Laurent-polynomial criterion for membrane indices

Identify a sufficient condition under which the membrane index of a Calabi–Yau fivefold is a Laurent polynomial in the equivariant variables.

Background

For local P³ geometries, numerical experiments show that some membrane indices appear to be Laurent polynomials while others exhibit more complicated poles. The behavior varies with the splitting of the bundle over P³.

The paper explicitly asks for a general geometric or equivariant criterion distinguishing these cases.