# Is there an explicit mapping between N free bosonic fields and the $SU(N)_1$ WZW model + free boson?

Witten's nonabelian bosonization tells us that $$N$$ free Dirac fields can by written in terms of an $$SU(N)_1$$ WZW model and one free boson. But bosonization also tells us that we could just as well write $$N$$ free Dirac fields as $$N$$ bosonic fields. The equivalence of the two models is well known.

Superficially, this equivalence is quite surprising, since at the classical level the bosonic fields would have $$N$$ real degrees of freedom and a non-linear sigma model with SU(N) + bosonic field will have $$N^2$$ degrees of freedom.

I was wondering if there was an explicit way to map fields between the two theories that elucidates this. Is there a dictionary relating the WZW theory field $$g$$ to the n bosonic fields? Or is there a simple way to see this fact without the fancy bosonization machinery?

• There are well-known free field realizations of WZW models. As far as I remember a fair number of them is discussed in the yellow book, but most likely only at the level of matching the operator content. Jun 28 '19 at 3:35