Denef–Moore D4–D6 generating-series conjecture

Let $X$ be a smooth projective Calabi–Yau $3$-fold, let $H$ be an ample divisor, and let $m e 0$ and $(\beta,n) e(0,0)$. Define the relevant D4, ideal-sheaf, stable-pair, and D6–anti-D6 generating series using the invariants and cutoff specified in the source.

Denef–Moore conjecture. (i) The invariant $\operatorname{DT}_H(0,mH,-\beta,-n)$ vanishes unless

$$-\frac{H^3}{24}m^3\le n+\frac{(\beta\cdot H)^2}{2mH^3}.$$

(ii) For any $\xi\ge1$, there are $\mu>0$, $\delta>0$, and a constant $m(\xi,\mu)>0$ depending only on $\xi,\mu$ such that for any $m>m(\xi,\mu)$,

$$\mathcal{Z}_{\rm D4}^{m}(x,y)=\left.\frac{\partial}{\partial z}\mathcal{Z}_{\rm D6-\overline{\rm D6}}^{m,\epsilon=\frac{\delta}{m^\xi}}(x,y,z)\right|_{z=-1}$$

modulo terms $x^ny^\beta$ satisfying

$$-\frac{H^3}{24}m^3\left(1-\frac{\mu}{m^\xi}\right)\le n+\frac{(\beta\cdot H)^2}{2mH^3}.$$

The formula is presented as a mathematical formulation of the D4–D6 relation appearing in Denef–Moore's work and in the study of the OSV conjecture.