A correspondence is established between the dynamics of the two-vortex system and the non-commutative Landau problem (NCLP) in its sub- (non-chiral), super- (chiral) and critical phases. As a result, a trivial permutation symmetry of the point vortices induces a weak-strong coupling duality in the NCLP. We show that quantum two-vortex systems with non-zero total vorticity can be generated by applying conformal bridge transformation to a two-dimensional quantum free particle or to a quantum vortex-antivortex system of zero total vorticity. The sub- and super-critical phases of the quantum NCLP are generated in a similar way from the 2D quantum free particle in a commutative or non-commutative plane. The composition of the inverse and direct transformations of the conformal bridge also makes it possible to link the non-chiral and chiral phases in each of these two systems.

Chile

Andrey Alcala

Mikhail Plyushchay

Weak-strong duality of the non-commutative Landau problem induced by a two-vortex permutation, and conformal bridge transformation

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.

Luis Inzunza

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 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 ( ).

Germany

Andreas Wipf

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

We investigate how the Lax-Novikov integral in the perfectly invisible PT-regularized zero-gap quantum conformal and superconformal mechanics systems affects on their (super)-conformal symmetries. We show that the expansion of the conformal symmetry with this integral results in a nonlinearly extended generalized Shrödinger algebra. The PT-regularized superconformal mechanics systems in the phase of the unbroken exotic nonlinear   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 4 super-Poincaré symmetry are described by nonlinearly super-extended Schrödinger algebra with the osp(2|2) sub-superalgebra. In the partially broken phase, the scaling dimension of all odd integrals is indefinite, and the osp(2|2) is not contained as a sub-superalgebra.

Spain

Juan Guilarte

Nonlinear symmetries of perfectly invisible PT-regularized conformal and superconformal mechanics systems

We investigate a special class of the P T $$ \mathcal{P}\mathcal{T} $$ -symmetric quantum models being perfectly invisible zero-gap systems with a unique bound state at the very edge of continuous spectrum of scattering states. The family includes the P T $$ \mathcal{P}\mathcal{T} $$ -regularized two particle Calogero systems (conformal quantum mechanics models of de Alfaro-Fubini-Furlan) and their rational extensions whose potentials satisfy equations of the KdV hierarchy and exhibit, particularly, a behaviour typical for extreme waves. We show that the two simplest Hamiltonians from the Calogero subfamily determine the fluctuation spectra around the P T $$ \mathcal{P}\mathcal{T} $$ -regularized kinks arising as traveling waves in the field-theoretical Liouville and SU(3) conformal Toda systems. Peculiar properties of the quantum systems are reflected in the associated exotic nonlinear supersymmetry in the unbroken or partially broken phases. The conventional N = 2 $$ \mathcal{N}=2 $$ supersymmetry is extended here to the N = 4 $$ \mathcal{N}=4 $$ nonlinear supersymmetry that involves two bosonic generators composed from Lax-Novikov integrals of the subsystems, one of which is the central charge of the superalgebra. Jordan states are shown to play an essential role in the construction.

Perfectly invisible P T $$ \mathcal{P}\mathcal{T} $$ -symmetric zero-gap systems, conformal field theoretical kinks, and exotic nonlinear supersymmetry

It is long known that the rational Calogero model describing n identical particles on a line with inverse-square mutual interaction potential is quantum superintegrable. We review the (nonlinear) algebra of the conserved quantum charges and the intertwiners which relate the Liouville charges at couplings g and g ±1. For integer values of g , these intertwiners give rise to additional conserved charges commuting with all Liouville charges and known since the 1990s. We give a direct construction of such a charge, the unique one being totally antisymmetric under particle permutations. It is of order $ \frac{1}{2} $ n ( n −1)(2 g −1) in the momenta and squares to a polynomial in the Liouville charges. With a natural $ \mathbb{Z} $ 2 grading, this charge extends the algebra of conserved charges to a nonlinear supersymmetric one. We provide explicit expressions for intertwiners, charges and their algebra in the cases of two, three and four particles.