  We present a numerical study of the massive two-flavor QED in two dimensions
with the gauge action proposed by L\"uscher, which allows only ``admissible''
gauge fields. We find that the admissibility condition does not allow any
topology changes by the local updation in Hybrid Monte Carlo algorithm so that
the configurations in each topological sector can be generated separately. By
developing a new method to sum over different topological sectors, we
investigate $\theta$ vacuum effects. Combining with domain-wall fermion action,
we obtain the fermion mass dependence and $\theta$ dependence of the meson
masses, which are consistent with the analytic results by mass perturbation in
the continuum theory.
