Minimal Immersions and the Spectrum of Supermembranes

We describe the minimal configurations of the compact D=11 Supermembrane and D-branes when the spatial part of the world-volume is a K\"ahler manifold. The minima of the corresponding hamiltonians arise at immersions into the target space minimizing the K\"ahler volume. Minimal immersions of particular K\"ahler manifolds into a given target space are known to exist. They have associated to them a symplectic matrix of central charges. We reexpress the Hamiltonian of the D=11 Supermembrane with a symplectic matrix of central charges, in the light cone gauge, using the minimal immersions as backgrounds and the $Sp\parent{2g,\mathbb{Z}}$ symmetry of the resulting theory, $g$ being the genus of the K\"ahler manifold. The resulting theory is a symplectic noncommutative Yang-Mills theory coupled with the scalar fields transverse to the Supermembrane. We prove that both theories are exactly equivalent. A similar construction may be performed for the Born-Infeld action. Finally, the noncommutative formulation is used to show that the spectrum of the reguralized Hamiltonian of the above mentioned D=11 Supermembrane is a discrete set of eigenvalues with finite multiplicity.