It is long known that quantum Calogero models feature intertwining operators, which increase or decrease the coupling constant by an integer amount, for any fixed number of particles. We name these as “horizontal” and construct “vertical” intertwiners, which $\textit{change the number of interacting particles}$ for a fixed but integer value of the coupling constant. The emerging structure of a grid of intertwiners exists only in the algebraically integrable situation (integer coupling) and allows one to obtain each Liouville charge from the free power sum in the particle momenta by iterated intertwining either horizontally or vertically. We present recursion formulæ for the intertwiners as a factorization problem for partial differential operators and prove their existence for small values of particle number and coupling. As a byproduct, a new basis of non-symmetric Liouville integrals appears, algebraically related to the standard symmetric one.

Chile

Francisco Correa

Luis Inzunza

Germany

Olaf Lechtenfeld

Coupling and particle number intertwiners in the Calogero model

Supersymmetric extensions of the 1D and 2D Swanson models are investigated by applying the conformal bridge transformation (CBT) to the first order Berry-Keating Hamiltonian multiplied by i and its conformally neutral enlargements. The CBT plays the role of the Dyson map that transforms the models into supersymmetric generalizations of the 1D and 2D harmonic oscillator systems, allowing us to define pseudo-Hermitian conjugation and a suitable inner product. In the 1D case, we construct a   <math> <mi>PT</mi> </math>  $$ \mathcal{PT} $$ -invariant supersymmetric model with N subsystems by using the conformal generators of supersymmetric free particle, and identify its complete set of the true bosonic and fermionic integrals of motion. We also investigate an exotic N = 2 supersymmetric generalization, in which the higher order supercharges generate nonlinear superalgebras. We generalize the construction for the 2D case to obtain the   <math> <mi>PT</mi> </math>  $$ \mathcal{PT} $$ -invariant supersymmetric systems that transform into the spin-1/2 Landau problem with and without an additional Aharonov-Bohm flux, where in the latter case, the well-defined integrals of motion appear only when the flux is quantized. We also build a 2D supersymmetric Hamiltonian related to the “exotic rotational invariant harmonic oscillator” system governed by a dynamical parameter γ. The bosonic and fermionic hidden symmetries for this model are shown to exist for rational values of γ.

Mikhail Plyushchay

Conformal bridge transformation,   <math> <mi>PT</mi> </math>  $$ \mathcal{PT} $$ - and supersymmetry

Non-relativistic conformally invariant systems in a rotating cosmic string (conical) spacetime are analyzed at the classical and quantum levels by means of the gravitoelectromagnetic interpretation of the background. Solutions of the equations of motion are found by employing a local canonical transformation, that leads to their natural interpretation in terms of Riemann surfaces. The cone parameter α and the angular velocity Ω of the background determine the existence of hidden symmetries. Globally defined higher order integrals associated with perihelion of geodesic orbits appear at rational values of α. For the harmonic oscillator system with frequency ω, the integrals responsible for the trajectory closure arise only for rational values of α and |γ| = |Ω/ω|, with |γ| = 1 corresponding to the Landau problem. We face a quantum anomaly problem since the hidden symmetry operators can only be constructed when α is integer. Such operators are non-local in the case of the free particle. For the harmonic oscillator, the symmetry generators are obtained with the help of the conformal bridge transformation. We also study a multi-particle version of the harmonic oscillator system with |γ| = 1 using the mean-field theory and find that the emerging vortex structure respects a singular point of the background.

Dynamics, symmetries, anomaly and vortices in a rotating cosmic string background

Hidden symmetries of non-relativistic   <math> <mi>so</mi> <mfenced> <mn>2</mn> <mn>1</mn> </mfenced> <mo>≅</mo> <mi>sl</mi> <mfenced> <mn>2</mn> <mi>ℝ</mi> </mfenced> </math>  $$ \mathfrak{so}\left(2,1\right)\cong \mathfrak{sl}\left(2,\mathrm{\mathbb{R}}\right) $$  invariant systems in a cosmic string background are studied using the conformal bridge transformation. Geometric properties of this background are analogous to those of a conical surface with a deficiency/excess angle encoded in the “geometrical parameter” α, determined by the linear positive/negative mass density of the string. The free particle and the harmonic oscillator on this background are shown to be related by the conformal bridge transformation. To identify the integrals of the free system, we employ a local canonical transformation that relates the model with its planar version. The conformal bridge transformation is then used to map the obtained integrals to those of the harmonic oscillator on the cone. Well-defined classical integrals in both models exist only at α = q/k with q, k = 1, 2, . . ., which for q &gt; 1 are higher-order generators of finite nonlinear algebras. The systems are quantized for arbitrary values of α; however, the well-defined hidden symmetry operators associated with spectral degeneracies only exist when α is an integer, that reveals a quantum anomaly.

Conformal bridge in a cosmic string background

We construct a nonunitary transformation that relates a given “asymptotically free” conformal quantum mechanical system <math><msub><mi>H</mi><mi>f</mi></msub></math> with its confined, harmonically trapped version <math><msub><mi>H</mi><mi>c</mi></msub></math>. In our construction, Jordan states corresponding to the zero eigenvalue of <math><msub><mi>H</mi><mi>f</mi></msub></math>, as well as its eigenstates and Gaussian packets, are mapped into the eigenstates, coherent states, and squeezed states of <math><msub><mi>H</mi><mi>c</mi></msub></math>, respectively. The transformation is an automorphism of the conformal <math><mrow><mrow><mi>s</mi><mi>l</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>R</mi><mo>)</mo></mrow></math> algebra of the nature of the fourth-order root of the identity transformation, to which a complex canonical transformation corresponds on the classical level being the fourth-order root of the spatial reflection. We investigate the one- and two-dimensional examples that reveal, in particular, a curious relation between the two-dimensional free particle and the Landau problem.

Mikhail S. Plyushchay

Andreas Wipf

Conformal bridge between asymptotic freedom and confinement

We study classical and quantum hidden symmetries of a particle with electric charge in the background of a Dirac monopole of magnetic charge subjected to an additional central potential ( ) = ( ) + ( ) with ( ) = $$ \frac{1}{2} $$ , 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, ( ) = 0. Introducing spin degrees of freedom via a very special spin-orbit coupling, we construct the $$ \mathfrak{sop} $$ (2 2) superconformal extension of the system with unbroken $$ \mathcal{N} $$ = 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 ( ) and the dynamics of a charged particle in a monopole background subjected to the potential ( ).

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

We study the Klein four-group (<math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></math>) 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 <math><mi>g</mi><mo>=</mo><mi>ν</mi><mo>(</mo><mi>ν</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>≥</mo><mo>−</mo><mn>1</mn><mo>/</mo><mn>4</mn></math>. We show that it undergoes a complete or partial (at half-integer <math><mi>ν</mi></math>) breaking on eigenstates of the system, and is the automorphism of the <math><mrow><mi>osp</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math> superconformal symmetry in super-extensions of the model by inducing a transformation between the exact and spontaneously broken phases of <math><mi>N</mi><mo>=</mo><mn>2</mn></math> Poincaré supersymmetry. We exploit the <math><msub><mi>K</mi><mn>4</mn></msub></math> 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 <math><mrow><mi>sl</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>R</mi><mo>)</mo></math> algebra as its symmetry. We show that at half-integer <math><mi>ν</mi></math>, the Jordan states associated with confluent Darboux transformations enter the construction, and the spectrum of rationally deformed AFF systems undergoes structural changes.

Klein four-group and Darboux duality in conformal mechanics

We study the spectrum generating closed nonlinear superconformal algebra that describes <math><mi>N</mi><mo>=</mo><mn>2</mn></math> superextensions of rationally deformed quantum harmonic oscillator and conformal mechanics models with coupling constant <math><mi>g</mi><mo>=</mo><mi>m</mi><mo>(</mo><mi>m</mi><mo>+</mo><mn>1</mn><mo>)</mo></math>, <math><mi>m</mi><mo>∈</mo><mi>N</mi></math>. It has a nature of a nonlinear finite <math><mi>W</mi></math> superalgebra being generated by higher-derivative integrals and generally contains several different copies of either deformed superconformal <math><mrow><mi>osp</mi></mrow><mo>(</mo><mn>2</mn><mo>|</mo><mn>2</mn><mo>)</mo></math> algebra in the case of superextended rationally deformed conformal mechanics models or deformed super-Schrödinger algebra in the case of the superextension of rationally deformed harmonic oscillator systems.

Hidden symmetries of rationally deformed superconformal mechanics

It is known that a single quantum harmonic oscillator is characterized by a hidden spectrum generating superconformal symmetry, but its origin has remained rather obscure. We show how this hidden superconformal symmetry can be derived by applying a nonlocal Foldy-Wouthuysen transformation to three extended systems with fermion degrees of freedom. The associated systems have essentially different nature from the point of view of conventional supersymmetric quantum mechanics, and generate the desired hidden symmetry in three different ways. We also trace out how the hidden superconformal symmetry of the quantum free particle system is produced in the limit of zero frequency.