THE SOLUTION SPACE OF THE UNITARY MATRIX MODEL STRING EQUATION AND THE SATO GRASSMANNIAN

The space of all solutions to the string equation of the symmetric unitary one-matrix model is determined. It is shown that the string equation is equivalent to simple conditions on points V1 and V2 in the big cell Gr(0) of the Sato Grassmannian Gr. This is a consequence of a well-defined continuum limit in which the string equation has the simple form [P, 2_] = 1, with P and 2_ 2 x 2 matrices of differential operators. These conditions on V1 and V2 yield a simple system of first order differential equations whose analysis determines the space of all solutions to the string equation. This geometric formulation leads directly to the Virasoro constraints L(n) (n greater-than-or-equal-to 0), where L(n) annihilate the two modified-KdV tau-functions whose product gives the partition function of the Unitary Matrix Model.