In joint work with A. Brini, it was conjectured that equivariant Gromov–Witten invariants of Calabi–Yau fivefolds are governed by so-called membrane indices. When the fivefold is a product of a Calabi–Yau threefold and the affine plane, the latter invariants agree with, or refine, Gopakumar–Vafa invariants. I will present evidence for the conjecture. Moreover, I will explain how the numerical correspondence informs us about a possible modular interpretation of M2-branes since ultimately their moduli space should give rise to membrane indices. Building on ideas of Nekrasov and Okounkov, I will propose a geometric interpretation of M2-branes in some concrete situations. I will then translate this proposal into formulas for Hodge integrals, which can be verified in certain limits. This is based on work in progress with A. Giacchetto and R. Pandharipande.