Fukaya-category generation and quiver correspondence for geometric bases

Establish that both the simple basis {α1,…,αk} of geometric roots (constructed in Proposition 4.x as a basis of H1(M,∂M;Z)) and the projective basis {β1,…,βk} (defined via edges and spokes on the Coxeter wheel) generate the Fukaya category of the Milnor fiber M of the simple curve singularity f(x,y). Further, show that the A∞ endomorphism algebra of {β1,…,βk} in this Fukaya category is quasi-isomorphic to the path algebra of the corresponding ADE quiver and that the Hom functor with respect to {β1,…,βk} sends geometric roots (equivalence classes of oriented edges and spokes) to indecomposable quiver representations up to shift.

Background

The paper introduces geometric root systems for simple curve singularities of ADE type by using arcs in Milnor fibers and the negative symmetrized Seifert form. Two natural bases of geometric roots are constructed: a ‘simple basis’ {αi} whose pairing reproduces the Cartan matrix, and a ‘projective basis’ {βi} whose monodromy orbits cover all roots and are realized by edges/spokes on a Coxeter wheel.

Building on these geometric models, the authors anticipate categorical consequences connecting the Fukaya category of the Milnor fiber to representation theory of ADE quivers. Specifically, they propose generation statements for both bases, an identification of the A∞ endomorphism algebra of the projective basis with the path algebra of the ADE quiver, and a Hom functor that maps geometric roots to indecomposable quiver representations.