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(2+1)D Exotic Newton-Hooke Symmetry, Duality and Projective Phase

Pedro D. Alvarez, Joaquim Gomis, Kiyoshi Kamimura, Mikhail S. Plyushchay

TL;DR

The paper develops a comprehensive framework for a (2+1)D particle with exotic Newton-Hooke symmetry by contracting AdS3 and realizing the symmetry nonlinearly. It uncovers three distinct phases dictated by two central charges, elucidates duality relations between subcritical and supercritical regimes, and shows how these phases manifest in both reduced-phase-space quantization and wave equations carrying projective representations. A rich structure emerges, including chiral decompositions, an enhanced SO(3)/SO(2,1) symmetry, and a nontrivial two-cocycle that governs the projective phase. The work also connects to flat-space limits, noncommutative geometry on the plane, and potential links to 3D gravity and BTZ black holes, while outlining avenues for supersymmetric extensions and further generalizations.

Abstract

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.

(2+1)D Exotic Newton-Hooke Symmetry, Duality and Projective Phase

TL;DR

The paper develops a comprehensive framework for a (2+1)D particle with exotic Newton-Hooke symmetry by contracting AdS3 and realizing the symmetry nonlinearly. It uncovers three distinct phases dictated by two central charges, elucidates duality relations between subcritical and supercritical regimes, and shows how these phases manifest in both reduced-phase-space quantization and wave equations carrying projective representations. A rich structure emerges, including chiral decompositions, an enhanced SO(3)/SO(2,1) symmetry, and a nontrivial two-cocycle that governs the projective phase. The work also connects to flat-space limits, noncommutative geometry on the plane, and potential links to 3D gravity and BTZ black holes, while outlining avenues for supersymmetric extensions and further generalizations.

Abstract

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.

Paper Structure

This paper contains 18 sections, 210 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Exotic NH${}_3$ as a contraction of AdS${}_3$
  3. Classical Lagrangian for a system with exotic NH${}_3$ symmetry
  4. Classical dynamics
  5. Constraints
  6. Non-chiral basis
  7. Chiral basis
  8. Symmetries
  9. Classical exotic Newton-Hooke symmetry
  10. SO(3) and SO(2,1) symmetry
  11. Reduced phase space quantization
  12. Duality
  13. Wave equations
  14. Chiral basis
  15. Wave equations in non-chiral basis and flat limit
  16. ...and 3 more sections

Figures (2)

  • Figure 1: Phases and duality.
  • Figure 2: Constant energy levels.