Bosonization of 2+1 dimensional fermions on the surface of topological insulators

Three-dimensional topological insulators can be described by an effective field theory involving two `hydrodynamic' Abelian gauge fields. The action contains a bulk topological BF term and a surface term, called loop model. This describes the massless 2+1 dimensional excitations and provides them with a semiclassical, yet non-trivial conformal invariant dynamics. Given that topological insulators are originally fermionic, this physical setting is ideal for realizing the bosonization of massless fermions in terms of gauge fields. Building on earlier analyses of the loop model, we find that fermions belong to the solitonic spectrum and can be described by Wilson lines, through the generalization of 1+1 dimensional vertex operators. Their correlation functions agree with conformal invariance. The bosonic loop model is then mapped into a fermionic theory by using the general construction of fermionic topological phases described in the literature. It requires the identification of the characteristic one-form $Z_2$ symmetry of the bosonic theory and its gauging, which originates the fermion number $(-1)^F$, the spin sectors and the time reversal symmetry obeying ${\cal T}^2=(-1)^F$. These results are detailed for the effective action and the partition function on the geometry $S^2\times S^1$.