Recover open Gromov–Witten and Welschinger invariants via mirror symmetry for Fano manifolds

Determine a mirror-symmetric method to recover the open Gromov–Witten invariants of a Lagrangian submanifold L ⊂ X and the Welschinger invariants of a real symplectic manifold X when X is a Fano manifold. The goal is to extract these enumerative invariants from the mirror Landau–Ginzburg side in a way analogous to the Calabi–Yau case, where such recovery is known for the quintic threefold and its real locus.

Background

The paper begins by motivating a longstanding question in mirror symmetry: in the Fano setting, how to obtain open Gromov–Witten and Welschinger invariants from the mirror Landau–Ginzburg model. While homological mirror symmetry relates the Fukaya category to matrix factorizations, specific Lagrangians such as RP^n can be trivial over characteristic zero fields, complicating direct extraction of invariants.

To overcome this, the authors develop a normed matrix factorization category equipped with non-Archimedean norms and a normed ∞-trace, allowing passage to a Novikov ring and defining numerical invariants expected to mirror open Gromov–Witten and Welschinger counts. The open problem asks for a principled recovery of these invariants across the Fano landscape, paralleling known Calabi–Yau results.