The Gopakumar-Vafa type invariants on Calabi-Yau 4-folds (which are non-trivial only for genus zero and one) are defined by Klemm-Pandharipande from Gromov-Witten theory, and their integrality is conjectured. In a previous work of Cao-Maulik-Toda, $\mathrm{DT}_4$ invariants with primary insertions on moduli spaces of one dimensional stable sheaves are used to give a sheaf theoretical interpretation of the genus zero GV type invariants. In this paper, we propose a sheaf theoretical interpretation of the genus one GV type invariants using descendent insertions on the above moduli spaces. The conjectural formula in particular implies nontrivial constraints on genus zero GV type (equivalently GW) invariants of CY 4-folds which can be proved by the WDVV equation.

中文翻译：

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通过后代插入在 Calabi-Yau 4-folds 上的 Gopakumar-Vafa 类型不变量

Calabi-Yau 4-fold 上的 Gopakumar-Vafa 型不变量（仅对属零和属非平凡）由来自 Gromov-Witten 理论的 Klemm-Pandharipande 定义，并且推测它们的完整性。在 Cao-Maulik-Toda 之前的工作中，$\mathrm{DT}_4$ 不变量在一维稳定滑轮的模空间上具有主要插入，用于给出属零 GV 类型不变量的层理论解释。在本文中，我们提出了使用上述模空间上的后代插入对属一 GV 类型不变量的层理论解释。推测公式特别暗示了对 CY 4 倍的属零 GV 类型（等效 GW）不变量的非平凡约束，这可以通过 WDVV 方程证明。