Finite sigma models and exact string solutions with Minkowski signature metric

We consider $2d$ sigma models with a $D=2+N$ - dimensional Minkowski signature target space metric having a covariantly constant null Killing vector. These models are UV finite. The $2+N$-dimensional target space metric can be explicitly determined for a class of supersymmetric sigma models with $N$-dimensional `transverse' part of the target space being homogeneous K\"ahler. The corresponding `transverse' sub-theory is an $n=2$ supersymmetric sigma model with the exact $\gb$-function coinciding with its one-loop expression. For example, the finite $D=4$ model has $O(3)$ supersymmetric sigma model as its `transverse' part. Moreover, there exists a non-trivial dilaton field such that the Weyl invariance conditions are also satisfied, i.e. the resulting models correspond to string vacua. Generic solutions are represented in terms of the RG flow in `transverse' theory. We suggest a possible application 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' are identified with the light cone coordinates of the $2+N$ - dimensional sigma model.