The coupling of conformal field theories to 2-d gravity may be studied in the conformal gauge. As an application, the results of Knizhnik, Polyakov and Zamolodchikov for the scaling dimensions of conformal fields are derived in a simple way. Their conjecture for the susceptibility exponent γ of strings is proven and extended to arbitrary genus surfaces. The result agrees with exact results from random lattice models.

In this paper, we rigorously construct Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov. We establish some of its fundamental properties like conformal covariance under PSL2(C)-action, Seiberg bounds, KPZ scaling laws, KPZ formula and the Weyl anomaly formula. We also make precise conjectures about the relationship of the theory to scaling limits of random planar maps conformally embedded onto the sphere.

We solve the puzzle of the disagreement between orthogonal polynomials methods and mean field calculations for random N × N matrices with a disconnected eigenvalue support. We show that the difference does not stem from a Z 2 symmetry breaking, but from the discreteness of the number of eigenvalues. This leads to additional terms (quasiperiodic in N) which must be added to the naive mean field expressions. Our result invalidates the existence of a smooth topological large N expansion and some postulated universality properties of correlators. We derive the large N expansion of the free energy for the general 2-cut case. From it we rederive by a direct and easy meanfield-like method the 2-point correlators and the asymptotic orthogonal polynomials. We extend our results to any number of cuts and to non-real potentials.