Finding and solving Calogero–Moser type systems using Yang–Mills gauge theories

Blom, Jonas; Langmann, Edwin

doi:10.1016/s0550-3213(99)00550-7

Cited by 13 publications

(30 citation statements)

References 16 publications

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“…It is a system of 'Sutherland type' since the interaction exhibits trigonometric dependence on the coordinates, which means that the 'particles' move on the circle. For G = su(r + 1), the explicit formula (3.46) reproduces precisely the integrable spin Sutherland Hamiltonian obtained previously by Blom and Langmann [18,19] and by Polychronakos [20] by means of different methods, and generalizes their system for arbitrary simple Lie algebras.…”

Section: The Examples Associated With γ = Id Gsupporting

confidence: 76%

“…Our present work was motivated mainly by questions stemming from the papers of Blom and Langmann [18,19] and Polychronakos [20], where a family of generalized spin Sutherland systems was derived by means of two different methods. In these systems the particle positions are coupled to spin variables living on N arbitrary coadjoint orbits of SU(n), and the Hamiltonian also involves N arbitrary scalar parameters, for any integers n > 1 and N > 1.…”

Section: Introductionmentioning

confidence: 99%

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Generalized spin Sutherland systems revisited

Fehér

Pusztai

2015

Nuclear Physics B

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We present generalizations of the spin Sutherland systems obtained earlier by Blom and Langmann and by Polychronakos in two different ways: from SU(n) Yang--Mills theory on the cylinder and by constraining geodesic motion on the N-fold direct product of SU(n) with itself, for any N>1. Our systems are in correspondence with the Dynkin diagram automorphisms of arbitrary connected and simply connected compact simple Lie groups. We give a finite-dimensional as well as an infinite-dimensional derivation and shed light on the mechanism whereby they lead to the same classical integrable systems. The infinite-dimensional approach, based on twisted current algebras (alias Yang--Mills with twisted boundary conditions), was inspired by the derivation of the spinless Sutherland model due to Gorsky and Nekrasov. The finite-dimensional method relies on Hamiltonian reduction under twisted conjugations of N-fold direct product groups, linking the quantum mechanics of the reduced systems to representation theory similarly as was explored previously in the N=1 case.Comment: 21 page

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Section: The Examples Associated With γ = Id Gsupporting

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Generalized spin Sutherland systems revisited

Fehér

Pusztai

2015

Nuclear Physics B

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“…In the simplest case τ = id and θ is the cyclic permutation automorphism. We have inspected this case by taking an arbitrary simple Lie algebra for A and taking K as the diagonal embedding of a Cartan subalgebra of A into G. Then it turned out that the dynamical r-matrix construction reproduces (for A = A n ) certain generalized spin Calogero models found earlier in [32,33] by different methods. These examples and their generalizations for affine Lie algebras will be further studied elsewhere.…”

Section: Discussionmentioning

confidence: 99%

Spin Calogero models obtained from dynamical r-matrices and geodesic motion

Fehér

Pusztai

2006

Nuclear Physics B

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We study classical integrable systems based on the Alekseev-Meinrenken dynamical r-matrices corresponding to automorphisms of self-dual Lie algebras, G. We prove that these r-matrices are uniquely characterized by a non-degeneracy property and apply a construction due to Li and Xu to associate spin Calogero type models with them. The equation of motion of any model of this type is found to be a projection of the natural geodesic equation on a Lie group G with Lie algebra G, and its phase space is interpreted as a Hamiltonian reduction of an open submanifold of the cotangent bundle T * G, using the symmetry arising from the adjoint action of G twisted by the underlying automorphism. This shows the integrability of the resulting systems and gives an algorithm to solve them. As illustrative examples we present new models built on the involutive diagram automorphisms of the real split and compact simple Lie algebras, and also explain that many further examples fit in the dynamical r-matrix framework. IntroductionThe study of integrable many body systems initiated by Calogero [1], Sutherland [2] and Moser [3] is popular since these systems have interesting applications in several branches of theoretical physics and involve fascinating mathematics. See, for example, the reviews in [4,5,6,7]. The generalized Calogero models associated to root systems by Olshanetsky and Perelomov [8], the spin Calogero models due to Gibbons and Hermsen [9], and the Ruijsenaars-Schneider models [10] are also very important. Many approaches were applied to prove the classical integrability of these models. The most general among these are the construction of a Lax representation for the equation of motion and the realization of the system as a 'projection' of some other system that is integrable 'obviously', for example the free geodesic motion on a Lie group or on a symmetric space (see [3,11,12] and the review [4]). The result of Babelon and Viallet [13] linking Liouville integrability to the form of the Poisson brackets (PBs) of the Lax matrix is also relevant for us, since the determination of the r-matrices encoding these PBs in Calogero models led eventually to the questions investigated in this paper.The r-matrices of the A n type spinless Calogero models found in [14,15,16] depend on the coordinates of the particles and are not unitary. They were re-derived by Avan, Babelon and Billey in [17] using Hamiltonian reduction of an auxiliary non-integrable spin extension of the Calogero model. The definition of the non-integrable model relies on a 'quasi-Lax operator', whose PBs (of the type (2.10) below) are encoded by certain unitary dynamical r-matrices that yield the Calogero r-matrices upon reduction. The unitary r-matrices in question were later recognized [18] to be identical to the fundamental solutions of the so-called classical dynamical Yang-Baxter equation (CDYBE) that appeared first in studies of the WZNW conformal field theory [19,20]. Subsequently, the geometric meaning of the CDYBE and the classification of its solutions ...

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“…This latter equation is gauge invariant if the matrix J = J(t) introduced here transforms under gauge transformations as J →J = R −1 J R. It turns indeed out [1] [11] that if one makes the assignment…”

Section: B Gauge Theory Approachmentioning

confidence: 99%

Goldfishing by gauge theory

Calogero

Langmann

2006

Journal of Mathematical Physics

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A new solvable many-body problem of goldfish type is identified and used to revisit the connection among two different approaches to solvable dynamical systems. An isochronous variant of this model is identified and investigated. Alternative versions of these models are presented. The behavior of the alternative isochronous model near its equilibrium configurations is investigated, and a remarkable Diophantine result, as well as related Diophantine conjectures, are thereby obtained.1

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Finding and solving Calogero–Moser type systems using Yang–Mills gauge theories

Cited by 13 publications

References 16 publications

Generalized spin Sutherland systems revisited

Generalized spin Sutherland systems revisited

Spin Calogero models obtained from dynamical r-matrices and geodesic motion

Goldfishing by gauge theory

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