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

A single Dirac cone of (free) electrons famously arises on the surface of a 3D topological insulator. Recent work proposed that these metallic surfaces can alternatively be described by quantum electrodynamics in $(2+1)$ dimensions (QED$_3$), where charge-neutral 'dual fermions' strongly couple to an emergent photon. We explicitly derive this duality via an exact, non-local mapping from electrons to dual fermions on the level of path integrals. This mapping allows us to construct Hamiltonians for exotic topological-insulator surface phases, and to derive the particle-hole-symmetric field theory of a half-filled Landau level. By running the duality 'in reverse' we can constrain scaling dimensions for operators in QED$_3$ and establish duality between bosonic topological insulator surfaces and QED$_3$ with two fermion flavors.