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The Solution Space of the Unitary Matrix Model String Equation and the Sato Grassmannian
High Energy Physics - Theory · Submitted 21 Dec 1991
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The space of all solutions to the string equation of the symmetric unitary one-matrix model is determined.
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The abstract, sentence by sentence
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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The space of all solutions to the string equation of the symmetric unitary one-matrix model is determined. | result 0.70 | 0.88 | 0.74 | asserted 1.00 |
| s1 | 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 $Gr$. | result 0.91 | 0.88 | 0.33 | asserted 1.00 |
| s2 | 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. | method 0.54 | 0.43 | 0.10 | asserted 1.00 |
| s3 | 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. | method 0.84 | 0.84 | 0.26 | asserted 1.00 |
| s4 | 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. | result 0.69 | 0.80 | 0.29 | asserted 1.00 |
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Original 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.
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Unitary one matrix models have been studied from various viewpoints: as a toy model of two-dimensional QCD [ref] , in relation to the string theories [ref] , as exactly solvable quantum theories [ref] , and in connection with integrable systems and their hierarchies [ref] .
Perturbation of multi-critical unitary matrix models, double scaling limits, and Argyres-Douglas theories · Introduction · 1.00
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