Independence conjecture for generalized Gopakumar–Vafa invariants

Let $X$ be a smooth projective Calabi–Yau threefold, let $v=(\beta,m)$ with $\beta eq 0$, let $\sigma\in U(X)$, and let $\gamma\in\operatorname{Chow}_X(\beta)$. Suppose that the moduli stack $\mathcal{M}_{\sigma}(v)$ is Calabi–Yau at $\gamma$, meaning that its virtual canonical bundle is trivial on an analytic neighborhood over $\gamma$. The invariant $\Phi_{\sigma}(\gamma,m)$ is independent of $\sigma$ and $m$.

This conjecture predicts wall-crossing invariance of the generalized Gopakumar–Vafa invariant, extending the expected independence of stability condition and the integer parameter. The paper establishes related results under additional hypotheses, including for local surfaces, but the stated general invariance remains open.