MATRIX MODELS WITHOUT MATRICES: FROM INTEGRABLE HIERARCHIES TO RECURSION RELATIONS

Virasoro constraints on integrable hierarchies and their consequences are studied using the formalism of dressing operators. The dressing-operator description allows one to perform entirely in intrinsically hierarchical terms a double-scaling limit which takes "discrete" (lattice) Virasoro-constrained hierarchies into continuum hierarchies subjected to their own Virasoro constraints. Certain equations derived as consequences of the constraints suggest an interpretation as recursion/loop equations, thus establishing a link with the field-theoretic description. Such a correspondence with two-dimensional gravity-coupled theories, which does not require going through the matrix formulation, is conjectured to hold for general integrable hierarchies of the r-matrix type (appropriately constrained). The example considered explicitly is that of the Virasoro-constrained Toda hierarchy which undergoes a scaling into the Virasoro-constrained KP hierarchy, which in turn can be reduced to N-KdV hierarchies subjected to a subset of the KP Virasoro constraints. The dressing-operator formulation also facilitates the analysis of symmetry algebras of constrained hierarchies. The Kac–Moody sl (N) algebra is identified as a symmetry of the N-KdV hierarchy, while for the Virasoro-constrained KP hierarchy its symmetry algebra is related to a member of the family of the W∞(J) algebras. In the supersymmetric case this method allows one to impose super-Virasoro constraints on the super-KP hierarchy consistently with all the SKP flows.