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.

中文翻译：

---

Calabi-Yau 4-fold 的曲线计数和 DT/PT 对应关系

摘要 最近，Cao-Maulik-Toda 定义了一个紧凑的 Calabi-Yau 4-fold X 的稳定对不变量。他们的不变量与 Klemm-Pandharipande 使用 Gromov-Witten 理论定义的 X 的 Gopakumar-Vafa 型不变量推测相关。在本文中，我们使用 Hilbert 曲线方案考虑 X 的曲线计数不变量，并推测将这些与 X 的稳定对不变量相关联的 DT/PT 对应关系。在紧凑情况下提供证据后，我们定义了复曲面 Calabi-Yau 的类似不变量4 倍。我们为这两种理论制定了一个顶点形式主义，并推测了（完全等变的）DT/PT 顶点之间的关系，我们在几种情况下对其进行了检查。这种关系意味着具有初级插入的环面 Calabi-Yau 4 折的 DT/PT 对应关系。