Supersymmetric Calogero–Moser–Sutherland models and Jack superpolynomials

Desrosiers, Patrick; Lapointe, Luc; Mathieu, Pierre

doi:10.1016/s0550-3213(01)00208-5

Cited by 57 publications

(157 citation statements)

References 60 publications

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“…This normalization ensures that the coefficient of the monomial θ {1,...,m} z Λ appearing in the expansion of m Λ is equal to 1. The supermonomial m (3,1;2,1) is given in (4) for N = 4.…”

Section: Basic Definitionsmentioning

confidence: 99%

“…A triangular decomposition, in terms of a superspace extension of the symmetric monomial functions (or supermonomials), must be imposed. Unique eigenfunctions, J Λ , defined according to such a triangular decomposition, were constructed in [4,5], and called Jack superpolynomials. They are indexed by superpartitions Λ = (Λ a ; Λ s ), composed of a partition Λ a with distinct parts, and a usual partition Λ s .…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, two distinct superpolynomials labeled by two different superpartitions built out of the same set of N integers (but distributed differently among the two partitions Λ a and Λ s ) have identical H n eigenvalues. As a result, the Jack superpolynomials J Λ of [4,5] are not orthogonal under the scalar product (19) with respect to which H n is self-adjoint.…”

Section: Introductionmentioning

confidence: 99%

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Jack Polynomials in Superspace

Desrosiers

Lapointe

Mathieu

2003

Commun. Math. Phys.

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113

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This work initiates the study of orthogonal symmetric polynomials in superspace. Here we present two approaches leading to a family of orthogonal polynomials in superspace that generalize the Jack polynomials. The first approach relies on previous work by the authors in which eigenfunctions of the supersymmetric extension of the trigonometric Calogero-Moser-Sutherland Hamiltonian were constructed. Orthogonal eigenfunctions are now obtained by diagonalizing the first nontrivial element of a bosonic tower of commuting conserved charges not containing this Hamiltonian. Quite remarkably, the expansion coefficients of these orthogonal eigenfunctions in the supermonomial basis are stable with respect to the number of variables. The second and more direct approach amounts to symmetrize products of non-symmetric Jack polynomials with monomials in the fermionic variables. This time, the orthogonality is inherited from the orthogonality of the non-symmetric Jack polynomials, and the value of the norm is given explicitly. *

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Section: Basic Definitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Jack Polynomials in Superspace

Desrosiers

Lapointe

Mathieu

2003

Commun. Math. Phys.

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113

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“…For this analysis, which is the content of section 2, we focus on the rational case. Our second goal is to provide a simple quasi-qualitative presentation of the main ideas underlying our recent construction of the supersymmetric trigonometric CMS (stCMS) eigenfunctions, that is, of the Jack superpolynomials [3,4,5]. This is the content of section 3.…”

Section: Introductionmentioning

confidence: 99%

“…On this space, the supersymmetric Hamiltonain reduces to an ordinary exchange-type Hamiltonian [3]. Stated differently: acting on the proper space, we can study many features of the supersymmetric model without even introducing fermionic degrees of freedom!…”

Section: Introductionmentioning

confidence: 99%

Supersymmetric Calogero-Moser-Sutherland models: Superintegrability structure and eigenfunctions

Desrosiers

Lapointe²,

Mathieu³

2004

Superintegrability in Classical and Quantum Systems

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We first review the construction of the supersymmetric extension of the quantum CalogeroMoser-Sutherland (CMS) models. We stress the remarkable fact that this extension is completely captured by the insertion of a fermionic exchange operator in the Hamiltonian: sCMS models (s for supersymmetric) are nothing but special exchange-type CMS models. Under the appropriate projection, the conserved charges can thus be formulated in terms of the standard Dunkl operators. This is illustrated in the rational case, where the explicit form of the 4N (N being the number of bosonic variables) conserved charges is presented, together with their full algebra. The existence of 2N commuting bosonic charges settles the question of the integrability of the srCMS model. We then prove its superintegrability by displaying 2N − 2 extra independent charges commuting with the Hamiltonian. In the second part, we consider the supersymmetric version of the trigonometric case (stCMS model) and review the construction of its eigenfunctions, the Jack superpolynomials. This leads to closed-form expressions, as determinants of determinants involving supermonomial symmetric functions. Here we focus on the main ideas and the generic aspects of the construction: those applicable to all models whether supersymmetric or not. Finally, the possible Lie superalgebraic structure underlying the stCMS model and its eigenfunctions is briefly considered.

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New $$ \mathcal{N} $$= 2 superspace Calogero models

Krivonos¹,

Lechtenfeld

Sutulin³

2020

J. High Energ. Phys.

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Starting from the Hamiltonian formulation of N = 2 supersymmetric Calogero models associated with the classical A n , B n , C n and D n series and their hyperbolic/trigonometric cousins, we provide their superspace description. The key ingredients include n bosonic and 2n(n−1) fermionic N = 2 superfields, the latter being subject to a nonlinear chirality constraint. This constraint has a universal form valid for all Calogero models. With its help we find more general supercharges (and a superspace Lagrangian), which provide the N = 2 supersymmetrization for bosonic potentials with arbitrary repulsive two-body interactions.

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Supersymmetric Calogero–Moser–Sutherland models and Jack superpolynomials

Cited by 57 publications

References 60 publications

Jack Polynomials in Superspace

Jack Polynomials in Superspace

Supersymmetric Calogero-Moser-Sutherland models: Superintegrability structure and eigenfunctions

New $$ \mathcal{N} $$= 2 superspace Calogero models

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