NONPERTURBATIVE 2-DIMENSIONAL QUANTUM-GRAVITY, KDV HIERARCHY AND SOLUTIONS OF THE STRING EQUATIONS

We investigate the relation between the multicritical one-matrix random model and the moduli space parametrizing the solutions of the KdV hierarchy and give a geometrical interpretation of the non-perturbative free parameters of 2D-QG. From our viewpoint the string equations, in the asymptotic limit, give the boundary conditions of the stationary generalized KdV equation which contains all the information about the theory. We thus study those stationary solutions, matching the asymptotic behaviour of the solutions of the string equations, which are real and free of singularities in the scaling variable. We also discuss the consequences of these requirements on the geometry of such solutions. © 1990.