Curious vanishing of local-surface Gromov–Witten invariants

Prove that the equivariant Gromov–Witten series of Z=Tot_S(L_1⊕L_2⊕O_S) vanishes in every effective curve class β for which the stable-map moduli spaces of Y=Tot_S(L_1⊕L_2) are proper in all genera, under the stated Calabi–Yau assumptions.

Background

The authors study local surfaces with two sufficiently negative line bundles and one trivial affine direction. Numerical experiments suggest that the additional affine direction and the induced Hodge-class insertion force the Gromov–Witten series to vanish.

The result is proved only under restrictive torus-action and divisor hypotheses; the general Calabi–Yau statement remains conjectural.

References

Suppose $(Z,T)$ is equivariantly Calabi--Yau and $\beta$ is an effective curve class in $Z$ so that $_g(Y,\beta)$ 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 (label conj: curious CY4 vanishing), Section 5.1