  We prove the absence of a mass gap and confinement in the Polyakov model with
massless complex fermions in any representation of the gauge group. A
$U(1)_{*}$ topological shift symmetry protects the masslessness of one dual
photon. This symmetry emerges in the IR as a consequence of the Callias index
theorem and abelian duality. For matter in the fundamental representation, the
infrared limits of this class of theories interpolate between weakly and
strongly coupled conformal field theory (CFT) depending on the number of
flavors, and provide an infinite class of CFTs in $d=3$ dimensions. The long
distance physics of the model is same as certain stable spin liquids. Altering
the topology of the adjoint Higgs field by turning it into a compact scalar
does not change the long distance dynamics in perturbation theory, however,
non-perturbative effects lead to a mass gap for the gauge fluctuations. This
provides conceptual clarity to many subtle issues about compact QED$_3$
discussed in the context of quantum magnets, spin liquids and phase fluctuation
models in cuprate superconductors. These constructions also provide new
insights into zero temperature gauge theory dynamics on $\R^{2,1}$ and
$\R^{2,1} \times S^1$. The confined versus deconfined long distance dynamics is
characterized by a discrete versus continuous topological symmetry.
