Gromov-Witten invariants and membrane indices of fivefolds via the topological vertex

We conjecture the existence of almost integer invariants governing the all-genus equivariant Gromov-Witten theory of Calabi-Yau fivefolds with a torus action. We prove the conjecture for skeletal, locally anti-diagonal torus actions by establishing a vertex formalism evaluating the Gromov-Witten invariants via the topological vertex of Aganagic, Klemm, Marino and Vafa. We apply the formalism in several examples.