The quantum tropical vertex

Gross, Pandharipande and Siebert have shown that the 2-dimensional Kontsevich- Soibelman scattering diagrams compute certain genus-zero log Gromov-Witten invariants of log Calabi-Yau surfaces. We show that the q-refined 2-dimensional Kontsevich-Soibelman scattering diagrams compute, after the change of variables q = e(i (h) over bar), generating series of certain higher-genus log Gromov-Witten invariants of log Calabi-Yau surfaces. 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 Chem-Simons theory. We also prove some new BPS integrality results and propose some other BPS integrality conjectures.