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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?

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  • $\begingroup$ Minor comment to the post (v1): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$ – Qmechanic Aug 22 '18 at 17:51
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    $\begingroup$ It's best to read the original paper by Witten. He explains this point very clearly $\endgroup$ – mike stone Aug 22 '18 at 18:35
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    $\begingroup$ As the answer of @Qmechanic indicates, you do need to append the WZW term to the plain chiral model, as, alone, this is deficient: it fails to reproduce the current algebraic features of the fermion theory bosonized. $\endgroup$ – Cosmas Zachos Apr 10 at 14:20
  • $\begingroup$ Remotely related. $\endgroup$ – Cosmas Zachos Apr 11 at 15:39
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Alas, the book report you relied on glosses over the crucial geometrical peculiarity/pathology of the plain chiral model, which makes it unsuitable for bozonizing free nonabelian fermions.

I'll emphasize the point @Qmechanic and the original paper make, without using light-cone language and by paring down all superfluous aspects to focus on a primitive bare minimum.

So just look at right-invariant, so left currents, but the argument will go through for the right currents, mutatis mutandis.

In 2d bosonization it is important that the conserved symmetry currents identified have the same geometric properties, over and above mere conservation. (Ultimately they have to satisfy the same current algebras, but these trail the geometry.)

A free fermion theory has a left-chiral current, $$ L^\mu= \bar \psi \gamma^\mu (1-\gamma^5)\psi, $$ crucially self-dual, $$ \tilde L^\mu \equiv \epsilon^{\mu \nu } L_\nu= \bar \psi \epsilon^{\mu \nu }\gamma_\nu (1-\gamma^5)\psi \\ = \bar \psi \gamma^5 \gamma^\mu (1-\gamma^5)\psi = \bar \psi \gamma^\mu (1-\gamma^5)\psi =L^\mu, $$ being, of course, conserved, $ \partial\cdot L= \partial\cdot \tilde L=0$.

It had been know for a long time that the conserved currents of the plain (Gürsey, 1960) chiral model fail this requirement miserably. That is, for example, the density $$ {\cal L} \propto \operatorname {Tr} \partial_\mu g^{-1} \partial^\mu g $$ has EOM $$ \partial_\mu (g\partial^\mu g^{-1}) =0 \Longrightarrow \\ \Box g+ g \partial_\mu g^{-1} \partial^{\mu} g= 0=\Box g + \partial_\mu g \partial^\mu g^{-1} g, $$ where $\partial g ~g^{-1}+ g \partial g^{-1}=0$.

The evident Noether current of a left transformation $g\mapsto U g$ is the (right-invariant) $$ L'_\mu= g \partial_\mu g^{-1}, $$ conserved by the above EOM, $$ \partial ^\mu L'_\mu =0, $$ (evident from the parity transform $g\leftrightarrow g^{-1}$ of the above EOM form.)

But it is zero-curvature, so not self-dual, $$ \partial \cdot \tilde L'= \epsilon^{\mu\nu} \partial_\mu g \partial_\nu g^{-1}= -L'\cdot \tilde L' . $$ (N.B. This is a central property in 2d nonabelian duality, and inapplicable/irrelevant to 4d, where these models have been used since the 60s.)

Witten chose to cut this Gordian knot by noting these two relations combine to one, $$ \partial^\mu L_\mu'' =0 $$ for a self-dual current, $$ L_\mu ''\equiv L_\mu ' + \tilde L'_\mu = \tilde L_\mu '' ~~. $$

  • But wait!This is not conserved, as seen above, for the chiral model. Could one, instead, augment the action so it is?

That is, how does one obtain an extended action with EOM $$ \partial g \cdot \partial g^{-1} + g \Box g^{-1} + \epsilon^{\mu \nu} \partial_\mu g \partial_\nu g^{-1}=0 ~~? $$ Witten noted this would actually be identical to the parity image right current conservation below!

The short bare-bones-simplified answer is augmenting the chiral action by his eponymous topological term for a special fixed coupling, extending the $g(x)$ s to $\bar g(y)$ s in a solid ball B with the $S^2$ sphere as a boundary. This is probably not the place to review this WZW term, whose analog summarizes the flavor-chiral anomalies of low-energy QCD. The crucial property is that the variation of the WZW term is a surface term in the ball, so it is local on the sphere (!). That is the full action for a special coupling is proportional to $$ \operatorname{Tr} \Biggl ( \int_{S^2} d^2x ~~ \partial^\mu g \partial_\mu g^{-1} -\frac{2}{3} \int_B d^3y ~~\epsilon^{ijk} \bar g ^{-1} \partial_i \bar g ~\bar g ^{-1} \partial_j \bar g ~\bar g ^{-1} \partial_k \bar g \Biggr ), $$ has variation $$ \bbox[yellow]{ 2\operatorname{Tr} \int_{S^2} d^2x ~ g\delta g^{-1} \Biggl ( \partial_\mu (g \partial^\mu g^{-1} )+\epsilon^{\mu \nu } \partial_\mu g \partial_\nu g^{-1} \Biggr )}, $$ yielding the above equations of motion. You haven't lived if you skipped this most instructive of exercises. This is how conserved self-dual currents are achieved.

The analog for the parity-image right currents holds for the opposite relative sign of the WZW term, since these are anti-self-dual, $R_\mu ''= g^{-1} \partial^\mu g -\epsilon^{\mu \nu } g^{-1} \partial_\nu g =-\tilde R''_\mu$.

Indeed, pursuing the current algebra confirms the wisdom of the choice of $L''$ and $R''$ and whence of the isospin (vector) and axial currents.

Once the geometry is thus fixed, the RG has no choice but stay put at a fixed point (geometrostasis), so the theory does not renormalize, just like the free fermion equivalent theory.

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For starters, the WZ-term is needed to get EOMs that factorize in right- & left-movers, similar to the dual fermionic theory, cf. eq. (15) in Ref. 1.

References:

  1. E. Witten, Non-Abelian Bosonization in Two Dimensions, Commun. Math. Phys. 92 (1984) 455.
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