Curve counting and S-duality

TL;DR

This paper studies moduli spaces of torsion sheaves on certain threefolds, deriving a wall crossing formula linking curve counts to D-brane invariants and exploring their modular properties via S-duality.

Contribution

It establishes smoothness of specific moduli spaces and derives a wall crossing formula connecting curve counts with D-brane invariants on threefolds satisfying the Bogomolov-Gieseker conjecture.

Findings

Moduli spaces of torsion sheaves are smooth bundles over Hilbert schemes.

Curve counts are expressed via D4-D2-D0 brane invariants.

Predicted modular properties relate to S-duality and Noether-Lefschetz theory.

Abstract

We work on a projective threefold $X$ which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macr\`i-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.