Enumerative geometry of Calabi-Yau 4-folds

TL;DR

This work develops an enumerative framework for curves on Calabi–Yau 4-folds via Gromov–Witten theory, proposing integer-valued genus-0 and genus-1 invariants $n_{0,\beta}$ and $n_{1,\beta}$ together with meeting invariants $m_{\beta_1,\beta_2}$ and formulating a conjectural integrality structure. It provides explicit constructions of these invariants, including an Aspinwall–Morrison–type transformation for genus 0 and a genus-1 formula that combines elliptic, rational, and nodal contributions, with holomorphic anomaly equations and mirror symmetry used to fix ambiguities. The paper demonstrates the approach through exact local CY4 calculations by localization and holomorphic anomaly methods, and by detailed studies of compact examples (sextic, quintic fibrations, and elliptic fibrations), verifying integrality up to high degrees and interpreting genus-1 data in terms of BPS state counting and bound states at thresholds. The results pave the way for a sheaf-theoretic interpretation of CY4 GW invariants and suggest universal higher-dimensional transformations that relate GW invariants to integer counts, with potential applications in string compactifications and higher-dimensional enumerative geometry.

Abstract

Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 4-folds. The main technique is to find exact solutions to moving multiple cover integrals. The resulting invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds. We conjecture the 4-fold invariants to be integers and expect a sheaf theoretic explanation. Several local Calabi-Yau 4-folds are solved exactly. Compact cases, including the sextic Calabi-Yau in CP5, are also studied. A complete solution of the Gromov-Witten theory of the sextic is conjecturally obtained by the holomorphic anomaly equation.

Figures (1)

• Figure 1: A double comb

Theorems & Definitions (5)

• Lemma 1
• Lemma 2
• Proposition 1
• Proposition 2
• Proposition 3