Holomorphic Quantization in Constant Curvature Backgrounds

We present a holomorphic quantization scheme for free point particles on two-dimensional constant curvature Riemannian backgrounds. The procedure is based on a Lagrangian embedding of the particle configuration space into a product of coadjoint orbits of the background isometry group. Examples are provided by particles on the plane, torus, sphere, and hyperbolic plane, with or without a monopole field. We elaborate the method by recovering the Hamiltonian spectrum and the wave functions on such spaces. As a by-product, we obtain a geometric and physical interpretation of Repka's result on the decomposition of tensor products of $\mathbf{SL}(2,\mathbb{R})$ discrete series representations.