The quantum tropical vertex
Pierrick Bousseau
Abstract
Gross, Pandharipande and Siebert have shown that the [math] –dimensional Kontsevich–Soibelman scattering diagrams compute certain genus-zero log Gromov–Witten invariants of log Calabi–Yau surfaces. We show that the [math] –refined [math] –dimensional Kontsevich–Soibelman scattering diagrams compute, after the change of variables [math] , generating series of certain higher-genus log Gromov–Witten invariants of log Calabi–Yau surfaces.\n¶ This result provides a mathematically rigorous realization of the physical derivation of the refined wall-crossing formula from topological string theory proposed by Cecotti and Vafa and, in particular, can be viewed as a nontrivial mathematical check of the connection suggested by Witten between higher-genus open A–model and Chern–Simons theory.\n¶ We also prove some new BPS integrality results and propose some other BPS integrality conjectures.