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I came across this question while reading Chapter 15 of Conformal Field Theory by Di Francesco.

So the action of the Weiss-Zumino-Witten(WZW) model is as follows:

$$S = \frac{1}{4a^2}\int d^2x {\rm Tr}'(\partial^{\mu}g^{-1}\partial_\mu g) + k\Gamma$$

where $g$ is in some representation of Lie Group $G$ and $\Gamma$ is the Weiss-Zumino-Witten term. The action is invariant under the transformation

$$g \rightarrow \Omega_1 g \Omega_2$$

where $\Omega_{1,2}$ belongs to $G$. So it has symmetry $G\times G$. But, it later pointed out that due to the holomorphic factorization of the $g$ field, this symmetry is actually to $G(z)\times G(\bar{z})$.

My question is, this enlargement from a global symmetry to a local symmetry is reminicent of gauge symmetry. But I think things are here are a bit different because here the symmetry is not a redundancy. Is there a way to make this contrast/comparison between symmetry in WZW model to gauge symmetry more precise? Is it because some special properties of gauge transformations in 2d?

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Kaixiang Su
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1 Answers1

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This enlargement of symmetry is related to gauge symmetry in the following way.

It was realised by Witten [1], and further developed in [2] (see also this physics SE answer and the standard Chern-Simons reference [3]), that you can view a WZW model with fields $g:\Sigma\to G$, as the boundary theory for a $G_k\times G_{-k}$ Chern-Simons theory ($k$ is the level), on $M$ where $\partial M = \Sigma$. Chern-Simons has a $G\times G$ gauge symmetry. When you place it on a manifold with a boundary and impose holomorphic and anti-holomorphic boundary conditions, the gauge transformations that do not vanish on the boundary become global symmetries. In particular, they give rise to an $\mathfrak{g}_k\oplus\bar{\mathfrak{g}}_k$ current algebra. This is the infinite algebra whose conserved charges correspond to integrals of the holomorphic/anti-holomorphic conserved currents times arbitrary holomorphic/anti-holomorphic functions.

References:

[1]: E. Witten, Quantum Field Theory and the Jones Polynomial

[2]: D.C.Cabra, G.L.Rossini , Explicit connection between conformal field theory and 2+1 Chern-Simons theory, arXiv:hep-th/9506054

[3]: S. Elitzur, G. W. Moore, A. Schwimmer, and N. Seiberg, Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory, Nucl. Phys. B 326 (1989) 108–134 (pdf without paywall).

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