The Exact Tachyon Beta - Function for the Wess - Zumino - Witten Model

  • Ian Jack, D . R . T . Jones
  • Published 1993


We derive an exact expression for the tachyon β-function for the Wess-Zumino-Witten model. We check our result up to three loops by calculating the three-loop tachyon β-function for a general non-linear σ-model with torsion, and then specialising to the case of the WZW model. The Wess-Zumino-Witten (WZW) model[1] is a particularly interesting example of a conformal field theory. Indeed it is currently believed that all rational conformal field theories may be derived from the WZW model by the Goddard-Kent-Olive (GKO) con-struction[2] (or equivalently by gauging[3]). The WZW model on a Lie group manifold G is parametrised by the level (which is constrained to be an integer for a compact G). The properties which characterise the conformal field theory–the central charge and the con-formal dimensions of the primary fields–have exact non-perturbative expressions in terms of the level and the Casimirs for the Lie algebra of G[4]. In general, we may consider perturbing the WZW model by adding a potential term to the action. This potential term is usually taken to be a primary field of the WZW model; however we would like to consider the case where this restriction is not applied. Our motivation for this comes from non-abelian Toda field theories[5], which have recently been receiving some attention[6] [7]. The action for these models consists precisely of a WZW model coupled to a potential term. In a recent paper[8] we discussed the conformal properties of the non-abelian Toda field theory at the quantum level. The conformal behaviour of the potential term is described by the tachyon β-function (in terminology derived from string theory) and therefore we needed an exact expression for this β-function. Our aim in this paper is to derive this. In the particular case where the potential is simply tr(g), where g ∈ G, the tachyon β-function is simply related to the conformal dimension of tr(g), and we shall exploit this to deduce the general form of the tachyon β-function. We shall then check this result by performing an explicit perturbative calculation up to three-loop order. We first obtain the result for a general non-linear σ-model, and then specialise to the case of the WZW model. In the WZW case, our perturbative calculation is similar to a three-loop calculation of the conformal dimension of tr(g) carried out some time ago[9]. However, we use a different prescription[10] [11] for continuing the two-dimensional alternating symbol to d dimensions in the …

Cite this paper

@inproceedings{Jack1993TheET, title={The Exact Tachyon Beta - Function for the Wess - Zumino - Witten Model}, author={Ian Jack and D . R . T . Jones}, year={1993} }