Spinning particles on curved spaces and constants of motion

Spinning particles, such as Dirac fermions, can be described by pseudo-classical mechanics models involving anticommuting c-numbers for the spin-degrees of freedom. The configuration space of spinning particles (spinning space) is an extension of an ordinary Riemannian manifold, parametrized by local coordinates (xμ), to a graded manifold parametrized by local coordinates (xμ, ψμ), with the first set of variables being Grassmann-even (commuting) and the second set of variables being Grassmann-odd (anticommuting). The symmetries of a spinning-particle model can be divided into two classes. In the first class, there are four independent generic symmetries which exist in any theory: 1. Proper-time translations generated by Hamiltonian; 2. Supersymmetry generated by supercharge; 3. Chiral symmetry generated by chiral charge; 4. Dual supersymmetry, generated by dual supercharge. In the second class, there are conserved quantities, called non-generic, which depend on the explicit form of the metric gμν(x). We shall deal with the non-generic constants of motion in connection with the generalized Killing equations, looking for the general features of the solutions. The constants of motion can be seen as extensions of the constants from the scalar case or new ones depending on the Grassmann-valued spin variables (ψμ). The general results are applied to the case of the four dimensional Euclidean Taub-NUT manifold. The motivation of this selection is two-fold. First of all, in the Taub-NUT geometry there are known to exist four Killing-Yano tensors. From this point of view, the spinning Taub-NUT space is an exceedingly interesting space to exemplify the effective construction of all conserved quantities in terms of geometric ones, namely Killing-Yano tensors. On the other hand, the Taub-NUT geometry is involved in many modern studies in physics. For example, the Kaluza-Klein monopole of Gross and Perry and of Sorkin was obtained by embedding the TaubNUT gravitational instanton into five-dimensional Kaluza-Klein theory. The same object has re-emerged in the study of monopole scattering. In the long distance limit, neglecting radiation, the relative motion of the BPS monopoles is described by the geodesics of this space. The dynamics of well-separated monopoles is completely soluble and has a Kepler type symmetry. (author)