Curious vanishing of local-surface Pandharipande–Thomas invariants

Prove that the K-theoretic Pandharipande–Thomas series of local surfaces Z=Tot_S(L_1⊕L_2⊕O_S) vanishes in every effective curve class satisfying the stated properness condition for disconnected stable-map moduli of the associated fourfold.

Background

This conjecture is presented as the Pandharipande–Thomas counterpart of the proposed Gromov–Witten vanishing and is motivated by the conjectural K-theoretic GW/PT correspondence.

The authors verify the vanishing numerically for several local surfaces and curve classes, while non-proper classes provide nonzero counterexamples to an unrestricted statement.

References

Suppose $(Z,T)$ is Calabi--Yau and let $\beta$ be an effective curve class in $Z$ so that the moduli space $\smash{_g\bullet(Y,\beta)}$ of stable maps to the fourfold with possibly disconnected domain and no contracted connected components is proper for all $g\geq 0$.

— Experiments with membranes, maps and sheaves  (2609.03152 - Holmes et al., 2 Sep 2026) in Conjecture curious CY4 vanishing PT (label conj: curious CY4 vanishing PT), Section 5.2