The quantum tropical vertex

Journal article

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^iℏ, 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.

Authors: Bousseau, P

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Mathematical Sciences Publishers
• Journal: Geometry & Topology
• Volume: 24
• Issue: 3
• Pages: 1297-1379
• Publication date: 2020-09-30
• Acceptance date: 2020-09-04
• DOI: 10.2140/gt.2020.24.1297
• EISSN: 1364-0380
• ISSN: 1465-3060
• Language: English
• Keywords: scattering diagrams; quantum tori; Gromov–Witten invariants