  We explicitly derive the duality between a free electronic Dirac cone and
quantum electrodynamics in $(2+1)$ dimensions (QED$_3$) with $N = 1$ fermion
flavors. The duality proceeds via an exact, non-local mapping from electrons to
dual fermions with long-range interactions encoded by an emergent gauge field.
This mapping allows us to construct parent Hamiltonians for exotic
topological-insulator surface phases, derive the particle-hole-symmetric field
theory of a half-filled Landau level, and nontrivially constrain QED$_3$
scaling dimensions. We similarly establish duality between bosonic topological
insulator surfaces and $N = 2$ QED$_3$.
