Boundary conformal field theories on random surfaces and the non-critical open string

We analyse boundary conformal field theories on random surfaces using the conformal gauge approach of David, Distler and Kawai, The crucial point is the choice of boundary conditions on the Liouville field, We discuss the Weyl anomaly cancellation for Polyakov's non-critical open bosonic string with Neumann, Dirichlet and free boundary conditions. Dirichlet boundary conditions on the Liouville field imply that the metric is discontinuous as the boundary is approached. We consider the semi-classical limit and argue how it singles out the free boundary conditions for the Liouville field, We define the open string susceptibility, the anomalous gravitational scaling dimensions and a new Yang-Mills Feynman mass critical exponent.