Unitary Matrix Models and Painlev\'{e} III

TL;DR

This paper explores the connection between full unitary matrix models, integrable equations, and Painlevé III, revealing new relationships through Toda and string equations, with implications for topological sigma models.

Contribution

It derives a special case of Painlevé III from unitary matrix models using Toda and string equations, and links Virasoro constraints to radial coordinates.

Findings

Derived a special Painlevé III equation from matrix models.

Established a relation between parameters akin to complex conjugation.

Connected matrix models to topological sigma model fusion processes.

Abstract

We discussed the full unitary matrix models from the view points of integrable equations and string equations. Coupling the Toda equations and the string equations, we derive a special case of the Painlev\'{e} III equation. From the Virasoro constrains, we can use the radial coordinate. The relation between $t_{1}$ and $t_{-1}$ is like the complex conjugate. %An implicit to the $t\bar{t}$ fusion of topological sigma model %is therby expected.