The quantum tropical vertex

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 $q=e^{i\hslash }$, 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.

v3: 69 pages, minor correction in Section 8.5 compared to the version published in Geometry and Topology

References in corpus (4)

• BPS states, knots and quivers
• Tropical refined curve counting from higher genera and lambda classes
• Local Gromov-Witten Invariants are Log Invariants
• Quantum mirrors of log Calabi-Yau surfaces and higher genus curve counting

Cited by in corpus (13)

• The canonical wall structure and intrinsic mirror symmetry
• Stable maps to Looijenga pairs
• Holomorphic anomaly equation for $({\mathbb{P}}^2,E)$ and the Nekrasov-Shatashvili limit of local ${\mathbb{P}}^2$
• Scattering diagrams, stability conditions, and coherent sheaves on ${\mathbb{P}}^2$
• Strong positivity for the skein algebras of the $4$-punctured sphere and of the $1$-punctured torus
• A proof of N.Takahashi's conjecture for $({\mathbb{P}}^2,E)$ and a refined sheaves/Gromov-Witten correspondence
• Refined floor diagrams from higher genera and lambda classes
• Stable maps to Looijenga pairs: orbifold examples
• Quivers and curves in higher dimension
• Real Log Curves in Toric Varieties, Tropical Curves, and Log Welschinger Invariants
• Enumerative geometry of surfaces and topological strings
• Tropical vertex and real enumerative geometry
• The log-open correspondence for two-component Looijenga pairs