Phase transitions in q -deformed 2D Yang-Mills theory and topological strings

We analyze large $N$ phase transitions for $U(N)$ $q$-deformed two-dimensional Yang-Mills theory on the sphere. We determine the phase diagram of the model and we show that, for small values of the deformation parameter, the theory exhibits a phase transition which is smoothly connected to the Douglas-Kazakov phase transition. For large values of the deformation parameter the phase transition is absent. By explicitly computing the one-instanton suppression factor in the weakly coupled phase, we also show that the transition is triggered by instanton effects. Finally, we present the solution of the model in the strongly coupled phase. Our analysis suggests that, on certain backgrounds, nonperturbative topological string theory has new phase transitions at small radius. From the point of view of gauge theory, it suggests a mechanism to smooth out large $N$ phase transitions.