A DK Phase Transition in q-Deformed Yang-Mills on S 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 S2. The strong coupling phase is characterized by the formation of a clump of eigenvalues in the associated matrix model of the Douglas-Kazakov (DK) type [1]. 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 θ angle. The Ooguri-Strominger-Vafa [2] relation of this theory to topological strings on the local Calabi-Yau O(−p)⊕O(p−2) → P1 via a chiral decompostion at large N [3], 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.