Phase transitions, double-scaling limit, and topological strings

TL;DR

This paper investigates phase transitions in topological strings on local Calabi-Yau threefolds, revealing a double-scaling limit governed by Painlevé I and proposing a new open/closed string duality.

Contribution

It provides the first analysis of phase transitions in this setting, solves the planar model, and introduces a double-scaling limit linked to Painlevé I, connecting perturbative and nonperturbative theories.

Findings

Identified a phase transition at small radius in the universality class of 2d gravity.

Established a double-scaling limit governed by Painlevé I equation.

Proposed a new open/closed duality relating Calabi-Yau backgrounds to open strings with framing.

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 Painlev\'e 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. We also give evidence for a new open/closed duality relating these Calabi--Yau backgrounds to open strings with framing.