Finiteσmodels and exact string solutions with Minkowski signature metric

We consider two-dimensional (2D) σ 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 σ models with the N-dimensional "transverse" part of the target space being σ homogeneous K\"ahler type. The corresponding "transverse" subtheory is an $n=2$ supersymmetric σ model with the exact β function coinciding with its one-loop expression. For example, the finite $D=4$ model has the O(3) sypersymmetric σ model as its "transverse" part. Moreover, there exists a nontrivial 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 renormalization group flow in "transverse" theory. We suggest a possible application of the constructed Weyl-invariant σ models to quantization of 2D gravity. They may be interpreted as "effective actions" of the quantum 2D dilaton gravity coupled to a (nonconformal) 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 σ model.