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The quantum tropical vertex

Pierrick Bousseau

Geometry & Topology 24 (2020) 1297–1379

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 = ei, 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 Chern–Simons theory.

We also prove some new BPS integrality results and propose some other BPS integrality conjectures.

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scattering diagrams, quantum tori, Gromov–Witten invariants
Mathematical Subject Classification 2010
Primary: 14N35
Received: 15 November 2018
Revised: 1 July 2019
Accepted: 4 September 2019
Published: 30 September 2020
Proposed: Jim Bryan
Seconded: Dan Abramovich, Richard P Thomas
Pierrick Bousseau
Department of Mathematics
Imperial College London
United Kingdom
Institute for Theoretical Studies
ETH Zürich