baxter equations and the Zamolodchikov model,” Physica D. Nonlinear Phenomena, vol. 18, no. 1–3, pp. 321–347, 1986. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  3. V. V. Bazhanov and R. J. Baxter, “New solvable lattice models in three dimensions,” Journal of Statistical Physics, vol. 69, no. 3-4, pp. 453–485, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. V. V. Bazhanov, A. N. Hibberd, and S. M. Khoroshkin, “Integrable structure of W3 conformal field theory, quantum Boussinesq theory and boundary affine Toda theory,” Nuclear Physics. B, vol. 622, no. 3, pp. 475–547, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. V. V. Bazhanov and S. Sergeev, “Zamolodchikov's Tetrahedron Equation and Hidden Structure of Quantum Groups,” Journal of Physics. A: Mathematical and General, vol. 39, pp. 3295–3310, 2006, http://arxiv.org/pdf/hep-th/0509181. View at: Google Scholar
  6. V. V. Bazhanov and Yu. G. Stroganov, “Conditions of commutativity of transfer-matrices on a multidimensional lattice,” Theoretical and Mathematical Physics, vol. 52, no. 1, pp. 685–691, 1982. View at: Publisher Site | Google Scholar
  7. H. J. de Vega, “Yang-Baxter algebras, integrable theories and Bethe ansatz,” International Journal of Modern Physics B, vol. 4, no. 5, pp. 735–801, 1990, Proceedings of the Conference on Yang-Baxter Equations. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. I. Korepanov, “Algebraic integrable dynamical systems, 2+1 dimensional models on wholly discrete space-time, and inhomogeneous models on 2-dimensional statistical physics,” preprint, http://arxiv.org/pdf/solv-int/9506003. View at: Google Scholar
  9. S. Sergeev, “3D symplectic map,” Physics Letters. A, vol. 253, no. 3-4, pp. 145–150, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. S. Sergeev, “Quantum 2+1 evolution model,” Journal of Physics. A: Mathematical and General, vol. 32, no. 30, pp. 5693–5714, 1999. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. S. Sergeev, “Auxiliary transfer matrices for three-dimensional integrable models,” Theoretical and Mathematical Physics, vol. 124, no. 3, pp. 391–409, 2000. View at: Google Scholar | MathSciNet
  12. S. Sergeev, “On exact solution of a classical 3D integrable model,” Journal of Nonlinear Mathematical Physics, vol. 7, no. 1, pp. 57–72, 2000. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  13. S. Sergeev, “Complex of three-dimensional solvable models,” Journal of Physics. A: Mathematical and General, vol. 34, no. 48, pp. 10493–10503, 2001, (Symmetries and integrability of difference equations (Tokyo, 2000)). View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  14. S. Sergeev, “Integrable three dimensional models in wholly discrete space-time,” in Integrable Structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory (Kiev, 2000), vol. 35 of NATO Sci. Ser. II Math. Phys. Chem., pp. 293–304, Kluwer Academic, Dordrecht, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  15. S. Sergeev, “Integrability of q-oscillator lattice model,” to appear in Physic Letters A, http://arxiv.org/pdf/nlin.SI/0509043. View at: Google Scholar
  16. S. Sergeev, V. V. Mangazeev, and Yu. G. Stroganov, “The vertex formulation of the Bazhanov-Baxter model,” Journal of Statistical Physics, vol. 82, no. 1-2, pp. 31–49, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  17. B. Sutherland, “Model for a multicomponent quantum system,” Physica Review B, vol. 12, no. 9, pp. 3795–3805, 1975. View at: Publisher Site | Google Scholar
  18. A. B. Zamolodchikov, “Tetrahedra equations and integrable systems in three-dimensional space,” Soviet Physics JETP, vol. 52, pp. 325–336, 1980, [Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki \textbf{79} (1980), 641–664]. View at: Google Scholar | MathSciNet
  19. A. B. Zamolodchikov, “Tetrahedron equations and the relativistic S-matrix of straight-strings in 2+1-dimensions,” Communications in Mathematical Physics, vol. 79, no. 4, pp. 489–505, 1981. View at: Publisher Site | Google Scholar | MathSciNet

Copyright © 2006 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Views | 0 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Order printed copiesOrder
 Sign up for content alertsSign up