( p, q) string actions

Abstract

We investigate a recently proposed nonperturbative formulation of two-dimensional quantum gravity coupled to ( p, q) minimal conformal matter. The coupled differential equations for the partition function summed over topologies are shown to follow from an action principle. The basic action for a ( p, q) model takes the general form S ( p, q) = ∫Res[ Q p/ q+ 1 + Σ k = 0 Σ α = 0 q − 2 t ( k), α Q k + ( α + 1)/ q ], where Q is a qth-order differential operator and the t ( k), α are sources for operator insertions. We illustrate our results with the explicit examples of the Ising (4,3) and tricritical Ising (5,4) models. The action S ( p, q) embodies the essential features of the problem (including the relation to generalized KdV hierarchies) in a most compact form.