Matrix models of two-dimensional gravity and Toda theory

Abstract

Recurrent relations for orthogonal polynomials, arising in the study of the one-matrix model of two-dimensional gravity, are shown to be equivalent to the equations of the Toda-chain hierarchy supplemented by additional Virasoro constraints. This is the case even before the continuum limit is taken. When the odd times are suppressed, the Volterra hierarchy arises, its continuum limit being the KdV hierarchy. The unitary-matrix model gives rise to a sort of quantum deformation of the Volterra equation. We call it the modified Volterra hierarchy, since in the analogous continuum limit it may turn into the mKdV. For multimatrix models the Toda lattice hierarchy of the type A ∞ appears. The time-dependent partition functions are given by τ-functions — the sections of the determinant bundle over the infinite-dimensional grassmannian, associated with Riemann surfaces of spectral parameters. However, because of the Virasoro constraints matrix models correspond to a very restricted subset of the grassmannian, intimately related to the W ∞ Lie subalgebra of UGL(∞). The continuum limit, leading to multicritical behavior is adequately described in terms of peculiar operators, defined as hypergeometric polynomials of the original matrix fields and resembling a sort of handle-gluing operator.