The Solution Space of the Unitary Matrix Model String Equation and the Sato Grassmannian

TL;DR

This paper characterizes all solutions to the string equation of the symmetric unitary one-matrix model using the Sato Grassmannian, revealing geometric conditions that lead to Virasoro constraints on the model's partition function.

Contribution

It establishes a geometric framework connecting the string equation to points in the Sato Grassmannian, deriving differential equations and Virasoro constraints for the unitary matrix model.

Findings

Solutions correspond to specific points in the Sato Grassmannian.

The string equation simplifies to a commutation relation involving differential operators.

Virasoro constraints are derived from the geometric formulation.

Abstract

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 $V_1$ and $V_2$ in the big cell $\Gr$ of the Sato Grassmannian $Gr$. This is a consequence of a well-defined continuum limit in which the string equation has the simple form $\lb \cp ,\cq_- \rb =\hbox{\rm 1}$, with $\cp$ and $\cq_-$ $2\times 2$ matrices of differential operators. These conditions on $V_1$ and $V_2$ 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 constraints $\L_n\,(n\geq 0)$, where $\L_n$ annihilate the two modified-KdV $\t$-functions whose product gives the partition function of the Unitary Matrix Model.