Stable Pair Invariants of Local Calabi–Yau 4-folds

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  • Yalong Cao
    Kavli Institute for the Physics and Mathematics of the Universe (WPI) ,The University of Tokyo Institutes for Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
  • Martijn Kool
    Mathematical Institute , Utrecht University, P.O. Box 80010, 3508 TA Utrecht, The Netherlands
  • Sergej Monavari
    Mathematical Institute , Utrecht University, P.O. Box 80010, 3508 TA Utrecht, The Netherlands

説明

<jats:title>Abstract</jats:title> <jats:p>In 2008, Klemm–Pandharipande defined Gopakumar–Vafa type invariants of a Calabi–Yau 4-folds $X$ 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 $\mathbb{P}^3$.</jats:p>

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