  We discuss the ordinary quantum Hall effect and a higher-dimensional cousin.
We consider the dimensional reduction of these effects to 1+1 and 3+1
space-time dimensions, respectively. After dimensional reduction, an axion
field appears, which plays the role of a space-time dependent difference of
chemical potentials of chiral modes. As applications, we sketch a theory of
quantum pumps and a mechanism for the generation of primeval magnetic fields in
the early universe.
