I was recently reading about non-Abelian bosonization, and I had a question concerning the Wess-Zumino term. In particular, I have been reading this short introduction, which states that

A non-abelian bosonisation introduced by Witten in 1983 allows to translate any fermi theory into local bose theory while having all of the original symmetries conserved.

To ensure that the field theory is scale invariant, we add the Wess-Zumino action

$\Gamma[g]=\frac{1}{24\pi}\int_B d^3 y \epsilon^{ijk}{\bf Tr}\left( g^{-1} \partial_i g g^{-1}\partial_j g g^{-1}\partial_k g \right)$

where $g\in SU(N)$. My question is when, precisely, do we need to include the Wess-Zumino term when we bosonize a theory of 1+1 D massless fermions? Does the Wess-Zumino term mainly needed to describe a nonlinear sigma model, or is it more general?

I was recently reading about non-Abelian bosonization, and I had a question concerning the Wess-Zumino term. In particular, I have been reading this short introduction by Ivan Karmazin, which states that

A non-abelian bosonisation introduced by Witten in 1983 allows to translate any fermi theory into local bose theory while having all of the original symmetries conserved.

To ensure that the field theory is scale invariant, we add the Wess-Zumino action

$\Gamma[g]=\frac{1}{24\pi}\int_B d^3 y \epsilon^{ijk}{\bf Tr}\left( g^{-1} \partial_i g g^{-1}\partial_j g g^{-1}\partial_k g \right)$

where $g\in SU(N)$. My question is when, precisely, do we need to include the Wess-Zumino term when we bosonize a theory of 1+1 D massless fermions? Does the Wess-Zumino term mainly needed to describe a nonlinear sigma model, or is it more general?