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arXiv · 1903.12171

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

Yalong Cao·Martijn Kool
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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 using a localization formula. 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.

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BibTeXRIS

Yalong Cao, Martijn Kool. 2019-03-28. Curve counting and DT/PT correspondence for Calabi-Yau 4-folds. https://doi.org/10.1016/j.aim.2020.107371

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