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The solution space of the unitary matrix model string equation and the Sato Grassmannian. (English) Zbl 0753.35073

Summary: 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 \(Gr^{(0)}\) of the Sato Grassmannian \(Gr\). This is a consequence of a well-defined continuum limit in which the string equation has the simple form \([{\mathcal P},{\mathcal I}_ -]=1\), with \({\mathcal P}\) and \({\mathcal I}_ -2\times 2\) 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 \(L_ n\) annihilates the two modified-KdV \(\tau\)-functions whose product gives the partition function on the Unitary Matrix Model.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
83C45 Quantization of the gravitational field
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