The Gopakumar–Vafa invariant formula via relative Hilbert schemes

Let $X$ be a Calabi–Yau threefold, fix $g\ge0$, and suppose $I_{1-g}(X,\beta)$ parametrizes ideals of local complete intersection curves of arithmetic genus $g$. Let $\mathcal{M}=I_{1-g}(X,\beta)$, let $\mathcal{C}\subset\mathcal{M}\times X$ be the universal subscheme, and let $\mathcal{C}^{[n]}$ be the relative Hilbert scheme of $n$ points in the family $\mathcal{C}/\mathcal{M}$, with $\mathcal{C}^{[0]}=\mathcal{M}$. Fix $\delta\le g$ and assume that $\mathcal{C}^{[n]}$ is smooth for $n\le\delta$. Under the additional hypotheses stated in the source, the Gopakumar–Vafa invariant conjecture.

$$\left(-1\right)^{\dim\mathcal{M}+\delta}n^{g-\delta}_\beta=e\left(\mathcal{C}^{[\delta]}\right)+\left(2g-2\delta\right)e\left(\mathcal{C}^{[\delta-1]}\right)+$$ $$\sum_{i=2}^\delta\frac1{i!}\left(2g-2\delta+2i-2\right)\left(2g-2\delta+i-3\right)\left(2g-2\delta+i-4\right)\cdots\left(2g-2\delta-1\right)e\left(\mathcal{C}^{[\delta-i]}\right).$$

Here $e$ denotes the topological Euler characteristic. The formula gives a way to recover Gopakumar–Vafa invariants from Euler characteristics of relative Hilbert schemes, extending computations from the physics literature to a broader geometric setting. The supplied text does not give enough information to determine whether the formula has been proved in full generality.