Unitary Matrix Models and Painlev\'{e} III

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.Comment: 10 pages, LaTeX, final version to appear in Mod. Phys. Lett.

Unitary Matrix Models and Phase Transition

We study the unitary matrix model with a topological term. We call the topological term the theta term. In the symmetric model there is the phase transition between the strong and weak coupling regime at $\lambda_{c}=2$. If the Wilson term is bigger than the theta term, there is the strong-weak coupling phase transition at the same $\lambda_{c}$. On the other hand, if the theta term is bigger than the Wilson term, there is only the strong coupling regime. So the topological phase transition disappears in this case.Comment: 9 pages, LaTeX, Comments about the topological phase transition are adde