The Gopakumar–Vafa/Pandharipande–Thomas correspondence

Let $X$ be a Calabi–Yau threefold, let $P_n(X,\beta)$ be the moduli space of stable pairs of class $\beta$ and Euler characteristic $n$, and let $P_{n,\beta}$ be its stable pair invariant. Define

$$Z_{PT}= \sum_{n\in\mathbb{Z}}\sum_{\beta\in H_2(X,\mathbb{Z})}P_{n,\beta}q^nQ^\beta.$$

Let $n^g_\beta$ denote the Gopakumar–Vafa invariant in class $\beta$ and genus $g$.

The Gopakumar–Vafa/Pandharipande–Thomas correspondence. One should have

$$Z_{PT}=\prod_{\beta\in H_2(X,\mathbb{Z})}\Biggl(\prod_{j=1}^\infty(1+(-1)^{j+1}q^jQ^\beta)^{jn^0_\beta}\cdot\prod_{g=1}^\infty\prod_{k=0}^{2g-2}\left(1+(-1)^{g-k}q^{g-1-k}Q^\beta\right)^{(-1)^{k+g}n^g_\beta\binom{2g-2}{k}}\Biggr).$$

This is the proposed relation between stable pair invariants and Gopakumar–Vafa invariants; the source presents it as a conjecture and does not give a resolution.