Family Floer SYZ conjecture for $A_n$ singularity

We resolve a mathematically precise SYZ conjecture for $A_n$ singularity by building a quantum-corrected T-duality between two singular torus fibrations related to the K\"ahler geometry of the $A_n$-smoothing and the Berkovich geometry of the $A_n$-resolution, respectively. Our approach involves novel computations that embody a non-archimedean version of the partition of unity, and it confirms the strategy that patching verified local singularity models brings global SYZ conjecture solutions (like K3 surfaces) within reach. There is also explicit extra evidence concerning the collision of singular fibers and braid group actions. On one hand, we address the central challenge of matching SYZ singular loci identified by Joyce. In reality, we construct not merely an isolated SYZ mirror fibration partner, but a parameter-dependent one that always keeps the matching singular loci plus integral affine structure, even when the collision of singular fibers occurs. On the other hand, our SYZ result displays a visible tie, regardless of the parameter choice, between the $(A_n)$-configuration of Lagrangian spheres occurred as vanishing cycles in the $A_n$-smoothing and the exceptional locus of rational $(-2)$-curves in the $A_n$-resolution, which aligns with the celebrated works of Khovanov, Seidel, and Thomas.