Airy Equation for the Topological String Partition Function in a Scaling Limit

TL;DR

This paper derives all-order perturbative free energies for topological string theory on Calabi-Yau threefolds using the polynomial holomorphic anomaly equations, revealing an Airy equation governing the partition function in a scaling limit.

Contribution

It introduces a novel scaling limit where the topological string partition function satisfies an Airy differential equation, connecting perturbative and non-perturbative aspects.

Findings

Partition function satisfies an Airy differential equation in the scaling limit.

Perturbative expansion corresponds to one solution of the Airy equation.

The other solution hints at non-perturbative structure and can be expanded at strong coupling.

Abstract

We use the polynomial formulation of the holomorphic anomaly equations governing perturbative topological string theory to derive the free energies in a scaling limit to all orders in perturbation theory for any Calabi-Yau threefold. The partition function in this limit satisfies an Airy differential equation in a rescaled topological string coupling. One of the two solutions of this equation gives the perturbative expansion and the other solution provides geometric hints of the non-perturbative structure of topological string theory. Both solutions can be expanded naturally around strong coupling.