Stable maps to Calabi–Yau fivefolds

Gromov–Witten invariants enumerate curves in a variety X via stable maps. However, degenerate contributions lead to substantial overcounting which makes these invariants far from optimal. When X is a Calabi–Yau threefold, a set of more fundamental curve counting invariants is provided by Gopakumar–Vafa invariants. I will propose a conjectural generalisation of this correspondence between Gromov–Witten and Gopakumar–Vafa invariants to the setting of Calabi–Yau fivefolds equipped with a torus action. I will demonstrate the conjecture in the setting of local curves. For a special type of torus action we will prove a closed-form formula for the local contribution of a smooth embedded curve and for general torus actions the validity of the formula will be translated into a conjectural formula for certain tautological integrals over the moduli of curves.