  The possible couplings of an axion to gauge fields depend on the global
structure of the gauge group. If the Standard Model gauge group is minimal, or
equivalently if fractionally charged color-singlet particles are forbidden,
then the QCD axion's Chern-Simons couplings to photons and gluons obey
correlated quantization conditions. Specifically, the photon coupling can have
a fractional part which is a multiple of 1/3, but which is determined by the
gluon coupling. A consequence of this result is that, among all theories with a
minimal gauge group and minimal axion coupling to gluons, the smallest possible
axion-photon amplitude $|g_{a\gamma\gamma}|$ arises for $E/N = 8/3$. This
provides a new motivation for experiments targeting this axion-photon coupling.
