A Class of String Backgrounds as a Semiclassical Limit of WZW Models

A class of string backgrounds associated with non semi-simple groups is obtained as a special large level limit of ordinary WZW models. The models have an integer Virasoro central charge and they include the background recently studied by Nappi and Witten. November 1993 In this note we discuss from an algebraic point of view a class of exact conformal field theories which are based on WZW models [1] on group manifolds which are neither compact nor semi-simple. The feature of these models is that the Virasoro central charge is independent of the levels of the affine Lie algebra and is thus an integer equal to the dimension of the group manifold. A non-linear sigma model Lagrangian representation is formally available for all WZW models. To a given group G corresponds an appropriate target space metric and torsion in the non-linear sigma model. In that setting the criteria for conformal invariance is the exact vanishing of the beta function of the theory. Usually it is difficult to verify such a condition in the absence of some underlying algebraic structure. For non-abelian groups neither the metric nor the torsion are flat but they are nevertheless related in a manner reflecting the parallelisability of the manifold [2]. The Virasoro central charge is not necessarily integral and this can be understood in a " free field representation " as arising from the necessity to project out states of the free fields by a Feigin-Fuks like construction. Recently [3] it has been shown that a WZW model based on a central extension of the Euclidean group in two dimensions has a central charge equal to four, independently of the level. This result was established by considering the beta functions in a non-linear sigma model description with the central charge four found in perturbation theory. Non perturbative arguments were also given. Reinterpreting the algebra as a Wigner contraction of an so(3) × u(1) algebra we are able to present a large class of models based on a certain contraction of the group G × H where the latter is a compact subgroup of the former, assumed compact and simple. For the affine Lie algebra the construction involves a correlated large x limit (x being the level) of G and H. Therefore the string backgrounds represented by the WZW models based on the centrally extended non semisimple Lie algebras are actually semiclassical limits of the appropriate simple …