Family Floer program and non-archimedean SYZ mirror construction

Abstract

Given a Lagrangian fibration, we provide a natural construction of a mirror Landau-Ginzburg model consisting of a rigid analytic space, a superpotential function, and a dual fibration based on Fukaya's family Floer theory. The mirror in the B-side is constructed by the counts of holomorphic disks in the A-side together with the non-archimedean analysis and the homological algebra of the $A_{\infty }$ structures. It fits well with the SYZ dual fibration picture and explains the quantum/instanton corrections and the wall crossing phenomenon. Instead of a special Lagrangian fibration, we only need to assume a weaker semipositive Lagrangian fibration to carry out the non-archimedean SYZ mirror reconstruction.

81 pages. Following the referee's suggestion, many secondary details, which should be recoverable by the reader, have been omitted to streamline the paper, while most essential details have been retained

References in corpus (9)

• Formes différentielles réelles et courants sur les espaces de Berkovich
• Family Floer cohomology and mirror symmetry
• Lagrangian Floer theory and mirror symmetry on compact toric manifolds
• Kuranishi structure, Pseudo-holomorphic curve, and Virtual fundamental chain: Part 1
• SYZ conjecture for Calabi-Yau hypersurfaces in the Fermat family
• Homological mirror symmetry without corrections
• Kuranishi structure, Pseudo-holomorphic curve, and virtual fundamental chain: Part 2
• Differential forms, Fukaya $A_{\infty }$ algebras, and Gromov-Witten axioms
• Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: I

Cited by in corpus (3)

• Family Floer mirror space for local SYZ singularities
• Gromov-Hausdorff distance between filtered $A_{\infty }$ categories 1: Lagrangian Floer theory
• Cyclic group actions on Fukaya categories and mirror symmetry