Non-perturbatively stable conformal minimal models coupled to two-dimensional quantum gravity

A study of two-dimensional quantum gravity coupled to the conformal minimal models (labelled by the positive integers (p,q)) is presented. Particular attention is paid to the understanding of non-perturbative issues. By studying certain random matrix models, a stable non-perturbative formulation is developed for the (2m - 1,2) sector. This formulation is understood in terms of the KP formalism, which appears to reproduce all of the physics of the matrix model description of the (2m - 1,2) sector. A generalisation of this successful non-perturbative formulation is presented for the KP description of the complete (p,q) sector. The resulting physics, where studied by example, is seen to be stable. However the KP description does not appear to reproduce all of the physics of the matrix model description of the gravitating (p,q) models beyond the (2m - 1,2) sector. It does not seem to be possible to subject the matrix model description of this sector to the above successful non-perturbative formulation. The success in formulating a non-perturbative description of the full set of gravitating (p,q) models is therefore incomplete. As these models are equivalent to certain types of toy string theories, the implication of these results for the program of finding a non-perturbative formulation of string theory are briefly discussed.