arXiv subjects
Publications and source records attributed to Armen Nersessian.
We propose analogs of the generalized MICZ-Kepler system on the three-dimensional sphere and the two-sheet hyperboloid. We construct their energy spectra and normalized wave functions and find that they depend on two quantum numbers, which suggests that the systems are minimally superintegrable.
We propose a simple model of two-dimensional N=2 superconformal mechanics with a spin-orbit interaction term and demonstrate that it inherits the Galilean symmetry of the initial free-particle system. We then propose a quaternionic counterpart of this system, which describes the four-dimensional N=4 superconformal mechanics$D(1,2|α). The bosonic part of the Hamiltonian describes a free particle on the cone, while the fermionic part necessarily includes the spin-orbit interaction term.
Two variants of $N=7$ superconformal mechanics with manifest $so(7)$ and $su(2) \times su(2)$ $R$-symmetries possessing exceptional $G(3)$ dynamical symmetry are presented. To construct these $G(3)$ theories, we developed two new versions of covariant embedding of the exceptional $g_2$ algebra into the $so(7)$ algebra and then the embedding of $so(7)$ into $so(8)$. As a result, two $N=8$ superconformal mechanics with the $OSp(8|2)$ and $F(4)$ superconformal algebras and $N=7, G(3)$ superconformal mechanics were constructed in a uniform way. The constants of the octonion multiplication play a key role in the construction of superconformal mechanics with manifest $so(7)$ symmetry.
We analyze that recently proposed clasical/quantum mechanical interpretation of Zernike system and establish its equivalence to the Higgs oscillator on sphere or pseudosphere (Lobachevsky plane). We show that the non-reality of the classical Zernike Hamiltonian is an insignificant artifact of imaginary gauge and can be eliminated with a canonical transformation. The quantum counterpart of this canonical transformation is a similarity transformation mapping the system to the quantum Higgs oscillator with integration measure depending on $α,β$ parameters. When $α=2 β$ it results in the Hermitian Hamiltonian describing a free particle on (pseudo)sphere, while deviation from this point leads to a pseudo-Hermitian system.
In the present paper we constructed the supercharges and Hamiltonians for all variants of superconformal mechanics associated with the superalgebras $osp(8|2), {\mathfrak F(4)}, osp(4^\star |4)$, and $su(1,1|4)$. The fermionic and bosonic fields involved were arranged into generators spanning $so(8), so(7), so(5)\oplus su(2)$ and $su(4) \oplus u(1)$ $R$-symmetry currents of the corresponding superconformal algebras. The bosonic and fermionic parts of these $R$-symmetry generators separately define the constants of motion and form the same algebras. The angular part of the supercharges defining the system have the structure ``$(R-symmetry\; generators)\, \times\, fermions$'' while the angular part of Hamiltonian is just a proper sum of full Casimir operators and its purely bosonic and fermionic parts. We also constructed the explicit embedding of the algebras $so(7), \, so(5) \times su(2)$, and $su(4)\times u(1)$ into $so(8)$, which provide the possibility to explicitly construct the corresponding supercharges.
We give a Hamiltonian formulation of the new model of $\mathcal{N}=8$ supersymmetric mechanics recently proposed by S.~Fedoruk and E.~Ivanov and show that it possesses the dynamical $\mathcal{N}=8$ superconformal symmetry $osp(8|2)$. The bosonic part of the Hamiltonian is just a free particle on eight-dimensional cone embedded in nine-dimensional pseudo-Euclidean space, while the fermionic part can be interpreted as a spin-orbit interaction term.
We consider the propagation of polarized light in the medium with an isotropic refraction index profile and show that polarization violates the additional symmetries of the medium. Then we suggest a scheme for the construction of polarization-dependent refraction index which restores all symmetries of the initial profile. We illustrate the proposed scheme on the examples of Luneburg and Maxwell's fisheye profiles.
Recently A.Galajinsky has suggested the N=1 supersymmetric extension of Euler top and made a few interesting observations on its properties [arXiv:2111.06083 [hep-th]]. In this paper we use the formulation of the Euler top as a system on complex projective plane, playing the role of phase space, i.e. as a one-dimensional mechanical system. Then we suggest the supersymmetrization scheme of the generic one-dimensional systems with positive Hamiltonian which yields a priori integrable family of N=2k supersymmetric Hamiltonians parameterized by N/2 arbitrary real functions.
We consider the propagation of polarized light in the medium with Maxwell fish eye refraction index profile. We show that polarization violates the additional symmetries of medium, so that ray trajectories no longer remain closed. Then we suggest a modified, polarization-dependent Maxwell fish eye refraction index which restores all symmetries of initial profile and yields closed trajectories of polarized light. Explicit expressions for the polarization dependent integrals of motion and the solutions of corresponding ray trajectories are presented.
We suggest the $su(1,N|M)$-superconformal mechanics formulated in terms of phase superspace given by the non-compact analogue of complex projective superspace $\mathbb{CP}^{N|M}$. We parameterized this phase space by the specific coordinates allowing to interpret it as a higher-dimensional super-analogue of the Lobachevsky plane parameterized by lower half-plane (Klein model). Then we introduced the canonical coordinates corresponding to the known separation of the "radial" and "angular" parts of (super)conformal mechanics. Relating the "angular" coordinates with action-angle variables we demonstrated that proposed scheme allows to construct the $su(1,N|M)$ supeconformal extensions of wide class of superintegrable systems. We also proposed the superintegrable oscillator- and Coulomb- like systems with a $su(1,N|M)$ dynamical superalgebra, and found that oscillator-like systems admit deformed $\mathcal{N}=2M$ Poincaré supersymmetry, in contrast with Coulomb-like ones.
We propose the description of superintegrable models with dynamical $so(1.2)$ symmetry, and of the generic superintegrable deformations of oscillator and Coulomb systems in terms of higher-dimensional Klein model (the non-compact analog of complex projective space) playing the role of phase space. We present the expressions of the constants of motion of these systems via Killing potentials defining the $su(N.1)$ isometries of the Kähler structure.
We study quantum properties of SU$(2|1)$ supersymmetric (deformed ${\cal N}=4$, $d=1$ supersymmetric) extension of the superintegrable Smorodinsky--Winternitz system on a complex Euclidian space $\mathbb{C}^N$. The full set of wave functions is constructed and the energy spectrum is calculated. It is shown that SU$(2|1)$ supersymmetry implies the bosonic and fermionic states to belong to separate energy levels, thus exhibiting the "even-odd" splitting of the spectra. The superextended hidden symmetry operators are also defined and their action on SU$(2|1)$ multiplets of the wave functions is given. An equivalent description of the same system in terms of superconformal SU$(2|1,1)$ quantum mechanics is considered and a new representation of the hidden symmetry generators in terms of the SU$(2|1,1)$ ones is found.
We examine additional symmetries of specific refraction index profiles that are used in the well-known phenomena of perfect imaging and cloaking.In the considered cases, the translation generator and the angular momentum are conserved. We express the ray trajectory parameters through the integrals of motion and observe the existence of a photon state with maximal angular momentum, which can be used as an optical resonator. Application in plasmons and the role of polarization are discussed, and the Spin Hall effect in an extended symmetry profile is predicted.
Based on the systematic Hamiltonian and superfield approaches we construct the deformed $\mathcal{N}=4,8$ supersymmetric mechanics on Kähler manifolds interacting with constant magnetic field, and study their symmetries. At first we construct the deformed $\mathcal{N}=4,8$ supersymmetric Landau problem via minimal coupling of standard (undeformed) $\mathcal{N}=4,8$ supersymmetric free particle systems on Kähler manifold with constant magnetic field. We show that the initial "flat" supersymmetries are necessarily deformed to $SU(2|1)$ and $SU(4|1)$ supersymmetries, with the magnetic field playing the role of deformation parameter, and that the resulting systems inherit all the kinematical symmetries of the initial ones. Then we construct $SU(2|1)$ supersymmetric Kähler oscillators and find that they include, as particular cases, the harmonic oscillator models on complex Euclidian and complex projective spaces, as well as superintegrable deformations thereof, viz. $\mathbb{C}^N$-Smorodinsky-Winternitz and $\mathbb{CP}^N$-Rosochatius systems. We show that the supersymmetric extensions proposed inherit all the kinematical symmetries of the initial bosonic models. They also inherit, at least in the case of $\mathbb{C}^N$ systems, hidden (non-kinematical) symmetries. The superfield formulation of these supersymmetric systems is presented, based on the worldline $SU(2|1)$ and $SU(4|1)$ superspace formalisms.
We construct the $\mathcal{N}=8$ supersymmetric mechanics with potential term whose configuration space is the special Kähler manifold of rigid type and show that it can be viewed as the Kähler counterpart of $\mathcal{N}=4$ mechanics related to "curved WDVV equations". Then, we consider the special case of the supersymmetric mechanics with the non-zero potential term defined on the family of $U(1)$-invariant one-(complex)dimensional special Kähler metrics. The bosonic parts of these systems include superintegrable deformations of perturbed two-dimensional oscillator and Coulomb systems.
We propose new superintegrable mechanical system on the complex projective space $\mathbb{CP}^N$ involving a potential term together with coupling to a constant magnetic fields. This system can be viewed as a $\mathbb{CP}^N$-analog of both the flat singular oscillator and its spherical analog known as "Rosochatius system". We find its constants of motion and calculate their (highly nonlinear) algebra. We also present its classical and quantum solutions. The system belongs to the class of "Kähler oscillators" admitting $SU(2|1)$ supersymmetric extension. We show that, in the absence of magnetic field and with the special choice of the characteristic parameters, one can construct $\mathcal{N}=4, d=1$ Poinacaré supersymmetric extension of the system considered.
We present ${\cal N}{=}\,4$ supersymmetric mechanics on $n$-dimensional Riemannian manifolds constructed within the Hamiltonian approach. The structure functions entering the supercharges and the Hamiltonian obey modified covariant constancy equations as well as modified Witten-Dijkgraaf-Verlinde-Verlinde equations specified by the presence of the manifold's curvature tensor. Solutions of original Witten-Dijkgraaf-Verlinde-Verlinde equations and related prepotentials defining ${\cal N}{=}\,4$ superconformal mechanics in flat space can be lifted to $so(n)$-invariant Riemannian manifolds. For the Hamiltonian this lift generates an additional potential term which, on spheres and (two-sheeted) hyperboloids, becomes a Higgs-oscillator potential. In particular, the sum of $n$ copies of one-dimensional conformal mechanics results in a specific superintegrable deformation of the Higgs oscillator.
We investigate dynamics of probe particles moving in the near-horizon limit of extremal Myers-Perry black holes in arbitrary dimensions. Employing ellipsoidal coordinates we show that this problem is integrable and separable, extending the results of the odd dimensional case discussed in arXiv:1703.00713. We find the general solution of the Hamilton-Jacobi equations for these systems and present explicit expressions for the Liouville integrals, discuss Killing tensors and the associated constants of motion. We analyze special cases of the background near-horizon geometry were the system possesses more constants of motion and is hence superintegrable. Finally, we consider near-horizon extremal vanishing horizon case which happens for Myers-Perry black holes in odd dimensions and show that geodesic equations on this geometry are also separable and work out its integrals of motion.