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?

  • $\begingroup$ 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. $\endgroup$ Jun 28 '19 at 3:35

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