Abelian and non-Abelian bosonization: The operator solution of the WZW $\sigma$ model

The complete equivalence between the Abelian and the non-Abelian bosonization formalisms for the treatment of $\mathrm{SU}\mathrm{}\left(N\right)\mathrm{}$ fermions in two dimensions is analyzed and the operator solution of the Wess-Zumino-Witten nonlinear $\sigma$ model, written in terms of the scalar fields of the non-Abelian construction, is obtained. The importance of the order and disorder operators is stressed. In particular, they are used to show that an adequate reinterpretation of Mandelstam's formula gives the fermion representation in the non-Abelian bosonization formalism.