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The solution space of the unitary matrix model string equation and the Sato Grassmannian

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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 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 (n≧0), whereL n annihilate the two modified-KdV τ-functions whose product gives the partition function of the Unitary Matrix Model.

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Communicated by N. Yu. Reshetikhin

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Anagnostopoulos, K.N., Bowick, M.J. & Schwarz, A. The solution space of the unitary matrix model string equation and the Sato Grassmannian. Commun.Math. Phys. 148, 469–485 (1992). https://doi.org/10.1007/BF02096545

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