RATIONAL THEORIES OF 2D-GRAVITY FROM THE 2-MATRIX MODEL

The correspondence claimed by Douglas between the multicritical regimes of the two-matrix model and 2d gravity coupled with (p, q) rational matter field, is worked out explicitly. We found the minimal (p, q) multicritical potentials U(X) and V(Y), which are polynomials of degree p and q, correspondingly. The loop averages W(X) and W(Y) are shown to satisfy the Heisenberg relations {W, X} = 1 and {W, Y} = 1 and essentially coincide with the canonical momenta P and Q. The operators X and Y create the two kinds of boundaries in the (p, q) model related by the duality (p, q) <-> (q, p). Finally, we present a closed expression for the two two-loop correlators, and interpret its scaling limit.