Topological symmetry, spin liquids and CFT duals of Polyakov model with massless fermions

Abstract

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.