Anomaly Cancellation in Supergravity with Fayet-Iliopoulos Couplings

We review and clarify the cancellation conditions for gauge anomalies which occur when N=1, D=4 supergravity is coupled to a Kahler non-linear sigma-model with gauged isometries and Fayet-Iliopoulos couplings. For a flat sigma-model target space and vanishing Fayet-Iliopoulos couplings, consistency requires just the conventional anomaly cancellation conditions. A consistent model with non-vanishing Fayet-Iliopoulos couplings is unlikely unless the Green-Schwarz mechanism is used. In this case the U(1) gauge boson becomes massive and the D-term potential receives corrections. A Green-Schwarz mechanism can remove both the abelian and certain non-abelian anomalies in models with a gauge non-invariant Kahler potential