Valenzuela, M (Valenzuela, Mauricio). Univ Talca, Inst Matemat & Fis, Talca, ChileA local supersymmetric action for a (2+1)-dimensional system including gravity, the electromagnetic field and a Dirac spin-1/2 field is presented. The action is a Chern-Simons form for a connection of the OSp(2|2) group. All the fields enter as parts of the connection, that transforms in the adjoint representation of the gauge group. The system is off-shell invariant under local (gauge) supersymmetry. Although the supersymmetry is locally realized, there is no spin-3/2 gravitino, and is therefore not supergravity. The fields do not necessarily form supersymmetric doublets of equal mass, and moreover, the fermion may acquire mass through the coupling with geometry, while the bosons - the U(1) field and the spin connection - remain massless
A particle system with a (2+1)D exotic Newton-Hooke symmetry is constructed by the method of nonlinear realization. It has three essentially different phases depending on the values of the two central charges. The subcritical and supercritical phases (describing 2D isotropic ordinary and exotic oscillators) are separated by the critical phase (one-mode oscillator), and are related by a duality transformation. In the flat limit, the system transforms into a free Galilean exotic particle on the noncommutative plane. The wave equations carrying projective representations of the exotic Newton-Hooke symmetry are constructed. *
Using a remarkable mapping from the original (3+1)dimensional Skyrme model to the Sine-Gordon model, we construct the first analytic examples of Skyrmions as well as of Skyrmions-anti-Skyrmions bound states within a finite box in 3+1 dimensional flat space-time. An analytic upper bound on the number of these Skyrmions-anti-Skyrmions bound states is derived. We compute the critical isospin chemical potential beyond which these Skyrmions cease to exist. With these tools, we also construct topologically protected time-crystals: time-periodic configurations whose time-dependence is protected by their non-trivial winding number. These are striking realizations of the ideas of Shapere and Wilczek. The critical isospin chemical potential for these time-crystals is determined.
We investigate the planar anisotropic harmonic oscillator with explicit rotational symmetry as a particle model with non-commutative coordinates. It includes the exotic Newton-Hooke particle and the non-commutative Landau problem as special, isotropic and maximally anisotropic, cases. The system is described by the same (2+1)-dimensional exotic Newton-Hooke symmetry as in the isotropic case, and develops three different phases depending on the values of the two central charges. The special cases of the exotic Newton-Hooke particle and non-commutative Landau problem are shown to be characterized by additional, so(3) or so(2, 1) Lie symmetry, which reflects their peculiar spectral properties. * 1 The case of an exotic non-relativistic string in 3+1 dimensions and the relation with the exotic particle in 2+1 dimensions has been recently studied in [9].to that of Landau problem in the non-commutative plane [5,14,15]. The noncommutative Landau problem (NLP) also develops three phases, the sub-and super-critical ones, separated by a critical, quantum Hall effect phase [15]. Therefore, these similarities indicate on a possible close relation between the (2+1)D exotic Newton-Hooke symmetry and the non-commutative Landau problem. The purpose of this article is to study in detail this relation by means of a planar exotic anisotropic harmonic oscillator with explicit spatial rotation symmetry as a particle model with non-commutative coordinates.The model of the anisotropic harmonic oscillator we propose [(2.2) below], includes the exotic Newton-Hooke particle and the non-commutative Landau problem as special, isotropic and maximally anisotropic, cases. We shaw that what distinguishes the exotic Newton-Hooke particle and non-commutative Landau problem as special cases, is a presence of the additional, so(3) or so(2, 1) Lie symmetry. In a generic case of commensurable frequencies, the exotic anisotropic harmonic oscillator is characterized, instead, by a nonlinear deformation of the indicated additional Lie symmetry. Like the exotic Newton-Hooke particle and non-commutative Landau problem, the anisotropic oscillator system develops the subcritical and supercritical phases, separated by a critical phase. The phase is defined by the values of the two central charges of the exotic Newton-Hooke algebra.The paper is organized as follows. In Section 2 we introduce a planar anisotropic harmonic oscillator with explicit rotational symmetry as a particle model with non-commutative coordinates, and establish its relation with the non-commutative Landau problem. In Section 3 we discuss the chiral form of the exotic Newton-Hooke symmetry of the system, and analyze its additional symmetries, which depend on the concrete values of the model parameters. In Section 4 we analyze the exotic Newton-Hooke symmetry in the non-chiral, space-time picture. Section 5 is devoted to the discussion and concluding remarks.
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