  We study the electromagnetic properties of dense QCD in the so-called
Magnetic Dual Chiral Density Wave phase. This inhomogeneous phase exhibits a
nontrivial topology that comes from the fermion sector due to the asymmetry of
the lowest Landau level modes. The nontrivial topology manifests in the
electromagnetic effective action via a chiral anomaly term $~\theta
F^{\mu\nu}\tilde{F}_{\mu\nu}$, with a dynamic axion field $\theta$ given by the
phase of the dual chiral density wave condensate. The coupling of the axion
with the electromagnetic field leads to several macroscopic effects that
include, among others, an anomalous, nondissipative Hall current, an anomalous
electric charge, magnetoelectricity, and the formation of a hybridized
propagating mode known as an axion polariton. Connection to topological
insulators and Weyls semimetals, as well as possible implications for heavy-ion
collisions and neutron stars are all highlighted.
