Quantum mechanics on noncommutative Riemann surfaces

We study the quantum mechanics of a charged particle on a constant curvature noncommutative Riemann surface in the presence of a constant magnetic field. We formulate the problem by considering quantum mechanics on the noncommutative AdS2 covering space and gauging a discrete symmetry group which defines a genus-g surface. Although there is no magnetic field quantization on the covering space, a quantization condition is required in order to have single-valued states on the Riemann surface. For noncommutative AdS2 and sub-critical values of the magnetic field the spectrum has a discrete Landau level part as well as a continuum, while for over-critical values we obtain a purely noncommutative phase consisting entirely of Landau levels