Phase transitions, double-scaling limit, and topological strings

Topological strings on Calabi--Yau manifolds are known to undergo phasetransitions at small distances. We study this issue in the case of perturbativetopological strings on local Calabi--Yau threefolds given by a bundle over atwo-sphere. This theory can be regarded as a q--deformation of Hurwitz theory,and it has a conjectural nonperturbative description in terms of q--deformed 2dYang--Mills theory. We solve the planar model and find a phase transition atsmall radius in the universality class of 2d gravity. We give strong evidencethat there is a double--scaled theory at the critical point whose all genusfree energy is governed by the Painlev\'e I equation. We compare the criticalbehavior of the perturbative theory to the critical behavior of itsnonperturbative description, which belongs to the universality class of 2dsupergravity. We also give evidence for a new open/closed duality relatingthese Calabi--Yau backgrounds to open strings with framing.Comment: 49 pages, 3 eps figures; section added on non-perturbative proposal and 2d gravity; minor typos correcte