Integrability of the holomorphic anomaly equations

TL;DR

This paper demonstrates that modularity and the gap condition render the holomorphic anomaly equations fully integrable for non-compact Calabi-Yau manifolds, enabling efficient solutions for topological string amplitudes across moduli space.

Contribution

It introduces a formalism leveraging modularity and the gap condition to solve holomorphic anomaly equations exactly for non-compact Calabi-Yau geometries.

Findings

Provides a method to compute topological string amplitudes using almost holomorphic modular forms.

Achieves holomorphic expansions at all points in moduli space, including large radius and conifold points.

Offers an efficient approach to higher genus string amplitudes in matrix models with multiple cuts.

Abstract

We show that modularity and the gap condition make the holomorphic anomaly equation completely integrable for non-compact Calabi-Yau manifolds. This leads to a very efficient formalism to solve the topological string on these geometries in terms of almost holomorphic modular forms. The formalism provides in particular holomorphic expansions everywhere in moduli space including large radius points, the conifold loci, Seiberg-Witten points and the orbifold points. It can be also viewed as a very efficient method to solve higher genus closed string amplitudes in the $\frac{1}{N^2}$ expansion of matrix models with more then one cut.