Dynamal (super)symmetries of monopoles and vortices

TL;DR

This paper reviews the dynamical (super)symmetries in monopole and vortex systems, revealing new algebraic structures and their implications for quantum spectra and reductions.

Contribution

It identifies and analyzes the superalgebraic symmetries in various monopole and vortex models, extending known symmetries and connecting them to physical spectra.

Findings

Dirac monopole lacks a smooth Runge-Lenz vector but has conformal $o(2,1)$ symmetry.

Self-dual monopoles exhibit an $su(2/2)$ supersymmetry algebra.

Asymptotic monopole systems show Kepler-type symmetry and helicity-supersymmetry.

Abstract

The dynamical (super)symmetries for various monopole systems are reviewed. For a Dirac monopole, no smooth Runge-Lenz vector can exist; there is, however, a spectrum-generating conformal $o(2,1)$ dynamical symmetry that extends into $osp(1/1)$ or $osp(1/2)$ for spin 1/2 particles. Self-dual 't Hooft-Polyakov-type monopoles admit an $su(2/2)$ dynamical supersymmetry algebra, which allows us to reduce the fluctuation equation to the spin zero case. For large $r$ the system reduces to a Dirac monopole plus an suitable inverse-square potential considered before by McIntosh and Cisneros, and by Zwanziger in the spin 0 case, and to the `dyon' of D'Hoker and Vinet for spin 1/2. The asymptotic system admits a Kepler-type dynamical symmetry as well as a `helicity-supersymmetry' analogous to the one Biedenharn found in the relativistic Kepler problem. Similar results hold for the Kaluza-Klein monopole of Gross-Perry-Sorkin. For the magnetic vortex, the N=2 supersymmetry of the Pauli Hamiltonian in a static magnetic field in the plane combines with the $o(2)\times o(2,1)$ bosonic symmetry into an $o(2)\times osp(1/2)$ dynamical superalgebra.