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

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 ${Ł}_n(n\ge 0)$, where ${Ł}_n$ annihilate the two modified-KdV $\t$-functions whose product gives the partition function of the Unitary Matrix Model.

21 pages

References in corpus (1)

• On the solutions to the string equation

Cited by in corpus (5)

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• String Equations for the Unitary Matrix Model and the Periodic Flag Manifold
• Large Representation Recurrences in Large N Random Unitary Matrix Models