Weaker hypotheses for membrane-index vanishing

Determine a weaker condition on chains of rational curves in Calabi–Yau fivefolds than the strip-geometry hypotheses that guarantees Ω_β≠0 only for curve classes consisting of a consecutive block of coefficients equal to one.

Background

The paper considers a proposed vanishing pattern for membrane indices of chains whose rational curves have normal bundles of type O(-1)⊕O(-1) or O(-2)⊕O. A counterexample shows that the broadest version is false.

The authors observe that the vanishing still appears in other examples and explicitly ask for weaker geometric assumptions ensuring it.