The quantum tropical vertex

Bousseau, Pierrick¹

¹ Department of Mathematics, Imperial College London, London, United Kingdom, Institute for Theoretical Studies, ETH Zürich, Zürich, Switzerland

Geometry & topology, Tome 24 (2020) no. 3, pp. 1297-1379.

Résumé

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

Détail

DOI : 10.2140/gt.2020.24.1297

Classification : 14N35
Keywords: scattering diagrams, quantum tori, Gromov–Witten invariants

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