Curve counting and S-duality

We work on a projective threefold $X$ which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-Toda, such as $\mathbb P^3$ or quintic threefold. prove certain moduli spaces 2-dimensional torsion sheaves are smooth bundles over Hilbert schemes ideal curves and points in $X$. When is Calabi-Yau this gives simple wall crossing formula expressing curve counts (and so ultimately Gromov-Witten invariants) terms D4-D2-D0 branes. These latter invariants predicted to have modular properties we discuss from point view S-duality Noether-Lefschetz theory.