Disk potential functions for quadrics (Q2693957)

A certain disk potential is a generating function counting pseudo-holomorphic disks bounded by a Lagrangian submanifold; it has played a pivotal role in symplectic topology and mirror symmetry. When a Lagrangian submanifold \(L\) is a torus and its disk potential is defined, the disk potential governs deformations of underlying Fukaya's \(A_\infty\)-algebra associated to \(L\) as it has been effectively employed to detect Lagrangian tori having nontrivial (deformed) Floer cohomology. In the context of SYZ mirror symmetry beyond Calabi-Yau manifolds, the disk potential provides a Landau-Ginzburg mirror of the ambient symplectic manifold. Computing disk potential functions is a nontrivial task as it requires classification of holomorphic disks and computation of counting invariants using certain transversality of moduli spaces. When it comes to the disk potentials, the most well-studied examples are toric manifolds and orbifolds. They have a Lagrangian torus fibration, consisting of free real torus orbits on a dense subset. In particular, for compact Fano toric manifolds, the classification shows that the disk potential agrees with the Givental-Hori-Vafa superpotential, which can compute the quantum cohomology ring as its Jacobian ring. Y. Kim computes the disk potential of Gelfand-Zeitlin monotone torus fiber in a quadric hypersurface via using toric degenerations, Lie theoretical mirror symmetry, and the structural result of the monotone Fukaya category.