Abstract
We introduce a family of classical integrable systems describing dynamics of M interacting glN integrable tops. It extends the previously known model of interacting elliptic tops. Our construction is based on the GLNR-matrix satisfying the associative Yang-Baxter equation. The obtained systems can be considered as extensions of the spin type Calogero-Moser models with (the classical analogues of) anisotropic spin exchange operators given in terms of the R-matrix data. In N = 1 case the spin Calogero-Moser model is reproduced. Explicit expressions for glNM -valued Lax pair with spectral parameter and its classical dynamical r-matrix are obtained. Possible applications are briefly discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Aminov, S. Arthamonov, A. Smirnov and A. Zotov, Rational Top and its Classical R-matrix, J. Phys.A 47 (2014) 305207 [arXiv:1402.3189] [INSPIRE].
A. Antonov, K. Hasegawa and A. Zabrodin, On trigonometric intertwining vectors and nondynamical R matrix for the Ruijsenaars model, Nucl. Phys.B 503 (1997) 747 [hep-th/9704074] [INSPIRE].
V.I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier16 (1966) 319.
L.A. Dikii, Hamiltonian systems connected with the rotation group, Funct. Anal. Appl.6 (1972) 326.
S.V. Manakov, Note on the integration of Euler’s equations of the dynamics of an n-dimensional rigid body, Funct. Anal. Appl.10 (1976) 328.
A.S. Mishenko, Integral geodesics of a flow on Lie groups, Funct. Anal. Appl.4 (1970) 232.
A.S. Mishenko and A.T. Fomenko, Euler equation on finite-dimensional Lie groups, Math. USSR Izv.12 (1978) 371.
E. Billey, J. Avan and O. Babelon, The r matrix structure of the Euler-Calogero-Moser model, Phys. Lett.A 186 (1994) 114 [hep-th/9312042] [INSPIRE].
E. Billey, J. Avan and O. Babelon, Exact Yangian symmetry in the classical Euler-Calogero-Moser model, Phys. Lett.A 188 (1994) 263 [hep-th/9401117] [INSPIRE].
I. Krichever, O. Babelon, E. Billey and M. Talon, Spin generalization of the Calogero-Moser system and the Matrix KP equation, Amer. Math. Soc. Transl.170 (1995) 83.
R.J. Baxter, Partition function of the eight vertex lattice model, Annals Phys.70 (1972) 193 [INSPIRE].
A.A. Belavin, Dynamical Symmetry of Integrable Quantum Systems, Nucl. Phys.B 180 (1981) 189 [INSPIRE].
M.P. Richey and C.A. Tracy, ℤnBaxter model: Symmetries and the Belavin parametrization, J. Stat. Phys.42 (1986) 311.
M. Bertola, M. Cafasso and V. Roubtsov, Noncommutative Painlevé Equations and Systems of Calogero Type, Commun. Math. Phys.363 (2018) 503 [arXiv:1710.00736] [INSPIRE].
F. Calogero, Solution of a three-body problem in one-dimension, J. Math. Phys.10 (1969) 2191 [INSPIRE].
F. Calogero, Solution of the one-dimensional N body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys.12 (1971) 419 [INSPIRE].
B. Sutherland, Exact results for a quantum many body problem in one-dimension, Phys. Rev.A 4 (1971) 2019 [INSPIRE].
B. Sutherland, Exact results for a quantum many body problem in one-dimension. II, Phys. Rev.A 5 (1972) 1372 [INSPIRE].
J. Moser, Three integrable Hamiltonian systems connnected with isospectral deformations, Adv. Math.16 (1975) 197 [INSPIRE].
I.V. Cherednik, Relativistically Invariant Quasiclassical Limits of Integrable Two-dimensional Quantum Models, Theor. Math. Phys.47 (1981) 422 [INSPIRE].
E. Corrigan and R. Sasaki, Quantum versus classical integrability in Calogero-Moser systems, J. Phys.A 35 (2002) 7017 [hep-th/0204039] [INSPIRE].
P. Etingof and O. Schiffmann, Twisted traces of intertwiners for Kac-Moody algebras and classical dynamical r-matrices corresponding to generalized Belavin-Drinfeld triples, Math. Res. Lett.6 (1999) 593 [math.QA/9908115].
P. Etingof and O. Schiffmann, Lectures on the dynamical Yang-Baxter equations, in Quantum Groups and Lie Theory , London Mathematical Society Lecture Note Series, volume 290, Cambridge University Press, Cambridge U.K. (2001), pp. 89-129 [math.QA/9908064].
L. Feher and B.G. Pusztai, Generalizations of Felder’s elliptic dynamical r matrices associated with twisted loop algebras of selfdual Lie algebras, Nucl. Phys.B 621 (2002) 622 [math/0109132] [INSPIRE].
S. Fomin and A.N. Kirillov, Quadratic algebras, Dunkl elements, and Schubert calculus, in Advances in geometry , Progress in Mathematics Series, volume 172, Birkhäuser, Boston Massachusetts (1999), pp. 147-182.
A.N. Kirillov, On Some Quadratic Algebras I \( \frac{1}{2} \): Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials, SIGMA12 (2016) 002 [arXiv:1502.00426].
J. Gibbons and T. Hermsen, A generalization of the Calogero-Moser systems, PhysicaD 11 (1984) 337.
S. Wojciechowski, An integrable marriage of the Euler equations with the Calogero-Moser system, Phys. Lett.A 111 (1985) 101.
A. Grekov and A. Zotov, On R-matrix valued Lax pairs for Calogero-Moser models, J. Phys. A 51 (2018) 315202 [arXiv:1801.00245] [INSPIRE].
A. Zotov, Calogero-Moser Model and R-Matrix Identities, Theor. Math. Phys.197 (2018) 1755.
F.D.M. Haldane, Exact Jastrow-Gutzwiller resonating valence bond ground state of the spin 1/2 antiferromagnetic Heisenberg chain with 1/R2exchange, Phys. Rev. Lett.60 (1988) 635 [INSPIRE].
B.S. Shastry, Exact solution of an S = 1/2 Heisenberg antiferromagnetic chain with long ranged interactions, Phys. Rev. Lett.60 (1988) 639 [INSPIRE].
V.I. Inozemtsev, On the connection between the one-dimensional S = 1/2 Heisenberg chain and Haldane-Shastry model, J. Stat. Phys.59 (1990) 1143 [INSPIRE].
A.P. Polychronakos, Lattice integrable systems of Haldane-Shastry type, Phys. Rev. Lett.70 (1993) 2329 [hep-th/9210109] [INSPIRE].
K. Hikami and M. Wadati, Integrable spin-12 particle systems with long-range interactions Phys. Lett.A 173 (1993) 263.
T. Krasnov and A. Zotov, Trigonometric Integrable Tops from Solutions of Associative Yang-Baxter Equation, Ann. Henri Poincaré20 (2019) 2671 [arXiv:1812.04209] [INSPIRE].
T. Krasnov and A. Zotov, Trigonometric Integrable Tops from Solutions of Associative Yang-Baxter Equation, Ann. Henri Poincaré20 (2019) 2671 [arXiv:1812.04209] [INSPIRE].
I. Krichever, Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles, Funct. Anal. Appl.14 (1980) 282.
A. Levin, M. Olshanetsky and A. Zotov, Hitchin Systems — Symplectic Hecke Correspondence and Two-dimensional Version, Commun. Math. Phys.236 (2003) 93 [nlin/0110045] [INSPIRE].
A. Levin, M. Olshanetsky and A. Zotov, Relativistic Classical Integrable Tops and Quantum R-matrices, JHEP07 (2014) 012 [arXiv:1405.7523] [INSPIRE].
A. Levin, M. Olshanetsky and A. Zotov, Classical integrable systems and soliton equations related to eleven-vertex R-matrix, Nucl. Phys.B 887 (2014) 400 [arXiv:1406.2995] [INSPIRE].
A. Levin, M. Olshanetsky and A. Zotov, Planck Constant as Spectral Parameter in Integrable Systems and KZB Equations, JHEP10 (2014) 109 [arXiv:1408.6246] [INSPIRE].
A. Levin, M. Olshanetsky and A. Zotov, Quantum Baxter-Belavin R-matrices and multidimensional Lax pairs for Painlevé VI, Theor. Math. Phys.184 (2015) 924 [arXiv:1501.07351] [INSPIRE].
A. Levin, M. Olshanetsky and A. Zotov, Noncommutative extensions of elliptic integrable Euler-Arnold tops and Painlevé VI equation, J. Phys.A 49 (2016) 395202 [arXiv:1603.06101] [INSPIRE].
A. Levin, M. Olshanetsky, A. Smirnov and A. Zotov, Characteristic Classes and Integrable Systems. General Construction, Commun. Math. Phys.316 (2012) 1 [arXiv:1006.0702] [INSPIRE].
A. Levin, M. Olshanetsky, A. Smirnov and A. Zotov, Calogero-Moser systems for simple Lie groups and characteristic classes of bundles, J. Geom. Phys.62 (2012) 1810 [INSPIRE].
A. Levin, M. Olshanetsky, A. Smirnov and A. Zotov, Characteristic Classes and Integrable Systems for Simple Lie Groups, arXiv:1007.4127 [INSPIRE].
A. Levin, M. Olshanetsky, A. Smirnov and A. Zotov, Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-trivial Bundles, SIGMA8 (2012) 095 [arXiv:1207.4386] [INSPIRE].
A. Levin, M. Olshanetsky, A. Smirnov and A. Zotov, Characteristic Classes of SL(N)-Bundles and Quantum Dynamical Elliptic R-Matrices, J. Phys.A 46 (2013) 035201 [arXiv:1208.5750] [INSPIRE].
A. Mironov, A. Morozov, Y. Zenkevich and A. Zotov, Spectral Duality in Integrable Systems from AGT Conjecture, JETP Lett.97 (2013) 45 [arXiv:1204.0913] [INSPIRE].
A. Mironov, A. Morozov, B. Runov, Y. Zenkevich and A. Zotov, Spectral Duality Between Heisenberg Chain and Gaudin Model, Lett. Math. Phys.103 (2013) 299 [arXiv:1206.6349] [INSPIRE].
A. Mironov, A. Morozov, B. Runov, Y. Zenkevich and A. Zotov, Spectral dualities in XXZ spin chains and five dimensional gauge theories, JHEP12 (2013) 034 [arXiv:1307.1502] [INSPIRE].
N. Nekrasov, Holomorphic bundles and many body systems, Commun. Math. Phys.180 (1996) 587 [hep-th/9503157] [INSPIRE].
A. Polishchuk, Classical Yang-Baxter equation and the A∞-constraint, Adv. Math.168 (2002) 56.
A.P. Polychronakos, Calogero-Moser models with noncommutative spin interactions, Phys. Rev. Lett.89 (2002) 126403 [hep-th/0112141] [INSPIRE].
A.P. Polychronakos, Generalized Calogero models through reductions by discrete symmetries, Nucl. Phys.B 543 (1999) 485 [hep-th/9810211] [INSPIRE].
A.P. Polychronakos, Physics and Mathematics of Calogero particles, J. Phys.A 39 (2006) 12793 [hep-th/0607033] [INSPIRE].
T. Schedler, Trigonometric solutions of the associative Yang-Baxter equation, Math. Res. Lett.10 (2003) 301 [math.QA/0212258].
A. Polishchuk, Massey products on cycles of projective lines and trigonometric solutions of the Yang-Baxter equations, in Algebra, Arithmetic, and Geometry , Progress in Mathematics Series, volume 270, Birkhäuser, Boston Massachusetts U.S.A. (2010), pp. 573-617 [math.QA/0612761].
I. Sechin and A. Zotov, R-matrix-valued Lax pairs and long-range spin chains, Phys. Lett.B 781 (2018) 1 [arXiv:1801.08908] [INSPIRE].
E.K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation, Fuct. Anal. Appl.16 (1982) 263.
A.G. Reyman and M.A. Semenov-Tian-Shansky, Lie algebras and Lax equations with spectral parameter on an elliptic curve, Zap. Nauchn. Semin. LOMI150 (1986) 104.
A. Weil, Elliptic functions according to Eisenstein and Kronecker, Springer-Verlag (1976).
D. Mumford, Tata Lectures on Theta I, Birkhäuser, Boston Massachusetts U.S.A. (1983).
D. Mumford, Tata Lectures on Theta II. Jacobian theta functions and differential equations, Birkhäuser, Boston Massachusetts U.S.A. (1984).
A. Smirnov, Degenerate Sklyanin algebras, Central Eur. J. Phys.8 (2010) 542 [arXiv:0903.1466] [INSPIRE].
A. Zotov and A. Levin, Integrable Model of Interacting Elliptic Tops, Theor. Math. Phys.146 (2006) 45 [INSPIRE].
A. Zotov and A. Smirnov, Modifications of bundles, elliptic integrable systems, and related problems, Theor. Math. Phys.177 (2013) 1281.
A. Zotov, Relativistic elliptic matrix tops and finite Fourier transformations, Mod. Phys. Lett.A 32 (2017) 1750169 [arXiv:1706.05601] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1905.07820
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Grekov, A., Sechin, I. & Zotov, A. Generalized model of interacting integrable tops. J. High Energ. Phys. 2019, 81 (2019). https://doi.org/10.1007/JHEP10(2019)081
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2019)081