  Planar superconductors, thin films with thickness comparable to the
superconducting coherence length, differ crucially from their bulk
counterparts. The Coulomb interaction is logarithmic up to distances exceeding
typical sample sizes and the Anderson-Higgs mechanism is ineffective to screen
the resulting infrared divergences of the resulting (2+1)-dimensional QED
because the Pearl length is also typically larger than sample sizes. As a
consequence, the system decomposes into superconducting droplets with the
typical size of the coherence length. We show that the two possible phases of
the system match the two known mechanisms by which (2+1)-dimensional QED cures
its infrared divergences, either by generating a mixed topological Chern-Simons
mass or by magnetic monopole instantons. The former is realized in
superconductors, the latter governs mirror-dual superinsulators. Planar
superconductors are thus described by a topological Chern-Simons gauge (TCSG)
theory which replaces the Ginzburg-Landau model in two dimensions. In the TCSG
model, the Higgs field is absent. Accordingly, in planar superconductors
Abrikosov vortices do not form, and only Josephson vortices with no normal core
can emerge.
