Curve counting and S-duality

TL;DR

The paper analyzes 2-dimensional torsion sheaves on a projective threefold $X$ under the Bogomolov--Gieseker framework, proving these moduli spaces are smooth bundles over Hilbert schemes of curves and points, and, in Calabi--Yau cases, deriving a simple wall-crossing formula connecting curve counts to D4-D2-D0 brane counts. Central to the method is a robust wall-crossing analysis in the space of weak stability conditions, which identifies semistable objects of Chern character $v_n$ with cokernels of sections from line bundles to rank-1 sheaves, i.e. Joyce--Song pairs, yielding an isomorphism $M_{X,H}(v_n) \cong \mathrm{JS}_n(v) \times \mathrm{Pic}_0(X)$. This linkage reduces the moduli to Joyce--Song data, enabling a clear path to modularity: the generating series of the D4-D2-D0 counts is predicted to be modular (or mock modular) via S-duality and Noether-- Lefschetz theory, with precise NL-geometric contributions and eta-factor refinements. The paper also clarifies the relation to Toda’s approach, highlighting a simpler Wall-crossing framework in the present Calabi--Yau context and outlining extensions to higher rank DT invariants. Collectively, these results provide a concrete bridge between curve counting, Gromov--Witten theory, and the modular structure anticipated from string dualities.

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.

Figures (4)

• Figure 1: $(b,w)$-plane and the projection $\Pi(E)$ when $\operatorname{ch}_0(E)>0$
• Figure 2: The line segments $\ell_i \cap U$ are walls for $E$.
• Figure 3: Walls for objects of class $v_n$
• Figure 4: Walls for objects of class $v$

Theorems & Definitions (29)

• Theorem 1
• Theorem 2
• Definition 2.1
• Remark 2.2
• Conjecture 2.3
• Conjecture BG$_n$
• Theorem 2.4: cf. Li, Liu
• proof
• Proposition 2.5: Wall and chamber structure
• Theorem 3.1