  Tha quantum electrodynamics of particles constrained to move on a plane is
not a fully dimensionally reduced theory because the gauge fields through which
they interact live in higher dimensions. By constraining the gauge field to the
surface of the bulk, we obtain a fully reduced planar Abelian Chern-Simons
Higgs model that can describe the vortex dynamics and second-order
superconducting-normal phase transitions in planar Chern-Simons
superconductors. Dual analyses performed before and after dimensional reduction
yield the same Lagrangian for describing the vortex dynamics, indicating the
self-consistency of our reduced theory. Compared to ordinary (2+1)-dimensional
electrodynamics, we obtain anomalous fermion statistical vortices, consistent
with results considering boundary effects. An additional electric charge
constraint and different Chern-Simons parameter constraints are also found,
which may help define a self-dual conformal field theory. Our renormalization
group analysis shows that the quantized critical exponent depends on the
Chern-Simons parameter. Quench disorder can bring more stable fixed points with
different dynamical critical exponents. If we dimensionally reduce to a curved
surface, our theory can also be extended to curved spacetimes, where geometric
flow will be introduced and compete with vortex flow.
