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

Understand

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.