The geometry of the SU(2) Kepler problem

Abstract

The SU (2) Kepler problem is defined and analyzed, which is a Hamiltonian system reduced from the conformal Kepler problem on T ∗( R 8 − { 0}) by the use of the symplectic SU (2) action lifted from the SU (2) left action on the SU (2) bundle R 8 − { 0} → R 5 − { 0}. This reduced system has a parameter μ ϵ su (2) coming from the value of the moment map associated with the symplectic SU (2) action. If μ ≠ 0, the phase space of this system have a bundle structure with base space T ∗( R 5 − { 0}) and fibre S 2. The fibre, a (co)adjoint orbit through μ for SU (2), represents the internal degrees of freedom, called the isospin, of the particle of this system. The SU (2) Kepler problem with μ ≠ 0 is then interpreted as describing the motion of a classical particle with isospin in the Newtonian potential plus a specific repulsive potential together with a Yang-Mills field. This Yang-Mills field is to be referred to as BPST Yang's monopole field in R 5 − { 0};, since it becomes the Belavin-Polyakov-Schwartz-Tyupkin instanton, restricted on S 4. If μ = 0, the SU (2) Kepler problem reduces to the ordinary Kepler problem. Like the ordinary Kepler problem, the Hamiltonian flows of the SU (2) Kepler problem of negative energy are all closed. It is shown in an explicit manner that the energy manifolds and isoenergetic orbit spaces for the SU (2) Kepler problem of negative energy are both homogeneous manifolds on which SU (4) acts transitively to the right; those homogeneous manifold are classified into two, according as the parameter μ is zero or not. For a certain value of μ, however, they contracts to the manifold which represents the set of all the equilibrium states. The isoenergetic orbit spaces are finally shown to be symplectomorphic to certain Kirillov-Konstant-Souriau coadjoint orbits for U (4), if μ is not the exceptional value mentioned above.