The quantum tropical vertex

TL;DR

This paper demonstrates that q-refined Kontsevich-Soibelman scattering diagrams compute higher genus log Gromov-Witten invariants of log Calabi-Yau surfaces, linking mathematical invariants with physical theories and confirming aspects of topological string theory.

Contribution

It provides a rigorous mathematical framework for the refined wall-crossing formula and connects higher genus invariants with physical string theory concepts.

Findings

q-refined diagrams compute higher genus invariants

Mathematical validation of the refined wall-crossing formula

New BPS integrality results and conjectures

Abstract

Gross-Pandharipande-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 $q=e^{i \hbar}$, 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-Vafa, and in particular can be seen as a non-trivial mathematical check of the connection suggested by Witten between higher genus open A-model and Chern-Simons theory. We also prove some new BPS integrality results and propose some other BPS integrality conjectures.