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

SIGMA 6 (2010), 013, 52 pages      arXiv:0909.3759      https://doi.org/10.3842/SIGMA.2010.013
Contribution to the Proceedings of the Workshop “Geometric Aspects of Discrete and Ultra-Discrete Integrable Systems”

Bethe Ansatz, Inverse Scattering Transform and Tropical Riemann Theta Function in a Periodic Soliton Cellular Automaton for An(1)

Atsuo Kuniba a and Taichiro Takagi b
a) Institute of Physics, University of Tokyo, Komaba, Tokyo 153-8902, Japan
b) Department of Applied Physics, National Defense Academy, Kanagawa 239-8686, Japan

Received September 21, 2009; Published online January 31, 2010

Abstract
We study an integrable vertex model with a periodic boundary condition associated with Uq(An(1)) at the crystallizing point q=0. It is an (n+1)-state cellular automaton describing the factorized scattering of solitons. The dynamics originates in the commuting family of fusion transfer matrices and generalizes the ultradiscrete Toda/KP flow corresponding to the periodic box-ball system. Combining Bethe ansatz and crystal theory in quantum group, we develop an inverse scattering/spectral formalism and solve the initial value problem based on several conjectures. The action-angle variables are constructed representing the amplitudes and phases of solitons. By the direct and inverse scattering maps, separation of variables into solitons is achieved and nonlinear dynamics is transformed into a straight motion on a tropical analogue of the Jacobi variety. We decompose the level set into connected components under the commuting family of time evolutions and identify each of them with the set of integer points on a torus. The weight multiplicity formula derived from the q=0 Bethe equation acquires an elegant interpretation as the volume of the phase space expressed by the size and multiplicity of these tori. The dynamical period is determined as an explicit arithmetical function of the n-tuple of Young diagrams specifying the level set. The inverse map, i.e., tropical Jacobi inversion is expressed in terms of a tropical Riemann theta function associated with the Bethe ansatz data. As an application, time average of some local variable is calculated.

Key words: soliton cellular automaton; crystal basis; combinatorial Bethe ansatz; inverse scattering/spectral method; tropical Riemann theta function.

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References

  1. Baxter R.J., Exactly solved models in statistical mechanics, Dover, 2007.
  2. Bethe H.A., Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkettee, Z. Phys. 71 (1931), 205-226.
  3. Date E., Tanaka S., Periodic multi-soliton solutions of Korteweg-de Vries equation and Toda lattice, Progr. Theoret. Phys. Suppl. (1976), no. 59, 107-125.
  4. Dubrovin B.A., Matveev V.A., Novikov S.P., Nonlinear equations of Korteweg-de Vries type, finite-band linear operators and Abelian varieties, Uspehi Mat. Nauk 31 (1976), no. 1, 55-136.
  5. Fulton W., Young tableaux. With applications to representation theory and geometry, London Mathematical Society Student Texts, Vol. 35, Cambridge University Press, Cambridge, 1997.
  6. Gardner C.S., Greene J.M., Kruskal M.D., Miura R.M., Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967), 1095-1097.
  7. Hatayama G., Hikami K., Inoue R., Kuniba A., Takagi T., Tokihiro T., The AM(1) automata related to crystals of symmetric tensors, J. Math. Phys. 42 (2001), 274-308, math.QA/9912209.
  8. Hatayama G., Kuniba A., Takagi T., Factorization of combinatorial R matrices and associated cellular automata, J. Statist. Phys. 102 (2001), 843-863, math.QA/0003161.
  9. Inoue R., Takenawa T., Tropical spectral curves and integrable cellular automata, Int. Math. Res. Not. 2008 (2008), Art ID. rnn019, 27 pages, arXiv:0704.2471.
  10. Inoue R., Takenawa T., A tropical analogue of Fay's trisecant identity and the ultra-discrete periodic Toda lattice, Comm. Math. Phys. 289 (2009), 995-1021, arXiv:0806.3318.
  11. Jimbo M., Miwa T., Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci. 19 (1983), 943-1001.
  12. Kang S.-J., Kashiwara M., Misra K.C., Miwa T., Nakashima T., Nakayashiki A., Affine crystals and vertex models, in Infinite Analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., Vol. 16, World Sci. Publ., River Edge, NJ, 1992, 449-484.
  13. Kang S.-J., Kashiwara M., Misra K.C., Miwa T., Nakashima T., Nakayashiki A., Perfect crystals of quantum affine Lie algebras, Duke Math. J. 68 (1992), 499-607.
  14. Kashiwara M., Crystalizing the q-analogue of universal enveloping algebras, Comm. Math. Phys. 133 (1990), 249-260.
  15. Kerov S.V., Kirillov A.N., Reshetikhin N.Yu., Combinatorics, the Bethe ansatz and representations of the symmetric group, Zap. Nauchn. Sem. LOMI 155 (1986), 50-64 (English transl.: J. Soviet Math. 41 (1988), no. 2, 916-924).
  16. Kirillov A.N., Reshetikhin N.Yu., The Bethe ansatz and the combinatorics of Young tableaux, Zap. Nauchn. Sem. LOMI 155 (1986), 65-115 (English transl.: J. Soviet Math. 41 (1988), no. 2, 925-955).
  17. Kirillov A.N., Sakamoto R., Relations between two approaches: rigged configurations and 10-eliminations, Lett. Math. Phys. 89, (2009) 51-65.
  18. Kirillov A.N., Schilling A., Shimozono M., A bijection between Littlewood-Richardson tableaux and rigged configurations, Selecta Math. (N.S.) 8, (2002) 67-135, math.CO/9901037.
  19. Kuniba A., Nakanishi T., Bethe equation at q=0, the Möbius inversion formula, and weight multiplicities. II. The Xn case, J. Algebra 251 (2002), 577-618, math.QA/0008047.
  20. Kuniba A., Sakamoto R., The Bethe ansatz in a periodic box-ball system and the ultradiscrete Riemann theta function, J. Stat. Mech. Theory Exp. 2006 (2006), no. 9, P09005, 11 pages, math.QA/0606208.
  21. Kuniba A., Sakamoto R., Combinatorial Bethe ansatz and ultradiscrete Riemann theta function with rational characteristics, Lett. Math. Phys. 80 (2007), 199-209, nlin.SI/0611046.
  22. Kuniba A., Sakamoto R., Combinatorial Bethe ansatz and generalized periodic box-ball system, Rev. Math. Phys. 20 (2008), 493-527, arXiv:0708.3287.
  23. Kuniba A., Sakamoto R., Yamada Y., Tau functions in combinatorial Bethe ansatz, Nuclear Phys. B 786 (2007), 207-266, math.QA/0610505.
  24. Kuniba A., Takagi T., Takenouchi A., Bethe ansatz and inverse scattering transform in a periodic box-ball system, Nuclear Phys. B 747 (2006), 354-397, math.QA/0602481.
  25. Kuniba A., Takenouchi A., Bethe ansatz at q=0 and periodic box-ball systems, J. Phys. A: Math. Gen. 39 (2006), 2551-2562, nlin.SI/0509001.
  26. Kuniba A., Takenouchi A., Periodic cellular automata and Bethe ansatz, in Differential Geometry and Physics, Nankai Tracts Math., Vol. 10, World Sci. Publ., Hackensack, NJ, 2006, 293-302, math-ph/0511013.
  27. Macdonald I., Symmetric functions and Hall polynomials, 2nd ed., Oxford University Press, New York, 1995.
  28. Mada J., Idzumi M., Tokihiro T., Path description of conserved quantities of generalized periodic box-ball systems, J. Math. Phys. 46 (2005), 022701, 19 pages.
  29. Mada J., Idzumi M., Tokihiro T., The box-ball system and the N-soliton solution of the ultradiscrete KdV equation, J. Phys. A: Math. Theor. 41 (2008), 175207, 23 pages.
  30. Mikhalkin G., Zharkov I., Tropical curves, their Jacobians and theta functions, in Curves and Abelian Varieties, Contemp. Math., Vol. 465, Amer. Math. Soc., Providence, RI, 2008, 203-230, math.AG/0612267.
  31. Nakayashiki A., Yamada Y., Kostka polynomials and energy functions in solvable lattice models, Selecta Math. (N.S.) 3 (1997), 547-599, q-alg/9512027.
  32. Okado M., Schilling A., Shimozono M., A crystal to rigged configuration bijection for nonexceptional affine algebras, in Algebraic Combinatorics and Quantum Groups, Editor N. Jing, World Sci. Publ., River Edge, NJ, 2003, 85-124, math.QA/0203163.
  33. Sakamoto R., Kirillov-Schilling-Shimozono bijection as energy functions of crystals, Int. Math. Res. Not. 2009 (2009), no. 4, 579-614, arXiv:0711.4185.
  34. Shimozono M., Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties, J. Algebraic Combin. 15 (2002), 151-187, math.QA/9804039.
  35. Stanley R.P., Enumerative combinatorics, Vol. 1, Cambridge Studies in Advanced Mathematics, Vol. 49, Cambridge University Press, Cambridge, 1997.
  36. Takagi T., Level set structure of an integrable cellular automaton, arXiv:0906.1410.
  37. Takahashi D., On some soliton systems defined by using boxes and balls, in Proceedings of the International Symposium on Nonlinear Theory and Its Applications (NOLTA '93), 1993, 555-558.
  38. Takahashi D., Satsuma J., A soliton cellular automaton, J. Phys. Soc. Japan 59 (1990), 3514-3519.
  39. Yamada Y., A birational representation of Weyl group, combinatorial R-matrix and discrete Toda equation, in Physics and Combinatorics (Nagoya, 2000), Editors A.N. Kirillov and N. Liskova, World Sci. Publ., River Edge, NJ, 2001, 305-319.
  40. Yoshihara D., Yura F., Tokihiro T., Fundamental cycle of a periodic box-ball system, J. Phys. A: Math. Gen. 36 (2003), 99-121, nlin.SI/0208042.
  41. Yura F., Tokihiro T., On a periodic soliton cellular automaton, J. Phys. A: Math. Gen. 35 (2002), 3787-3801, nlin.SI/0112041.

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