  The idea of statistical transmutation plays a crucial role in descriptions of
the fractional quantum Hall effect. However, a recently conjectured duality
between a critical boson and a massless 2-component Dirac fermion extends this
notion to gapless systems. This duality sheds light on highly non-trivial
problems such as the half-filled Landau level, the superconductor-insulator
transition, and surface states of strongly coupled topological insulators.
Although this boson-fermion duality has undergone many consistency checks, it
has remained unproven. We describe the duality in a non-perturbative fashion
using an exact UV mapping of partition functions on a 3D Euclidean lattice.
