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

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

1991-12-21

arxiv.org/abs/hep-th/9112066v1

Commun.Math.Phys. 148 (1992) 469-486

Physics

High Energy Physics

High Energy Physics - Theory

21 pages

Scientific paper

10.1007/BF02096545

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.

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