Bosonization in three spatial dimensions and a 2-form gauge theory

We describe a 3d analog of the Jordan-Wigner transformation which maps an arbitrary fermionic system on a 3d spatial lattice to a 2-form Z₂ gauge theory with an unusual Gauss law. An important property of this map is that it preserves the locality of the Hamiltonian. The map depends explicitly on the choice of a spin structure of the spatial manifold. We give examples of 3d bosonic systems dual to free fermions. We also describe the corresponding Euclidean lattice models, which is analogous to the Steenrod square term in (3+1)D [compared to the Chern-Simon term in (2+1)D].