Universal Curve Counting in Calabi-Yau 4-folds

Abstract

Recently, Pardon introduced a universal curve counting framework, which allows him to show that curve counting on 3-folds is "generated" by local curves (in a precise sense). We implement this universal curve counting program for Calabi-Yau 4-folds, where a significant technical difficulty comes from the dependence of the virtual dimension on the genus, which leads to more complicated generators than just local curves. Our main techniques are combinations of deformation-theoretic and homotopy-theoretic methods with an adaptation of Pardon's generic transversality results, which allow us to establish a full set of (computable) generators for CY4 curve counting. As a result, we prove (a slight correction of) the Gromov-Witten/Pandharipande-Thomas correspondence with primary descendents for CY4-folds, formulated by Cao, Maulik, and Toda. From this, we additionally obtain integrality of Gopakumar-Vafa style invariants for CY4-folds, as conjectured by Klemm-Pandharipande.