N-point g-loop vertex for a free bosonic theory with vacuum charge Q

Starting fron the N-point vertex on the sphere and using the sewing procedure we construct the N-point g-loop vertex for a free bosonic theory with vacuum charge Q. We then show that, when this vertex is saturated with N highest weight states, it gives their correlation function on an arbitrary Riemann surface of genus g. We also extend our formalism to the case of a free scalar field compactified on a circle, which is related to the Coulomb gas description of minimal models.