Curve counting and S-duality

Abstract

We work on a projective threefold X which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as P3 or the quintic threefold. We prove certain moduli spaces of 2-dimensional torsion sheaves on X are smooth bundles over Hilbert schemes of ideal sheaves of curves and points in X . When X is Calabi-Yau this gives a simple wall crossing formula expressing curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of D4-D2-D0 branes. These latter invariants are predicted to have modular properties which we discuss from the point of view of S-duality and Noether-Lefschetz theory.