×

On discrete three-dimensional equations associated with the local Yang-Baxter relation. (English) Zbl 0862.58033

Summary: The local Yang-Baxter equation (YBE), introduced by Maillet and Nijhoff, is a proper generalization to three dimensions of the zero curvature relation. Recently, Korepanov has constructed an infinite set of integrable three-dimensional lattice models, and has related them to solutions to the local YBE. The simplest Korepanov model is related to the star-triangle relation in the Ising model. In this letter the corresponding discrete equation is derived. In the continuous limit it leads to a differential three-dimensional equation, which is symmetric with respect to all permutations of the three coordinates. A similar analysis of the star-triangle transformation in electric networks leads to the discrete bilinear equation of Miwa, associated with the BKP hierarchy.
Some related operator solutions to the tetrahedron equation are also constructed.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
39A10 Additive difference equations
82B23 Exactly solvable models; Bethe ansatz
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Yang, C. N.: Some exact results for the many-body problem in one dimension with repulsive delta-function interaction,Phys. Rev. Lett. 19 (1967), 1312–1314. · Zbl 0152.46301 · doi:10.1103/PhysRevLett.19.1312
[2] Baxter, R. J.: Partition function of the eight-vertex lattice model,Ann. Phys. 70 (1972), 193–228. · Zbl 0236.60070 · doi:10.1016/0003-4916(72)90335-1
[3] Faddeev, L. D.: Quantum completely integrable models in field theory,Sov. Sci. Rev. C 1 (1980), 107–155. · Zbl 0569.35064
[4] Maillet, J. M. and Nijhoff, F. W.: Integrability for multidimensional lattice models,Phys. Lett. B 224 (1989), 389–396. · doi:10.1016/0370-2693(89)91466-4
[5] Maillet, J. M.: Integrable systems and gauge theories,Nuclear Phys. (Proc. Suppl.) 18B (1990), 212–241. · Zbl 0957.57503
[6] Korepanov, I. G.: A dynamical system connected with inhomogeneous 6-vertex model,Zap. Nauch. Semin. POMI 215 (1994), 178–196. · Zbl 0907.58028
[7] Korepanov, I. G.: Algebraic integrable dynamical systems, 2+1-dimensional models in wholly discrete space-time, and inhomogeneous models in 2-dimensional statistical physics, solvint/9506003.
[8] Miwa, T.: On Hirota’s Difference Equations,Proc. Japan Acad. A 58, (1982), 9–12. · Zbl 0508.39009 · doi:10.3792/pjaa.58.9
[9] Baxter, R. J.:Exactly Solved Models in Statistical Mechanics, Academic Press, New York, 1982. · Zbl 0538.60093
[10] Matveev, V. B. and Smirnov, A. O.: Some comments on the solvable chiral Potts model,Lett. Math. Phys. 19 (1990), 179–185. · Zbl 0715.14024 · doi:10.1007/BF01039310
[11] Faddeev, L. D.: Current-like variables in massive and massless integrable models, Lectures delivered at the International School of Physics ’Enrico Fermi’, held in Villa Monastero, Varenna, Italy, 1994.
[12] Hirota, R.: Discrete analogue of a generalized Toda equation,J. Phys. Soc. Jpn. 50(11) (1981), 3785–3791. · doi:10.1143/JPSJ.50.3785
[13] Zamolodchikov, A. B.:Comm. Math. Phys. 79 (1981), 489–505. · doi:10.1007/BF01209309
[14] Sergeev, S. M.: private communication, 1995.
[15] Korepanov, I. G.: private communication, 1995.
[16] Kashaev, R. M. and Reshetikhin, N. Yu.: Affine Toda field theory as a 3-dimensional integrable system, Preprint ENSLAPP-L-548/95. · Zbl 1056.81066
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.