Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

Abstract

We construct special Lagrangian fibrations for log Calabi-Yau surfaces, and scattering diagrams from Lagrangian Floer theory of the fibres. Then we prove that the scattering diagrams recover the scattering diagrams of Gross-Pandharipande-Siebert and the canonical scattering diagrams of Gross-Hacking-Keel. With an additional assumption on the non-negativity of boundary divisors, we compute the disc potentials of the Lagrangian torus fibres via a holomorphic/tropical correspondence. As an application, we provide a version of mirror symmetry for rank two cluster varieties.

61 pages, 9 figures. comments are welcome; v2. exposition improved, accepted version

References in corpus (2)

• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Fano mirror periods from the Frobenius structure conjecture

Cited by in corpus (1)

• Smoothing, scattering, and a conjecture of Fukaya