Duality of a compact topological superconductor model and the Witten effect

We consider a compact abelian Higgs model in 3+1 dimensions with a topological axion term and construct its dual theories for both bulk and boundary at strong coupling. The model may be viewed as describing a superconductor with magnetic monopoles, which can also be interpreted as a field theory of a topological Mott insulator. We show that this model is dual to a non-compact topological field theory of particles and vortices. It has exactly the same form of a model for superconducting cosmic strings with an axion term. We consider the duality of the boundary field theory at strong coupling and show that in this case $\theta$ is quantized as $-8\pi n\mathrm{/}m$ where $n$ and $m$ are the quantum numbers associated to electric and magnetic charges. These topological states lack a non-interacting equivalent.