  In this paper general abelian gauge field theories interacting with matter
fields are quantized on a closed and orientable Riemann surface $\Sigma$. The
approach used is that of small perturbations around topologically nontrivial
classical solutions $A^I_\mu$ of the Maxwell equations. The propagator of the
gauge fields and the form of the fields $A^I_\mu$ are explicitly computed in
the Feynman gauge. Despite of the fact that the quantization procedure
presented here is mainly perturbative, in the case of the Schwinger model, or
two dimensional quantum electrodynamics, it is also possible to derive
nonperturbative results. As an example, we show that, at very short distances,
the electromagnetic forces on a Riemann surface vanish up to zero modes.
