Bilinear Functional Equations in 2D Quantum Gravity

The microscopic theories of quantum gravity related to integrable lattice models can be constructed as special deformations of pure gravity. Each such deformation is defined by a second order differential operator acting on the coupling constants. As a consequence, the theories with matter fields satisfy a set of constraints inherited from the integrable structure of pure gravity. In particular, a set of bilinear functional equations for each theory with matter fields follows from the Hirota equations defining the KP (KdV) structure of pure gravity.