Topological strings in Calabi–Yau manifolds undergo phase transitions at small distances that signal the onset of quantum geometry. In most models the universality class of the transition is the c=1 string at selfdual radius. In this talk I analyze these transitions for topological strings on bundles over a sphere, which are described by a q–deformation of Hurwitz theory. Their critical behavior turns out to be in the universality class of pure 2d gravity, and one can define a double–scaled theory at the critical point governed by the Painleve I equation. These theories have moreover a conjectural nonperturbative description in terms of q–deformed Yang–Mills theory, which exhibit a generalized Douglas–Kazakov transition. I speculate on the implications of this description for the nonperturbative completion of pure 2d gravity.

CERN