Joyce–Song multiple cover conjecture for generalized Donaldson–Thomas invariants

Let $X$) be a smooth projective Calabi–Yau $3$-fold over $\mathbb{C}$, so that

$$\bigwedge^3T_X^{\vee}\cong\mathcal{O}_X,\qquad H^1(X,\mathcal{O}_X)=0.$$

For $n\in\mathbb{Z}$ and $\beta\in H_2(X,\mathbb{Z})$, let $N_{n,\beta}\in\mathbb{Q}$ be the generalized Donaldson–Thomas invariant counting one-dimensional semistable sheaves $F$ with $\chi(F)=n$ and $[F]=\beta$. Write $k\mid(n,\beta)$ when $k$ divides both $n$ and $\beta$.

Joyce–Song multiple cover conjecture. We have

$$N_{n,\beta}=\sum_{k\geq 1,\,k\mid(n,\beta)}\frac{1}{k^2}N_{1,\beta/k}.$$

This conjecture is equivalent to Pandharipande–Thomas's strong rationality conjecture for the generating series of rank-one DT-type invariants, and is motivated by the expected GW/DT correspondence. The paper reduces its local form for curves with at worst nodal singularities to local trees of smooth rational curves and proves it in some cases.