Hidden symmetries of rationally deformed superconformal mechanics

2019

Phys. Rev. D

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We study the Klein four-group (K 4 ) symmetry of the time-dependent Schrödinger equation for the conformal mechanics model of de Alfaro-Fubini-Furlan (AFF) with confining harmonic potential and coupling constant g = ν(ν + 1) ≥ −1/4. We show that it undergoes a complete or partial (at half-integer ν) breaking on eigenstates of the system, and is the automorphism of the osp(2, 2) superconformal symmetry in super-extensions of the model by inducing a transformation between the exact and spontaneously broken phases of N = 2 Poincaré supersymmetry. We exploit the K 4 symmetry and its relation with the conformal symmetry to construct the dual Darboux transformations which generate spectrally shifted pairs of the rationally deformed AFF models. Two distinct pairs of intertwining operators originated from Darboux duality allow us to construct complete sets of the spectrum generating ladder operators that identify specific finite-gap structure of a deformed system and generate three distinct related versions of nonlinearly deformed sl(2, R) algebra as its symmetry. We show that at half-integer ν, the Jordan states associated with confluent Darboux transformations enter the construction, and the spectrum of rationally deformed AFF systems undergoes structural changes.constant g = ν(ν + 1) ≥ −1/4 1 . Its non-relativistic conformal symmetry and supersymmetric extensions [5,6,7,8,9,10] find a variety of interesting applications including the particles dynamics in black hole backgrounds [11,12,13,14,15,16], cosmology [17,18,19], non-relativistic AdS/CFT correspondence [20,21,22,23,24], QCD confinement problem [25,26], and physics of Bose-Einstein condensates [27,28].On the other hand, the time-dependent Schrödinger equation for the AFF conformal mechanics model reveals a discrete Klein four-group symmetry generated by transformation of the parameter ν → −ν − 1, and by the spatial Wick rotation x → ix accompanied by the time reflection t → −t. In the picture of the stationary Schrödinger equation the time reflection transforms into the change of the eigenvalue's sign E → −E. The discrete symmetry K 4 , however, turns out to be completely broken at the level of the quantum states when ν is not a half-integer number : application of the group generators to physical eigenstates produces formal eigenstates which do not satisfy the necessary boundary conditions. In the case of half-integer values of the parameter ν the K 4 discrete symmetry breaks partially, and transformation ν → −ν − 1, as we shall see, turns out to be a true symmetry nontrivially realized on the spectrum of the system. In the physics of anyons, where the AFF model is used to generate the transmutation of statistics, halfinteger values of ν correspond to the two-particle system of identical fermions [29,30,31]. In the context of the problem we consider here, even though the new solutions with arbitrary value of ν generated by transformations of the discrete group are not acceptable from the physical point of view, the analogs of such non-physical states in other q...

Section: Introductionsupporting

confidence: 80%

Klein four-group and Darboux duality in conformal mechanics

Inzunza

2019

Phys. Rev. D

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“…In the second part, we added spin degrees of freedom by introducing a spin-orbit coupling of a special, unique form that guarantees a very peculiar degeneracy of energy levels and gives rise to the superconformal osp(2|2) symmetry. By two different dimensional reduction schemes this three-dimensional supersymmetric system produces the one-dimensional superconformal extensions of the AFF model [37] in unbroken and spontaneously broken phases of N = 2 Poincaré supersymmetry [17].…”

Section: Discussion and Outlookmentioning

confidence: 99%

“…The system with spin degrees of freedom gives rise to an N = 2 supersymmetric system characterized by the osp(2|2) superconformal symmetry. Applying two different dimensional reduction schemes to the obtained superconformal system produces in one case the one-dimensional superconformal extension of the AFF model with harmonic trap in the phase of the unbroken N = 2 Poincaré supersymmetry, while in the other case gives us the same system but in the spontaneously broken phase [65,16,17,63]. In this context, it would be interesting to look for three-dimensional generalizations of the one-dimensional rationally deformed superconformal systems constructed recently in [16,17,63] by using dual Darboux transformations.…”

Section: Discussion and Outlookmentioning

confidence: 99%

“…For more details see refs. [16,17,63]. The independence and dependence of energy levels on ℓ is reminiscent of two subsets of states in our three-dimensional system with infinitely degenerate energy levels due to their independence on the quantum number j and finitely degenerate, depending on j energy eigenvalues.…”

Section: Dimensional Reductionmentioning

confidence: 98%

“…The (anti)-commutators not displayed here do vanish. This superalgebra is identified as the osp(2|2) superconformal symmetry which appears in systems like one-dimensional harmonic super-oscillator or the superconformal mechanics model with a confining term [65,16,17,63]. Therefore our construction maybe considered as generalization of the three-dimensional versions of these systems in the monopole background.…”

Section: The Osp(2|2) Superconformal Extensionmentioning

confidence: 99%

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Hidden symmetry and (super)conformal mechanics in a monopole background

Inzunza

Wipf

2020

J. High Energ. Phys.

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We study classical and quantum hidden symmetries of a particle with electric charge e in the background of a Dirac monopole of magnetic charge g subjected to an additional central potential V (r) = U (r) + (eg) 2 /2mr 2 with U (r) = 1 2 mω 2 r 2 , similar to that in the one-dimensional conformal mechanics model of de Alfaro, Fubini and Furlan (AFF). By means of a non-unitary conformal bridge transformation, we establish a relation of the quantum states and of all symmetries of the system with those of the system without harmonic trap, U (r) = 0. Introducing spin degrees of freedom via a very special spin-orbit coupling, we construct the osp(2|2) superconformal extension of the system with unbroken N = 2 Poincaré supersymmetry and show that two different superconformal extensions of the one-dimensional AFF model with unbroken and spontaneously broken supersymmetry have a common origin. We also show a universal relationship between the dynamics of a Euclidean particle in an arbitrary central potential U (r) and the dynamics of a charged particle in a monopole background subjected to the potential V (r).To solve the vector equation in (2.4) in the case J 2 > ν 2 , which we will assume in what follows, we decompose n into the component parallel to the angular momentum and the orthogonal component,whereĴ is the unit vector in the direction of J . Since the parallel component n is constant, we conclude that the orthogonal component n ⊥ (t) has constant length and thus describes a circle in the plane orthogonal to J , n ⊥ (t) = n ⊥ (0) cos ϕ(t) +Ĵ × n ⊥ (0) sin ϕ(t) .(2.15) Here, θ, ϕ and ψ are the Euler angles, 0 ≤ ϕ, ψ < 2π, 0 ≤ θ < π, and J i J i = K a K a = J 2 . The common eigenstates of J 2 , J 3 and K 3 satisfying relations

Nonlinear Supersymmetry as a Hidden Symmetry

Integrability, Supersymmetry and Coherent States

2019

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Nonlinear supersymmetry is characterized by supercharges to be higher order in bosonic momenta of a system, and thus has a nature of a hidden symmetry. We review some aspects of nonlinear supersymmetry and related to it exotic supersymmetry and nonlinear superconformal symmetry. Examples of reflectionless, finite-gap and perfectly invisible PT -symmetric zero-gap systems, as well as rational deformations of the quantum harmonic oscillator and conformal mechanics, are considered, in which such symmetries are realized.