Noncritical Strings, RG Flows and Holography

We derive an RG flow equation that is satisfied by the regularized partition function for noncritical strings in background fields. The flow refers to change in the position of a ``boundary’’ in the liouville direction. The boundary is required to regularize the ultraviolet divergences in the partition function coming from integration over world-sheets of arbitrarily small area. From the point of view of the target space effective gravitational action that the partition function evaluates on-shell, the boundary regularizes {\it infrared} divergences coming from the infinite volume of the liouville direction. The RG flow equation that we obtain looks very much like the Hamilton-Jacobi constraint equation that an on-shell gauge-fixed gravitational action must satisfy.