Multicritical complex matrix models and non-perturbative two-dimensional quantum gravity

Abstract

The one-complex-matrix model is analysed by orthogonal polynomials and saddle point methods. For the double scaled limit, multicritical matter theories arising from the perturbative phase have 1 2 the free energy of that derived from the hermitian matrices. However, certain other potentials yield a new hierarchy of “string equations”. These string equations are the most general equations compatible with the same operator content, KdV flows and asymptotic expansions of the m-critical points in hermitian matrix models. In particular, the non-perturbative solution of the m = 2 equation, corresponding to 2D quantum gravity, yields the same genus expansion as the cosmological constant μ tends to +∞, but tends to zero as μ → −∞. Semi-classical analysis shows that this solution is real and stable at this level.