Dynamal (super)symmetries of monopoles and vortices

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.

Minor corrections.18 pages, no figures. Based on a review talk given at the International Symposium on Advanced Topics in Quantum Physics, Shanxi'92. Ed. J.-Q. Liang, M.-L. Wang, S.-N. Qiao, D.C. Su. pp. 283- 293. Science Press, Beijing (1993) [Tours Preprint N. 47/92], and on Tours Preprint N. 60/93 (1993) (unpublished)

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