Exact bosonization in two spatial dimensions and a new class of lattice gauge theories

We describe a 2d analog of the Jordan-Wigner transformation which maps an arbitrary fermionic system on a 2d lattice to a lattice gauge theory while preserving the locality of the Hamiltonian. When the space is simply-connected, this bosonization map is an equivalence. We describe several examples of 2d bosonization, including free fermions on square and honeycomb lattices and the Hubbard model. We describe Euclidean actions for the corresponding lattice gauge theories and find that they contain Chern-Simons-like terms. Finally, we write down a fermionic dual of the gauged Ising model (the Fradkin-Shenker model).