Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1992-05-18
Phys.Lett.B288:279-285,1992
Physics
High Energy Physics
High Energy Physics - Theory
15 pages [Complete revision. The main statement of the previous version is generalised to the case of an arbitrary ``transvers
Scientific paper
10.1016/0370-2693(92)91104-H
We consider a two - dimensional Minkowski signature sigma model with a $2+N$ - dimensional target space metric having a null Killing vector. It is shown that the model is finite to all orders of the loop expansion if the dependence of the ``transverse" part of the metric $\ggij (u,x)$ on the light cone coordinate $u$ is subject to the standard renormalization group equation of the $N$ - dimensional sigma model, $ {d\ggij\over du} = \gb_{ij} =R_{ij} + ... $. In particular, we discuss the `one - coupling' case when $\ggij(u,x)$ is a metric of an $N$ - dimensional symmetric space $\gij(x)$ 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 $\gij$ being the metric of the $N$ - sphere is determined by the form of the $\gb$ - 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 prove 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 also remark on possible connections with some $2d$ quantum gravity models.
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