Phase transitions, double-scaling limit, and topological strings

Abstract

Topological strings on Calabi-Yau manifolds are known to undergo phase transitions at small distances. We study this issue in the case of perturbative topological strings on local Calabi-Yau threefolds given by a bundle over a two-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 2D Yang-Mills theory. We solve the planar model and find a phase transition at small radius in the universality class of 2D gravity. We give strong evidence that there is a double-scaled theory at the critical point whose all-genus free energy is governed by the Painlevé I equation. We compare the critical behavior of the perturbative theory to the critical behavior of its nonperturbative description, which belongs to the universality class of 2D supergravity, and we comment on possible implications for nonperturbative 2D gravity. We also give evidence for a new open/closed duality relating these Calabi-Yau backgrounds to open strings with framing.

Topological Strings on Local Curves

“…In this note we review some properties of topological string theory on X p discovered in [8,24,25]. We summarize them briefly in this introduction by emphasizing both their connections to general aspects of topological string theory, as well as their idiosyncrasies.…”

“…The study of topological string theory on X p has led to many insights. For example, the conjecture relating toric backgrounds to matrix models stated in [24] and further refined in [5] was motivated to a large extent by the genus zero solution on X p presented in [8]. These backgrounds have been an important testing ground for recent techniques and ideas, but one has to keep in mind that they are rather unconventional in many respects.…”

“…This is established in section 2.2. Most of the properties above can be seen to be inherited by this Hurwitz model (like in particular the critical behavior and the instanton effects describing large order), and we refer for this to the original papers [8,25].…”

“…In the case of topological string theory on local curves, however, the singular behavior is described by the c = 0 string, i.e. by two-dimensional gravity [8]. Phase transitions on X p are described in section 3.…”

“…The backgrounds X p do not have standard mirror manifolds (see however [16] for some progress along this direction). However, a mirror geometry can be extracted by studying the saddle-point of the sum over partitions [8] and it is encoded in a complex curve which we will call the spectral curve of the model. This is similar to the way in which the Seiberg-Witten curve emerges from the sum over partitions in Nekrasov's computation [26,27].…”