Bosonization of three-dimensional non-abelian fermion field theories

Abstract

We discuss bosonization in three dimensions of an SU (N ) massive Thirring model in the low-energy regime. We find that the bosonized theory is related (but not equal) to SU (N ) Yang-Mills-Chern-Simons gauge theory. For free massive fermions bosonization leads, at low energies, to the pure SU (N ) (level k = 1) Chern-Simons theory.

Non-abelian bosonization in two and three dimensions

“…The resulting bosonization action coincides with that obtained using a completely different approach [4], based in the use of an interpolating Lagrangian [39]- [41]. The advantage of the present method lies in the fact that the BRST symmetry can be formulated in arbitrary dimensions while the interpolating Lagrangian, which replaces the role of this symmetry in decoupling auxiliary and bosonic fields is in principle applicable only in odddimensional spaces.…”

“…This result, advanced in [4] using a completely different approach, is the natural extension of the abelian result [1]- [3]. In this last case the bosonic theory corresponds, in the large fermion mass limit, to a Chern-Simons theory while in the massless case it coincides with the abelian non-local action discussed in [20], [23].…”

“…The addition of δG allows us to make contact at this point with the effective action discussed in refs. [4], [41]. Indeed, after the shift…”

On bosonization in 3 dimensions

“…As it was stressed in [6,7], the possibility of finding exact bosonization rules (in this functional approach), depends on our ability to compute the fermionic determinant in the presence of a background field exactly. Thus in 3 dimensions we must use an approximation scheme.…”

“…Thus in 3 dimensions we must use an approximation scheme. The one presented in [6,7] amounts to expanding the corresponding effective action in powers of ∂ m . The question presents itself about how to extend this approximation in order to include cases where the derivative expansion is no longer valid, as it is indeed the case for massless fermions.…”