  Much insight into the dynamics of quantum field theories can be gained by
studying the relationship between field theories in different dimensions. An
interesting observation is that when two theories are related by dimensional
reduction on a compact surface, their 't Hooft anomalies corresponding to
continuous symmetries are also related: the anomaly polynomial of the
lower-dimensional theory can be obtained by integrating that of the
higher-dimensional one on the compact surface. Naturally, this relation only
holds if both theories are even dimensional. This raises the question of
whether similar relations can also hold for the case of anomalies in discrete
symmetries, which might be true even in odd dimensions. The natural
generalization to discrete symmetries is that the anomaly theories, associated
with the lower and higher dimensional theories, would be related by reduction
on the compact surface. We explore this idea for compactifications of 5d
superconformal field theories (SCFTs) to 3d on Riemann surfaces with
global-symmetry fluxes. In this context, it can be used both as a check for
these compactification constructions and for discovering new anomalies in the
5d SCFTs. This opens the way to applying the same idea of dimensional reduction
of the anomaly theory to more general types of compactifications.
