Gravitationally dresed RG flows and zig-zag invariant strings

We propose a world-sheet realization of the zigzag-invariant bosonic and fermionic strings as a perturbed Wess-Zumino-Novikov-Witten model at large negative level $k$ on a group manifold $G$ coupled to 2D gravity. In the large $k$ limit the zigzag symmetry can be obtained as a result of a self-consistent solution of the gravitationally dressed RG equation. The only solution found for simple group is $G=SL(2)$. More general target-space geometries can be obtained via tensoring of various cosets based on SL(2). In the supersymmetric case the zigzag symmetry fixes the maximal target-space dimension of the confining fermionic string to be seven.