A class of finite two-dimensional sigma models and string vacua

Abstract

We consider a two-dimensional Minkowski signature sigma model with a (2 + N)-dimensional target space metric having a null Killing vector. It is found that the model is finite to all orders of the loop expansion if the dependence of the “transverse” part of the metric gij(u, x) on the light cone coordinate u is subject to the standard renormalization group equation of the N-dimensional sigma model, dgij/du=βijg=Rij+…. In particular, we discuss the “one-coupling” case when gij(u, x) is a metric of an N-dimensional symmetric space γij(x) multiplied by a function f(u). The theory is finite if f(u) is equal to the “running” coupling of the symmetric space sigma model (with u playing the role of the RG “time”). For example, the geometry of space-time with γij being the metric of the N-sphere is determined by the form of the β-function of the O(N + 1) model. The “asymptotic freedom” limit of large u corresponds to the weak coupling limit of small (2 + N)-dimensional curvature. We show that there exists a dilaton field which together with the (2 + N)-dimensional metric solves the sigma model Weyl invariance conditions. The resulting backgrounds thus represent new tree-level string vacua. We remark on possible connections with some 2D quantum gravity models.