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 pointsV ₁ andV ₂ 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 matrices of differential operators. These conditions onV ₁ andV ₂ 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 constraintsL _n (n0), whereL _n annihilate the two modified-KdV -functions whose product gives the partition function of the Unitary Matrix Model.