Remarks on the geometric quantization of Landau levels

In this note, we resume the geometric quantization approach to the motion of a charged \r\nparticle on a plane, subject to a constant magnetic field perpendicular to the latter, by \r\nshowing directly that it gives rise to a completely integrable system to which we may \r\napply holomorphic geometric quantization. In addition, we present a variant employing a \r\nsuitable vertical polarization and we also make contact with Bott’s quantization, enforcing the property “quantization commutes with reduction”, which is known to hold under quite general conditions. We also provide an interpretation of translational symmetry breaking in terms of coherent states and index theory. Finally, we give a representation \r\ntheoretic description of the lowest Landau level via theuse of an S^1-equivariant Dirac operator.