I learned in a class I took that there is a duality between a $D=3$ superfluid and $D=3$ Maxwell theory ($D$ being the dimension of spacetime). In the Euclidean formulation, the action of the superfluid in the symmetry broken phase the compact boson, $$S[\theta] = \frac{K}{2}\int d^3x (\mathrm{d} \theta)^2$$ where $\theta$ is a compact 0-form. Adding a probe 1-form gauge field $A$ and auxiliary compact 1-form $a$, one can write the equivalent action $$S[\theta,A,a] = \int d^3x \, \frac{K}{2} (\mathrm{d} \theta + A )^2 + \frac{i}{2\pi} a \wedge \mathrm{d} A $$ Integrating out $a$ fixes $\mathrm{d}A=0$, so it becomes pure gauge. Instead, integrating $\theta$ then $A$ (or making a gauge transform $A\to-\mathrm{d}\theta$ to remove $\theta$ then integrating out $A$), we obtain the Maxwell action $$S[a] = \frac{1}{8\pi^2 K} \int \mathrm{d}a \wedge \star \mathrm{d} a.$$ In principle this is a demonstration of particle-vortex duality. My understanding is that it relates (1) the vortex strings of the superfluid to the electric charge worldlines of the gauge theory, and (2) the massless Goldstone mode of the superfluid to the photon of the gauge theory.
My question is, what does this duality tell me about the effects of instantons and permanent confinement (the Polyakov mechanism) in the U(1) gauge theory in $D=3$? The superfluid has a gapless Goldstone mode corresponding to the photon of the dual Maxwell theory, but the Maxwell theory is permanently confined, meaning that the photon is gapped! Why is this not an inconsistency? How can a duality relate a gapped phase and a gapless phase?
