Particle in a self-dual dyon background: Hidden free nature and exotic superconformal symmetry

Abstract

We show that a nonrelativistic particle in a combined field of a magnetic monopole and $1/{r}^{2}$ potential reveals a hidden, partially free dynamics when the strength of the central potential and the charge-monopole coupling constant are mutually fitted to each other. In this case the system admits both a conserved Laplace-Runge-Lenz vector and a dynamical conformal symmetry. The supersymmetrically extended system corresponds then to a background of a self-dual or anti-self-dual dyon. It is described by a quadratically extended Lie superalgebra $D(2,1;\ensuremath{\alpha})$ with $\ensuremath{\alpha}=1/2$, in which the bosonic set of generators is enlarged by a generalized Laplace-Runge-Lenz vector and its dynamical integral counterpart related to Galilei symmetry, as well as by the chiral ${\mathbb{Z}}_{2}$-grading operator. The odd part of the nonlinear superalgebra comprises a complete set of $24=2\ifmmode\times\else\texttimes\fi{}3\ifmmode\times\else\texttimes\fi{}4$ fermionic generators. Here a usual duplication comes from the ${\mathbb{Z}}_{2}$-grading structure; the second factor can be associated with a triad of scalar integrals---the Hamiltonian, the generator of special conformal transformations, and the squared total angular momentum vector, while the quadruplication is generated by a chiral spin vector integral which exits due to the (anti-)self-dual nature of the electromagnetic background.