A DK Phase Transition in q-Deformed Yang-Mills on S² and Topological Strings

We demonstate the existence of a large $N$ phase transition with respect to the 't Hooft coupling in q-deformed Yang-Mills theory on $S^2$. The strong coupling phase is characterized by the formation of a clump of eigenvalues in the associated matrix model of Douglas-Kazakov (DK) type (hep-th/9305047). By understanding this in terms of instanton contributions to the q-deformed Yang-Mills theory, we gain some insight into the strong coupling phase as well as probe the phase diagram at nonzero values of the $\theta$ angle. The Ooguri-Strominger-Vafa relation (hep-th/0405146) of this theory to topological strings on the local Calabi-Yau $\mathcal{O}(-p) \oplus \mathcal{O}(p-2) \to \mathbb{P}^1$ via a chiral decompostion at large $N$ hep-th/0411280, motivates us to investigate the phase structure of the trivial chiral block, which corresponds to the topological string partition function, for $p>2$. We find a phase transition at a different value of the coupling than in the full theory, indicating the likely presence of a rich phase structure in the sum over chiral blocks.