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We consider $2d$ sigma models with a $D=2+N$ - dimensional Minkowskisignature target space metric having a covariantly constant null Killingvector. These models are UV finite. The $2+N$-dimensional target space metriccan be explicitly determined for a class of supersymmetric sigma models with$N$-dimensional `transverse' part of the target space being homogeneousK\"ahler. The corresponding `transverse' sub-theory is an $n=2$ supersymmetricsigma model with the exact $\gb$-function coinciding with its one-loopexpression. For example, the finite $D=4$ model has $O(3)$ supersymmetric sigmamodel as its `transverse' part. Moreover, there exists a non-trivial dilatonfield such that the Weyl invariance conditions are also satisfied, i.e. theresulting models correspond to string vacua. Generic solutions are representedin terms of the RG flow in `transverse' theory. We suggest a possibleapplication of the constructed Weyl invariant sigma models to quantisation of$2d$ gravity. They may be interpreted as `effective actions' of the quantum$2d$ dilaton gravity coupled to a (non-conformal) $N$-dimensional `matter'theory. The conformal factor of the $2d$ metric and $2d$ `dilaton' areidentified with the light cone coordinates of the $2+N$ - dimensional sigmamodel.Comment: 24 pages, harvmac, Imperial/TP/92-93/

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

Finite<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>σ</mml:mi></mml:math>models and exact string solutions with Minkowski signature metric

Finiteσmodels and exact string solutions with Minkowski signature metric