  Particle-vortex duality is a powerful theoretical tool that has been used to
study bosonic systems. Here we propose an analogous duality for Dirac fermions
in 2+1 dimensions. The physics of a single Dirac cone is proposed to be
described by a dual theory, QED3 with a dual Dirac fermion coupled to a gauge
field. This duality is established by considering two alternate descriptions of
the 3d topological insulator (TI) surface. The first description is the usual
Dirac cone surface state. The second description is accessed via an
electric-magnetic duality of the bulk TI coupled to a gauge field, which maps
it to a gauged topological superconductor. This alternate description
ultimately leads to a new surface theory - dual QED3. The dual theory provides
an explicit derivation of the T-Pfaffian state, a proposed surface topological
order of the TI, which is simply the paired superfluid state of the dual
fermions. The roles of time reversal and particle-hole symmetry are exchanged
by the duality, which connects some of our results to a recent conjecture by
Son on particle-hole symmetric quantum Hall states.
