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

Explicit derivation of duality between a free Dirac cone and quantum electrodynamics in (2+1) dimensions