Stable pair invariants of local Calabi-Yau 4-folds
Abstract
In 2008, Klemm-Pandharipande defined Gopakumar-Vafa type invariants of a Calabi-Yau 4-fold X using Gromov-Witten theory. Recently, Cao-Maulik-Toda proposed a conjectural description of these invariants in terms of stable pair theory. When X is the total space of the sum of two line bundles over a surface S, and all stable pairs are scheme theoretically supported on the zero section, we express stable pair invariants in terms of intersection numbers on Hilbert schemes of points on S. As an application, we obtain new verifications of the Cao-Maulik-Toda conjectures for low degree curve classes and find connections to Carlsson-Okounkov numbers. Some of our verifications involve genus zero Gopakumar-Vafa type invariants recently determined in the context of the log-local principle by Bousseau-Brini-van Garrel. Finally, using the vertex formalism, we provide a few more verifications of the Cao-Maulik-Toda conjectures when thickened curves contribute and also for the case of local P3.