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The quantum tropical vertex

Pierrick Bousseau

Geometry & Topology 24 (2020) 1297–1379
Abstract

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.

Keywords
scattering diagrams, quantum tori, Gromov–Witten invariants
Mathematical Subject Classification 2010
Primary: 14N35
References
Publication
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
Authors
Pierrick Bousseau
Department of Mathematics
Imperial College London
London
United Kingdom
Institute for Theoretical Studies
ETH Zürich
Zürich
Switzerland