Disk counting via family Floer theory

Given a family of Lagrangian tori with full quantum corrections, the non-archimedean SYZ mirror construction uses the family Floer theory to construct a non-archimedean analytic space with a global superpotential. In this talk, we will first briefly review the construction. Then, we will apply it to the Gross’s fibrations. As an application, we can compute all the non-trivial open GW invariants for a Chekanov-type torus in CPn or CPr×CPn−r. When n=2, r=1, we retrieve the previous results of Auroux an Chekanov-Schlenk without finding the disks explicitly. It is also compatible with the Pascaleff-Tonkonog’s work on Lagrangian mutations.