Lifting holomorphic disks from flag varieties to basic affine spaces

Let $G$ be a complex reductive algebraic group, the complexification of a compact Lie group $K$. Consider a holomorphic principal $G$-bundle whose total space contains a $K$-invariant Lagrangian submanifold $L$. We develop a method for lifting holomorphic disks from the base with boundary on $L\mathrm{/}K$ to the principal $G$-bundle. We show that the Gross–Hacking–Keel–Kontsevich superpotential restricted to a distinguished class of seeds of the basic affine space is obtained by lifting holomorphic disks in the corresponding flag manifold.