  We show that axions interacting with abelian gauge fields obtain a potential
from loops of magnetic monopoles. This is a consequence of the Witten effect:
the axion field causes the monopoles to acquire an electric charge and alters
their energy spectrum. The axion potential can also be understood as a type of
instanton effect due to a Euclidean monopole worldline winding around its dyon
collective coordinate. We calculate this effect, which has features in common
with both nonabelian instantons and Euclidean brane instantons. To provide
consistency checks, we argue that this axion potential vanishes in the presence
of a massless charged fermion and that it is robust against the presence of
higher-derivative corrections in the effective Lagrangian. Finally, as a first
step toward connecting with particle phenomenology and cosmology, we discuss
the regime in which this potential is important in determining the dark matter
relic abundance in a hidden sector containing an abelian gauge group,
monopoles, and axions.
