Enumerative geometry of Calabi–Yau 5-folds

Abstract

Gromov–Witten theory is used to define an enumerative geometry of curves in Calabi–Yau 5-folds. We find recursions for meeting numbers of genus 0 curves, and we determine the contributions of moving multiple covers of genus 0 curves to the genus 1 Gromov–Witten invariants. The resulting invariants, conjectured to be integral, are analogous to the previously defined BPS counts for Calabi–Yau 3 and 4-folds. We comment on the situation in higher dimensions where new issues arise.

Two main examples are considered: the local Calabi–Yau $\mathbb{P}^2$ with normal bundle $\oplus_{i=1}^{3} \mathcal{O} (-1)$ and the compact Calabi–Yau hypersurface $X_7 \subset \mathbb{P}^6$. In the former case, a closed form for our integer invariants has been conjectured by G. Martin. In the latter case, we recover in low degrees the classical enumeration of elliptic curves by Ellingsrud and Strömme.