Large Order Behaviour of 2 D Gravity Coupled to d < 1 Matter

We discuss the large order behaviour and Borel summability of the topological expansion of models of 2D gravity coupled to general (p, q) conformal matter. In a previous work it was proven that at large order k the string susceptibility had a generic a k Γ(2k − 1 2) behaviour. Moreover the constant a, relevant for the problem of Borel summability, was determined for all one-matrix models. We here obtain a set of equations for this constant in the general (p, q) model. String equations can be derived from the construction of two differential operators P, Q satisfying canonical commutation relations [P, Q] = 1. We show that the equation for a is determined by the form of the operators P, Q in the spherical or semiclassical limits. The results for the general one-matrix models are then easily recovered. Moreover, since for the (p, q) string models such p = (2m + 1)q ± 1 the semiclassical forms of P, Q are explicitly known, the large order behaviour is completely determined. This class contains all unitary (q +1, q) models for which the answer is specially simple. As expected we find that the topological expansion for unitary models is not Borel summable.