  We study the properties of 2+1d conformal field theories (CFTs) in a
background magnetic field. Using generalized particle-vortex duality, we argue
that in many cases of interest the theory becomes gapped, which allows us to
make a number of predictions for the magnetic response, background monopole
operators, and more. Explicit calculations at large N for Wilson-Fisher and
Gross-Neveu CFTs support our claim, and yield the spectrum of background
(defect) monopole operators. Finally, we point out that other possibilities
exist: certain CFTs can become metallic in a magnetic field. Such a scenario
occurs for a Dirac fermion coupled to a Chern-Simons gauge field, where a
non-Fermi liquid is argued to emerge.
