J. Zinn-Justin

The multicritical points of the $O(N)$ invariant $N$ vector model in the large $N$ limit are reexamined. Of particular interest are the subtleties involved in the stability of the phase structure at critical dimensions. In the limit $N \to \infty$ while the coupling $g \to g_c$ in a correlated manner (the double scaling limit) a massless bound state $O(N)$ singlet is formed and powers of $1/N$ are compensated by IR singularities. The persistence of the $N \to \infty$ results beyond the leading order is then studied with particular interest in the possible existence of a phase with propagating small mass vector fields and a massless singlet bound state. We point out that under certain conditions the double scaled theory of the singlet field is non-interacting in critical dimensions.

We review recent progress in 2D gravity coupled to $d<1$ conformal matter, based on a representation of discrete gravity in terms of random matrices. We discuss the saddle point approximation for these models, including a class of related $O(n)$ matrix models. For $d<1$ matter, the matrix problem can be completely solved in many cases by the introduction of suitable orthogonal polynomials. Alternatively, in the continuum limit the orthogonal polynomial method can be shown to be equivalent to the construction of representations of the canonical commutation relations in terms of differential operators. In the case of pure gravity or discrete Ising--like matter, the sum over topologies is reduced to the solution of non-linear differential equations (the Painlev\'e equation in the pure gravity case) which can be shown to follow from an action principle. In the case of pure gravity and more generally all unitary models, the perturbation theory is not Borel summable and therefore alone does not define a unique solution. In the non-Borel summable case, the matrix model does not define the sum over topologies beyond perturbation theory. We also review the computation of correlation functions directly in the continuum formulation of matter coupled to 2D gravity, and compare with the matrix model results. Finally, we review the relation between matrix models and topological gravity, and as well the relation to intersection theory of the moduli space of punctured Riemann surfaces.

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 $a^k\Gamma(2k-\ud)$ 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 $p=(2m+1)q\pm1$ 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. \preprint{SPhT/92-163}, Plain-TeX, macro harvmac