Enumeration of curves in Calabi–Yau fivefolds

• Applicant: Yannik Schüler-Hammer, Ph.D.
• Subject Area: Mathematics
• Term: since 2026
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 576663726

Enumerative geometry is concerned with counting geometric objects of a certain type. Gromov–Witten theory provides a framework to enumerate complex curves contained within a fixed algebraic ambient space (a variety). Since the 1990s, the development of this subject has proceeded hand in hand with advances in mathematical physics, particularly topological string theory. As a string moves through space-time, it sweeps out a real two-dimensional surface — that is, a complex curve. Path integrals in topological string theory thus acquire a mathematical incarnation as Gromov–Witten invariants. The most extensively studied setting is when the ambient variety, that is space-time, is a complex Calabi–Yau threefold. Here, the interplay between enumerative geometry and mathematical physics has sparked several influential conjectures. These include the Maulik–Nekrasov–Okounkov–Pandharipande conjecture, which equates Gromov–Witten invariants with sheaf-theoretic counts, and the Gopakumar–Vafa conjecture, which predicts more fundamental invariants underlying Gromov–Witten theory. This project will provide evidence for an equivariant generalisation of these conjectures in the setting of Calabi–Yau fivefolds with a group action. When the Calabi–Yau fivefold is taken to be the product of a threefold with the complex plane equipped with a special choice of torus action, the proposed generalisations reduce to the original threefold conjectures. To date, no evidence for the generalised conjectures exists beyond this product case. The project aims to remedy this gap by providing a verification in a class of basic yet highly representative geometries and by developing foundational techniques needed for establishing the conjectures for toric varieties and local curves. This will be achieved by generalising known methods from three to five dimensions, including developing a vertex formalism for the Gromov–Witten theory of toric fivefolds. As a by-product, the formalism will yield new formulae for Hodge integrals on the moduli space of curves. Beyond algebraic geometry, the results of this project have implications for mathematical physics. For instance, one of the Hodge integrals can be identified with the index of 11-dimensional supergravity. Moreover, the project contributes to the mathematical foundations of topological M-theory: the insights gained on the enumeration of curves in fivefolds impose non-trivial constraints on any possible mathematical characterisation of M2-branes, certain dynamical objects in M-theory. In specific cases, rigorous modular interpretations for M2-branes will be proposed.

• DFG Programme: WBP Fellowship
• International Connection: Switzerland
• Host: Professor Dr. Rahul Pandharipande