String vacuum backgrounds with covariantly constant null Killing vector and 2d quantum gravity

TL;DR

This paper explores 2D sigma models with specific target space metrics, linking solutions to RG flows and interpreting them as quantum gravity models coupled to transverse theories, including special conformal cases.

Contribution

It provides a systematic analysis of solutions with covariantly constant null Killing vectors, connecting conformal invariance conditions to RG flows in transverse space and interpreting the models as quantum gravity coupled to sigma models.

Findings

Solutions expressed via RG flow in transverse space

Identification of conformal factor with light cone coordinate

Reproduction of known flat transverse space solutions

Abstract

We consider a $2d$ sigma model with a $2+N$ - dimensional Minkowski signature target space metric having a covariantly constant null Killing vector. We study solutions of the conformal invariance conditions in $2+N$ dimensions and find that generic solutions can be represented in terms of the RG flow in $N$ - dimensional ``transverse space'' theory. The resulting conformal invariant sigma model is interpreted as a quantum action of the $2d$ scalar (``dilaton") quantum gravity model coupled to a (non-conformal) `transverse' sigma model. The conformal factor of the $2d$ metric is identified with a light cone coordinate of the $2+N$ - dimensional sigma model. We also discuss the case when the transverse theory is conformal (with or without the antisymmetric tensor background) and reproduce in a systematic way the solutions with flat transverse space known before.