Curve counting and DT/PT correspondence for Calabi-Yau 4-folds

Abstract

Recently, Cao-Maulik-Toda defined stable pair invariants of a compact Calabi-Yau 4-fold X. Their invariants are conjecturally related to the Gopakumar-Vafa type invariants of X defined using Gromov-Witten theory by Klemm-Pandharipande. In this paper, we consider curve counting invariants of X using Hilbert schemes of curves and conjecture a DT/PT correspondence which relates these to stable pair invariants of X.After providing evidence in the compact case, we define analogous invariants for toric Calabi-Yau 4-folds. We formulate a vertex formalism for both theories and conjecture a relation between the (fully equivariant) DT/PT vertex, which we check in several cases. This relation implies a DT/PT correspondence for toric Calabi-Yau 4-folds with primary insertions.