  We study in detail how four-dimensional local anomalies manifest themselves
when the theory is compactified on a circle. By integrating out the
Kaluza-Klein modes in a way that preserves the four-dimensional symmetries in
the UV, we show that the three-dimensional theory contains field-dependent
Chern-Simons terms that appear at one-loop. These vanish if and only if the
four-dimensional anomaly is canceled, so the anomaly is not lost upon
compactification. We extend this analysis to situations where anomalies are
canceled through a Green-Schwarz mechanism. We then use these results to show
automatic cancellation of local anomalies in F-theory compactifications that
can be obtained as a limit of M-theory on a smooth Calabi-Yau fourfold with
background flux.
