Extend Family Floer and non-abelianization to general (non-exact) Betti Lagrangians

Establish for general Betti Lagrangians L ⊂ T* S (not necessarily exact, e.g., including meromorphic spectral curves with O(−1) ends) that the Family Floer functor F: Loc1(L) → Locn(S) can be defined and is invariant under appropriate isotopies, and prove that, in the adiabatic limit, F is equivalent to the non‑abelianization functor ΦW determined by a compatible spectral network W. This extends Theorem \ref{thm:FamilyFloer_NonAbelianization}, which is proven for exact Betti Lagrangians.

Background

The paper proves Theorem \ref{thm:FamilyFloer_NonAbelianization} for exact Betti Lagrangians: (1) Family Floer theory yields a functor F: Loc(L) → Loc(S), and (2) in the adiabatic limit, this functor agrees with the non‑abelianization functor ΦW extracted from a compatible spectral network W. This provides a Floer‑theoretic construction of non‑abelian parallel transport for exact fillings.

For non‑exact Betti Lagrangians (notably those coming from meromorphic spectral curves), the authors currently compute only infinitesimal parallel transport (very short line segments) and do not yet have a full Family Floer construction nor the equivalence to ΦW. Extending the theorem to this general setting would bridge real and holomorphic contexts and remove the exactness restriction.