The spherical sector of the Calogero model as a reduced matrix model

Being introduced four decades ago [1], it continues to attract much interest due to its rich internal structure and numerous applications. So far, various integrable extensions have been constructed and studied, in particular, for the trigonometric potentials [2], for particles with spins [3], for supersymmetric systems [4], and for other Lie algebras [5]. An important feature of all rational Calogero models is the dynamical conformal symmetry so(1, 2) ≡ sl(2, R), defined by the Hamiltonian (1) together with the dilatation D = i p i q i and conformal boost K = i q 2 i /2 generators, {H, D} = 2H, {K, D} = -2K, {H, K} = D.

Due to this symmetry one can give an elegant explanation of the superintegrability property of the conformal invariant integrable systems [6] (initially observed by Wojcechowski in Calogero model [7]). The “radial” and “spherical” parts of rational Calogero models can be separated in the Hamiltonian

with the “spherical part” corresponding to the Casimir element of the conformal algebra. Hence, the whole information about the conformal mechanics is encoded in its “spherical part”, given by the Hamiltonian system

This system is of its own interest since it describes a multi-center generalization of the (N -1)-dimensional Higgs oscillator [8]. In the quantum case and for special discrete values of the coupling constant it can be mapped to free-particle systems on the sphere [9]. However, the connection between the constants of motion of the initial conformal mechanics and its spherical sector is highly complicated [6,10]. In particular, it is unclear up to now how to construct the Liouville constants of motion of the spherical sector from the ones of the full conformal mechanics.

On the other hand, the rational Calogero model can be easily constructed from the free Hermitian matrix model via a Hamiltonian reduction [11]. In this way, we get a transparent explanation of its integrability property and the Lax pair formulation. Hence, it is natural to try to explore the matrix origin of the spherical sector of the Calogero model in order to find the matrix-model origin of its constants of motion. In that case we shall immediately get the constants of motion of the spin-Calogero model as well. One may expect that the matrix model formulation of the spherical sector of the Calogero model can simplify the study of its constants of motion. Moreover, such a formulation, being purely algebraic, might establish new relations between the spherical sector of the Calogero model and other algebraic integrable systems, for instance lattice spin systems.

The investigation of the spherical sector of the rational Calogero model and of its constants of motion at the matrix-model level is the goal of the present paper. First we recall the formulation of the Liouville integrals of the initial Calogero model at the matrix-model level in terms of U (N )-invariant polynomials (and of SU (N )-invariant polynomials for the Calogero model with the center of mass excluded) corresponding to the highest states of the conformal algebra. Then we observe that the constants of motion of the spherical system are described by SU (N ) × SL(2, R) singlets. This allows us to reduce the study of the algebra of invariants of the spherical sector to a purely algebraic computation of SU (N ) invariant tensors. To simplify the calculations, we develop an appropriate diagrammatic technique illustrated by numerous examples. We present explicit expressions for all functionally independent constants of motion up to sixth order in momenta as well as recover the results obtained in [10] by the use of standard methods. Finally, we establish a relation of the developed diagrammatic technique with the valence-bond basis introduced by Temperley and Lieb [12].

The paper is arranged as follows. In Section 2 we give a brief description of the matrix-model formulation of the rational Calogero model, including the description of the reduction procedure and the exclusion of the center of mass at the matrix-model frame. Then we develop a similar formulation for the spherical sector of the Calogero model. In Section 3 we develop the diagrammatic technique for the formulation of the constants of motion of the spherical sector of the Calogero model and find by its use all functionally independent constants of motion up to sixth order in momenta. In Section 4, considering the free-particle limit, we rederive, by the use of our technique, the constants of motion obtained in [10] by standard methods. In Section 5 we establish a correspondence between our technique and the valence-bond basis developed by Temperley and Lieb. An interesting future task concerns the relation of the symmetries of the spherical sector of the Calogero model with W -and Hecke algebras.

We recall that the Calogero model (1) has N Liouville constants of motion [13][14][15]

given in terms of the Lax m