The non-perturbative topological string: from resurgence to wall-crossing of DT invariants

TL;DR

This paper explores the resurgence properties of topological string theory, linking alien calculus to wall-crossing phenomena and Donaldson-Thomas invariants, with numerical analysis on specific Calabi-Yau geometries.

Contribution

It introduces a differential operator for alien derivatives, establishing an algebraic link to wall-crossing and providing numerical evidence on the Borel plane structure.

Findings

Alien derivatives form an algebra isomorphic to the Kontsevich-Soibelman Lie algebra.

Borel singularities correspond to bound states involving D4-branes.

Identification of D2-brane decay in the Borel plane matching theoretical predictions.

Abstract

We study the resurgence structure of the topological string partition function, with an emphasis on the Borel analysis of the instanton amplitudes. To this end, we introduce a differential operator that implements the pointed alien derivative when acting on the topological string partition function and its iterated alien derivatives. We show that the algebra of alien derivatives is isomorphic to the Kontsevich-Soibelman Lie algebra, thus establishing a direct link between the resurgence of the topological string and wall-crossing of generalized Donaldson-Thomas invariants. Numerically, we continue the exploration of the Borel plane of the quintic and local $\mathbb{P}^2$. For the latter, we identify Borel singularities due to bound states involving D4-branes, and match the associated Stokes constants to the appropriate Donaldson-Thomas invariants. Finally, we identify the manifestation of a D2-brane decay in the Borel plane, and match to theoretical predictions.