The full phase space of a model in the Calogero–Ruijsenaars family

Fehér, L.; Görbe, Tamás

doi:10.1016/j.geomphys.2016.04.018

Cited by 8 publications

(6 citation statements)

References 21 publications

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“…We arrive to this expression for I ± k through the following chain of arguments. It is known that the Calogero-Moser-Sutherland models and their RS generalisations can be obtained at the classical level through the hamiltonian or Poisson reduction applied to a system exhibiting free motion on one of the suitably chosen initial finite-or infinite-dimensional phase spaces [16]- [23], [3][4][5][6]. For instance, the RS model with the rational potential is obtained by the hamiltonian reduction of the cotangent bundle T * G = G ⋉ G , where G ia Lie group and G is its Lie algebra.…”

Section: Introductionmentioning

confidence: 99%

“…The Ruijsenaars-Schneider (RS) models [1,2] continue to provide an outstanding theoretical laboratory for the study of various aspects of Liouville integrability, both at the classical and quantum level, see, for instance, [5][6][7][8][9][10]. Also, new interesting applications of these type of models were recently found in conformal field theories [11].…”

Section: Introductionmentioning

confidence: 99%

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Quantum trace formulae for the integrals of the hyperbolic Ruijsenaars-Schneider model

Arutyunov

Klabbers

Olivucci

2019

J. High Energ. Phys.

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We conjecture the quantum analogues of the classical trace formulae for the integrals of motion of the quantum hyperbolic Ruijsenaars-Schneider model. This is done by departing from the classical construction where the corresponding model is obtained from the Heisenberg double by the Poisson reduction procedure. We also discuss some algebraic structures associated to the Lax matrix in the classical and quantum theory which arise upon introduction of the spectral parameter.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quantum trace formulae for the integrals of the hyperbolic Ruijsenaars-Schneider model

Arutyunov

Klabbers

Olivucci

2019

J. High Energ. Phys.

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“…However, if we leave the type A n , we can observe that until recent works by Fehér and collaborators (see e.g. [FG,FK2,FK3,FK4,FM] mostly in the real case), integrable systems in the trigonometric RS family are generally devised using only a suitable Lax matrix, as they originally appear in [RS], without geometric perspectives.…”

Section: Introductionmentioning

confidence: 99%

Spin versions of the complex trigonometric Ruijsenaars–Schneider model from cyclic quivers

Fairon

2019

Journal of Integrable Systems

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We study multiplicative quiver varieties associated to specific extensions of cyclic quivers with m ≥ 2 vertices. Their global Poisson structure is characterised by quasi-Hamiltonian algebras related to these quivers, which were studied by Van den Bergh for an arbitrary quiver. We show that the spaces are generically isomorphic to the case m = 1 corresponding to an extended Jordan quiver. This provides a set of local coordinates, which we use to interpret integrable systems as spin variants of the trigonometric Ruijsenaars-Schneider system. This generalises to new spin cases recent works on classical integrable systems in the Ruijsenaars-Schneider family.

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“…The main difference is that the reduction is performed on a space which may not be symplectic, and the moment map takes values in the Lie group rather than the Lie algebra. Not attempting at a comprehensive review, we refer the reader to some of the more recent papers [P,FK1,FK2,FK3,M,FG], where the Ruijsenaars-Schneider model and its variants are treated by the method of (quasi-)Hamiltonian reduction, and where further references can be found. Let us also mention an alternative geometric approach to many-body problems by Krichever [Kr1], in which the Lax matrix structure play the central role instead, and the Hamiltonian picture is derived from that, cf.…”

Section: Introductionmentioning

confidence: 99%

Multiplicative quiver varieties and generalised Ruijsenaars–Schneider models

Chalykh

Fairon

2017

Journal of Geometry and Physics

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We study some classical integrable systems naturally associated with multiplicative quiver varieties for the (extended) cyclic quiver with m vertices. The phase space of our integrable systems is obtained by quasi-Hamiltonian reduction from the space of representations of the quiver. Three families of Poisson-commuting functions are constructed and written explicitly in suitable Darboux coordinates. The case m = 1 corresponds to the tadpole quiver and the Ruijsenaars-Schneider system and its variants, while for m > 1 we obtain new integrable systems that generalise the Ruijsenaars-Schneider system. These systems and their quantum versions also appeared recently in the context of supersymmetric gauge theory and cyclotomic DAHAs [BEF, BFN1, BFN2, KN], as well as in the context of the Macdonald theory [CE]. of the double affine Hecke algebra (DAHA) in type A. Inside the cyclotomic DAHA there are three natural commutative subalgebras and they give rise to quantum integrable systems, in the same way as the usual DAHA can be used to produce the Macdonald-Ruijsenaars operators. The classical versions of these systems correspond to the q = 1 limit of the cyclotomic DAHA (cf.[O] for the case of the usual DAHA), and this leads to the multiplicative quiver varieties for the cyclic quiver. Thus, the integrable systems constructed in [BEF] coincide (on the classical level) with those constructed by us. The interpretation of these integrbale systems via the cyclotomic DAHA in [BEF] allows to explain their relationship to the twisted Macodnald-Ruijsenaars systems from [CE] in type A. Our methods are quite different in comparison, and they allow us to find explicit formulas for the corresponding classical Hamiltonians and integrate the Hamiltonian flows (the approach via the cyclotomic DAHA in [BEF] is less explicit). Curiously, these Hamiltonians become much simpler under the Cherednik-Fourier transform. In this form they appeared in the work of Braverman, Finkelberg, and Nakajima [BFN1, BFN2] on the quantized Coulomb branch of quiver gauge theories, see also a related work of Kodera and Nakajima [KN]. This can also be seen from our formulas at the classical level, when the Cheredink-Fourier transform becomes the angle-action transform studied by Ruijsenaars [R]. See Section § 5 below for more details. Apart from being more explicit compared to [BEF], our approach also has an advantage of being better suited for studying spin versions of the Ruisjenaars-Scheider system and its generalisations; this will be a subject of a future work.The structure of the paper is as follows. In Section § 2 we first describe the general formalism of double Poisson brackets and quasi-Poisson algebras due to Van den Bergh [VdB1], and then exemplify it for the multiplicative quiver varieties. Section § 3 looks at the tadpole quiver, explaining how to obtain the hyperbolic Ruijsenaars-Schneider system by quasi-Hamiltonian reduction. In Section § 4 we consider the multiplicative quiver varieties (Calogero-Moser spaces) for the framed cyclic quiver wi...

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The full phase space of a model in the Calogero–Ruijsenaars family

Cited by 8 publications

References 21 publications

Quantum trace formulae for the integrals of the hyperbolic Ruijsenaars-Schneider model

Quantum trace formulae for the integrals of the hyperbolic Ruijsenaars-Schneider model

Spin versions of the complex trigonometric Ruijsenaars–Schneider model from cyclic quivers

Multiplicative quiver varieties and generalised Ruijsenaars–Schneider models

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