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 ${\mathbb{P}}^3$ 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.

Referee's corrections implemented; journal version. 25 pages, 4 figures

References in corpus (6)

• A Farey Tail for Attractor Black Holes
• Donaldson-Thomas invariants via microlocal geometry
• Quantum geometry, stability and modularity
• Rank $r$ DT theory from rank $0$
• Stability conditions on Calabi-Yau double/triple solids
• Explicit formulae for rank zero DT invariants and the OSV conjecture

Cited by in corpus (6)

• Rank $r$ DT theory from rank $1$
• Scaling Black Holes and Modularity
• Rank $r$ DT theory from rank $0$
• Mock modularity at work, or black holes in a forest
• On the Bogomolov-Gieseker inequality for hypersurfaces in the projective spaces
• Stability condition on Calabi-Yau threefold of complete intersection of quadratic and quartic hypersurfaces