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

TL;DR

This paper introduces a class of finite two-dimensional sigma models with null Killing vectors, showing their connection to string vacua and RG flow, and explores their geometric and quantum properties.

Contribution

It demonstrates the finiteness of a class of 2D sigma models with specific target space metrics and constructs new string vacua using solutions to Weyl invariance conditions.

Findings

Models are finite to all orders if transverse metric depends on light cone coordinate via RG equations.

In the one-coupling case, the model's finiteness relates to the RG flow of symmetric space sigma models.

Existence of a dilaton field ensures solutions satisfy string theory Weyl invariance, leading to new string backgrounds.

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 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.