Superintegrable Model Research Articles

On KP-integrable skew Hurwitz τ-functions and their β-deformations

We extend the old formalism of cut-and-join operators in the theory of Hurwitz τ-functions to description of a wide family of KP-integrable skew Hurwitz τ-functions, which include, in particular, the newly discovered interpolating WLZZ models. Recently, the simplest of them was related to a superintegrable two-matrix model with two potentials and one external matrix field. Now we provide detailed proofs, and a generalization to a multi-matrix representation, and propose the β-deformation of the matrix model as well. The general interpolating WLZZ model is generated by a W-representation given by a sum of operators from a one-parametric commutative sub-family (a commutative subalgebra of w∞). Different commutative families are related by cut-and-join rotations. Two of these sub-families (‘vertical’ and ‘45-degree’) turn out to be nothing but the trigonometric and rational Calogero-Sutherland Hamiltonians, the ‘horizontal’ family is represented by simple derivatives. Other families require an additional analysis.

Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras

Starting from a purely algebraic procedure based on the commutant of a subalgebra in the universal enveloping algebra of a given Lie algebra, the notion of algebraic Hamiltonians and the constants of the motion generating a polynomial symmetry algebra is proposed. The case of the special linear Lie algebra is discussed in detail, where an explicit basis for the commutant with respect to the Cartan subalgebra is obtained, and the order of the polynomial algebra is computed. It is further shown that, with an appropriate realization of , this provides an explicit connection with the generic superintegrable model on the -dimensional sphere and the related Racah algebra R(n). In particular, we show explicitly how the models on the two-sphere and three-sphere and the associated symmetry algebras can be obtained from the quadratic and cubic polynomial algebras generated by the commutants defined in the enveloping algebra of and , respectively. The construction is performed in the classical (or Poisson-Lie) context, where the Berezin bracket replaces the commutator.

Embedding of the Racah algebra R(n) and superintegrability

The rank-1 Racah algebra R(3) plays a pivotal role in the theory of superintegrable systems. It appears as the symmetry algebra of the 3-parameter system on the 2-sphere from which all second-order conformally flat superintegrable models in 2D can be obtained by means of suitable limits and contractions. A higher rank generalisation of R(3), the so-called rank n−2 Racah algebra R(n), has been considered recently and showed to be the symmetry algebra of the general superintegrable model on the (n−1)-sphere. In the present work, we show that such an algebraic structure naturally arises as embedded inside a larger quadratic algebra characterising nD superintegrable models with non-central terms. This is shown both in classical and quantum mechanics through suitable (symplectic or differential) realisations of the Racah and additional generators. Among the main results, we present an explicit construction of the complete symmetry algebras for two families of n-dimensional maximally superintegrable models, the Smorodinsky–Winternitz system and the generalised Kepler–Coulomb system. For both families, the underlying symmetry algebras are higher-rank quadratic algebras containing the Racah algebra R(n) as subalgebra. High-order algebraic relations among the generators of the full quadratic algebras are also obtained both in the classical and quantum frameworks. These results should shed new light to the further understanding of the structures of quadratic algebras in the context of superintegrable systems.

Polynomial algebras from su(3) and a quadratically superintegrable model on the two sphere

Construction of superintegrable systems based on Lie algebras have been introduced over the years. However, these approaches depend on explicit realisations, for instance as a differential operators, of the underlying Lie algebra. This is also the case for the construction of their related symmetry algebra which take usually the form of a finitely generated quadratic algebra. These algebras often display structure constants which depend on the central elements and in particular on the Hamiltonian. In this paper, we develop a new approach reexamining the quadratically superintegrable system on the two-sphere for which a symmetry algebra is known to be the Racah algebra R(3). Such a model is related to the 59 two dimensional quadratically superintegrable systems on conformally flat spaces via contractions and limits. We demonstrate that using further polynomials of degree 2, 3 and 4 in the enveloping algebra of su(3) one can generate an algebra based only on abstract commutation relations of su(3) Lie algebra without explicit constraints on the representations or realisations. This construction relies on the maximal Abelian subalgebra, also called MASA, which are the Cartan generators and their commutant. We obtain a new six-dimensional cubic algebra where the structure constant are integer numbers which reduce from a quartic algebra for which the structure constant depend on the Cartan generator and the Casimir invariant. We also present other form of the symmetry algebra using the quadratic and cubic Casimir invariants of su(3). It reduces as the known quadratic Racah algebra R(3) only when using an explicit realization. This algebraic structure describes the symmetry of the quadratically superintegrable systems on the 2 sphere. We also present a contraction to another six-dimensional cubic algebra which would corresponding to the symmetry algebra of a Smorodinsky–Winternitz model.

Superintegrability and the dual [formula omitted] Hahn algebra in superconformal quantum mechanics

A two-dimensional superintegrable system of singular oscillators with internal degrees of freedom is identified and exactly solved. Its symmetry algebra is seen to be the dual −1 Hahn algebra which describes the bispectral properties of the polynomials with the same name that are essentially the Clebsch–Gordan coefficients of the superconformal algebra osp(1|2). It is also shown how this superintegrable model is obtained under dimensional reduction from a set of uncoupled harmonic oscillators in four dimensions.

The generalized Racah algebra as a commutant

The Racah algebra R(n) of rank (n − 2) is obtained as the commutant of the subalgebra of in oscillator representations of the universal algebra of . This result is shown to be related in a Howe duality context to the definition of R(n) as the algebra of Casimir operators arising in recouplings of n copies of . These observations provide a natural framework to carry out the derivation by dimensional reduction of the generic superintegrable model on the (n − 1) sphere which is invariant under R(n).

Bargmann and Barut-Girardello models for the Racah algebra

The Racah algebra and its higher rank extension are the algebras underlying the univariate and multivariate Racah polynomials. In this paper, we develop two new models in which the Racah algebra naturally arises as symmetry algebra, namely, the Bargmann model and the Barut-Girardello model. We show how both models are connected with the superintegrable model of Miller et al. The Bargmann model moreover leads to a new realization of the Racah algebra of rank n as n-variable differential operators. Our conceptual approach also allows us to rederive the basis functions of the superintegrable model without resorting to separation of variables.

The Racah algebra as a commutant and Howe duality

The Racah algebra encodes the bispectrality of the eponym polynomials. It is known to be the symmetry algebra of the generic superintegrable model on the -sphere. It is further identified as the commutant of the subalgebra of in oscillator representations of the universal algebra of the latter. How this observation relates to the Racah problem and the superintegrable model on the -sphere is discussed on the basis of the Howe duality associated to the pair .

A higher rank Racah algebra and the Laplace–Dunkl operator

A higher rank generalization of the (rank one) Racah algebra is obtained as the symmetry algebra of the Laplace–Dunkl operator associated to the root system. This algebra is also the invariance algebra of the generic superintegrable model on the n-sphere. Bases of Dunkl harmonics are constructed explicitly using a Cauchy–Kovalevskaia theorem. These bases consist of joint eigenfunctions of labelling Abelian subalgebras of the higher rank Racah algebra. A method to obtain expressions for both the connection coefficients between these bases and the action of the symmetries on these bases is presented.

Mathcal{P}mathcal{T} deformation of angular Calogero models

The rational Calogero model based on an arbitrary rank-n Coxeter root system is spherically reduced to a superintegrable angular model of a particle moving on Sn−1 subject to a very particular potential singular at the reflection hyperplanes. It is outlined how to find conserved charges and to construct intertwining operators. We deform these models in a mathcal{P}mathcal{T} -symmetric manner by judicious complex coordinate transformations, which render the potential less singular. The mathcal{P}mathcal{T} deformation does not change the energy eigenvalues but in some cases adds a previously unphysical tower of states. For integral couplings the new and old energy levels coincide, which roughly doubles the previous degeneracy and allows for a conserved nonlinear supersymmetry charge. We present the details for the generic rank-two (A2, G2) and all rank-three Coxeter systems (AD3, BC3 and H3), including a reducible case (A1⊗ 3).

A superintegrable model with reflections on Sn−1 and the higher rank Bannai–Ito algebra

A quantum superintegrable model with reflections on the (n − 1)-sphere is presented. Its symmetry algebra is identified with the higher rank generalization of the Bannai–Ito algebra. It is shown that the Hamiltonian of the system can be constructed from the tensor product of n representations of the superalgebra and that the superintegrability is naturally understood in that setting. The separated solutions are obtained through the Fischer decomposition and a Cauchy–Kovalevskaia extension theorem.

The tetrahexahedric angular Calogero model

We investigate the spherical reduction of the rational An−1 Calogero model. It defines a quantum superintegrable model of a particle trapped in one of n! spherical (n−2)-simplices on Sn−2 in a special Sn-symmetric potential. The energy levels (including degeneracy) and eigenstates are given, and the construction of conserved charges and Hamiltonian intertwiners outlined. We describe superintegrable complex PT deformations which remove the trapping walls and in some cases create a ℤ2-graded (“supersymmetric”) extension of the spectrum. Details are worked out for the cases of n=3 and n=4.

Quadratic algebra for superintegrable monopole system in a Taub-NUT space

We introduce a Hartmann system in the generalized Taub-NUT space with Abelian monopole interaction. This quantum system includes well known Kaluza-Klein monopole and MIC-Zwanziger monopole as special cases. It is shown that the corresponding Schrödinger equation of the Hamiltonian is separable in both spherical and parabolic coordinates. We obtain the integrals of motion of this superintegrable model and construct the quadratic algebra and Casimir operator. This algebra can be realized in terms of a deformed oscillator algebra and has finite dimensional unitary representations (unirreps) which provide energy spectra of the system. This result coincides with the physical spectra obtained from the separation of variables.

A SUPERINTEGRABLE MODEL WITH REFLECTIONS ON S^3 AND THE RANK TWO BANNAI-ITO ALGEBRA

A quantum superintegrable model with reflections on the three-sphere is presented. Its symmetry algebra is identified with the rank-two Bannai-Ito algebra. It is shown that the Hamiltonian of the system can be constructed from the tensor product of four representations of the superalgebra osp(1|2) and that the superintegrability is naturally understood in that setting. The exact separated solutions are obtained through the Fischer decomposition and a Cauchy-Kovalevskaia extension theorem.

A new family of N dimensional superintegrable double singular oscillators and quadratic algebra Q(3) ⨁ so(n) ⨁ so(N-n)

We introduce a new family of N dimensional quantum superintegrable models consisting of double singular oscillators of type (n, N-n). The special cases (2,2) and (4,4) have previously been identified as the duals of 3- and 5-dimensional deformed Kepler–Coulomb systems with u(1) and su(2) monopoles, respectively. The models are multiseparable and their wave functions are obtained in (n, N-n) double-hyperspherical coordinates. We obtain the integrals of motion and construct the finitely generated polynomial algebra that is the direct sum of a quadratic algebra Q(3) involving three generators, so(n), so(N-n) (i.e. Q(3) ⨁ so(n) ⨁ so(N-n)). The structure constants of the quadratic algebra itself involve the Casimir operators of the two Lie algebras so(n) and so(N-n). Moreover, we obtain the finite dimensional unitary representations (unirreps) of the quadratic algebra and present an algebraic derivation of the degenerate energy spectrum of the superintegrable model.

The Bannai-Ito algebra and some applications

The Bannai-Ito algebra is presented together with some of its applications. Its relations with the Bannai-Ito polynomials, the Racah problem for the sl−1(2) algebra, a superintegrable model with reflections and a Dirac-Dunkl equation on the 2-sphere are surveyed.

The Racah algebra and superintegrable models

The universal character of the Racah algebra will be illustrated by showing that it is at the center of the relations between the Racah polynomials, the recoupling of three su(1,1) representations and the symmetries of the generic second-order superintegrable model on the 2-sphere.

The Bannai–Ito algebra and a superintegrable system with reflections on the two-sphere

A quantum superintegrable model with reflections on the two-sphere is introduced. Its two algebraically independent constants of motion generate a central extension of the Bannai–Ito algebra. The Schrödinger equation separates in spherical coordinates and its exact solutions are presented. It is further observed that the Hamiltonian of the system arises in addition of three representations of the sl−1(2) algebra (the dynamical algebra of the one-dimensional parabosonic oscillator). The contraction from the two-sphere to the Euclidean plane yields the Dunkl oscillator in two dimensions and its Schwinger–Dunkl symmetry algebra sd(2).

A superintegrable finite oscillator in two dimensions with SU(2) symmetry

A superintegrable finite model of the quantum isotropic oscillator in two dimensions is introduced. It is defined on a uniform lattice of triangular shape. The constants of the motion for the model form an SU(2) symmetry algebra. It is found that the dynamical difference eigenvalue equation can be written in terms of creation and annihilation operators. The wavefunctions of the Hamiltonian are expressed in terms of two known families of bivariate Krawtchouk polynomials, those of Rahman and those of Tratnik. These polynomials form bases for SU(2) irreducible representations. It is further shown that the pair of eigenvalue equations for each of these families is related to each other by an SU(2) automorphism. A finite model of the anisotropic oscillator that has wavefunctions expressed in terms of the same Rahman polynomials is also introduced. In the continuum limit, when the number of grid points tends to infinity, standard two-dimensional harmonic oscillators are obtained. The analysis provides the N → ∞ limit of the bivariate Krawtchouk polynomials as a product of one-variable Hermite polynomials.

Eigenvectors in the superintegrable model I: generators

In order to calculate correlation functions of the chiral Potts model, one only needs to study the eigenvectors of the superintegrable model. Here we start this study by looking for eigenvectors of the transfer matrix of the periodic τ2(tq) model which commutes with the chiral Potts transfer matrix. We show that the degeneracy of the eigenspace of τ2(tq) in the Q = 0 sector is 2r, with r = (N − 1)L/N when the size of the transfer matrix L is a multiple of N. We introduce chiral Potts model operators, different from the more commonly used generators of quantum group . From these we can form the generators of a loop algebra . For this algebra, we then use the roots of the Drinfeld polynomial to give new explicit expressions for the generators representing the loop algebra as the direct sum of r copies of the simple algebra .