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2020 The quantum tropical vertex
Pierrick Bousseau
Geom. Topol. 24(3): 1297-1379 (2020). DOI: 10.2140/gt.2020.24.1297

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.

Citation

Download Citation

Pierrick Bousseau. "The quantum tropical vertex." Geom. Topol. 24 (3) 1297 - 1379, 2020. https://doi.org/10.2140/gt.2020.24.1297

Information

Received: 15 November 2018; Revised: 1 July 2019; Accepted: 4 September 2019; Published: 2020
First available in Project Euclid: 6 October 2020

zbMATH: 07256607
MathSciNet: MR4157555
Digital Object Identifier: 10.2140/gt.2020.24.1297

Subjects:
Primary: 14N35

Keywords: Gromov–Witten Invariants , quantum tori , scattering diagrams

Rights: Copyright © 2020 Mathematical Sciences Publishers

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