Quantized point vortex theories on a compact Riemann surface of arbitrary genus (in the zero total vorticity case) are investigated. By taking meromorphic functions thereon as order parameters and resorting to the Weil-Kostant, Abel, Riemann and Riemann-Roth theorems, a natural phase space and Hamiltonian for the vortex-antivortex configurations is exhibited, leading to explicit vortex-antivortex coherent states wave functions via geometric quantization. Furthermore, a relationship between point and smooth vorticities is established by means of Green functions associated to divisors on a Riemann surface and Poincare duality, thereby yielding a natural regularization of the singular theory.
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