Family Floer SYZ mirror algorithm for the Grassmannian $Gr(2,4)$

Abstract

We give an explicit non-archimedean SYZ construction for the Landau-Ginzburg mirror of $Gr(2,4)$. This work is complementary to the approach of Hong-Kim-Lau \cite{hong2023immersed} to SYZ mirror symmetry for Grassmannians, while we focus on a more concrete fibration-level realization of the SYZ picture. Starting from a Lagrangian fibration on the A-side, we explicitly construct a non-archimedean analytic mirror fibration inside the Berkovich analytification of the Langlands dual Grassmannian on the B-side. We show that the two fibrations have identical smooth and singular loci and induce the same integral affine structure on the smooth locus. Moreover, the natural disk-counting Landau-Ginzburg superpotential agrees with the Marsh-Rietsch superpotential. While the construction is guided by the family Floer viewpoint, the proof proceeds mainly through explicit geometric constructions and does not rely on Floer-theoretic arguments. Thus, the Langlands-dual mirror and its superpotential are realized explicitly within a single framework, providing concrete geometric evidence for the SYZ principle.