Correspondence of Donaldson-Thomas and Gopakumar-Vafa invariants on local Calabi-Yau 4-folds over V5 and V22

Kiryong Chung; Sanghyeon Lee; Joonyeong Won

doi:10.1016/j.geomphys.2023.105082

Correspondence of Donaldson-Thomas and Gopakumar-Vafa invariants on local Calabi-Yau 4-folds over V₅ and V₂₂

Kiryong Chung, Sanghyeon Lee, Joonyeong Won

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We compute Gromov-Witten (GW) and Donaldson-Thomas (DT) invariants (and also descendant invariants) for local CY 4-folds over Fano 3-folds, V₅ and V₂₂ up to degree 3. We use torus localization for GW invariants computation, and use classical results for Hilbert schemes on V₅ and V₂₂ for DT invariants computation. From these computations, one can check correspondence between DT and Gopakumar-Vafa (GV) invariants conjectured by Cao-Maulik-Toda in genus 0. Also we can compute genus 1 GV invariants via the conjecture of Cao-Toda, which turned out to be 0. These fit into the fact that there are no smooth elliptic curves in V₅ and V₂₂ up to degree 3.

Original language	English
Article number	105082
Journal	Journal of Geometry and Physics
Volume	197
DOIs	https://doi.org/10.1016/j.geomphys.2023.105082
State	Published - Mar 2024

Bibliographical note

Publisher Copyright:
© 2023 Elsevier B.V.

Keywords

Donaldson-Thomas invariants
Fano varieties
Gopakumar-Vafa invariants
Gromov-Witten invariants

Access to Document

10.1016/j.geomphys.2023.105082

Cite this

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title = "Correspondence of Donaldson-Thomas and Gopakumar-Vafa invariants on local Calabi-Yau 4-folds over V5 and V22",

abstract = "We compute Gromov-Witten (GW) and Donaldson-Thomas (DT) invariants (and also descendant invariants) for local CY 4-folds over Fano 3-folds, V5 and V22 up to degree 3. We use torus localization for GW invariants computation, and use classical results for Hilbert schemes on V5 and V22 for DT invariants computation. From these computations, one can check correspondence between DT and Gopakumar-Vafa (GV) invariants conjectured by Cao-Maulik-Toda in genus 0. Also we can compute genus 1 GV invariants via the conjecture of Cao-Toda, which turned out to be 0. These fit into the fact that there are no smooth elliptic curves in V5 and V22 up to degree 3.",

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T1 - Correspondence of Donaldson-Thomas and Gopakumar-Vafa invariants on local Calabi-Yau 4-folds over V5 and V22

AU - Chung, Kiryong

AU - Lee, Sanghyeon

AU - Won, Joonyeong

PY - 2024/3

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N2 - We compute Gromov-Witten (GW) and Donaldson-Thomas (DT) invariants (and also descendant invariants) for local CY 4-folds over Fano 3-folds, V5 and V22 up to degree 3. We use torus localization for GW invariants computation, and use classical results for Hilbert schemes on V5 and V22 for DT invariants computation. From these computations, one can check correspondence between DT and Gopakumar-Vafa (GV) invariants conjectured by Cao-Maulik-Toda in genus 0. Also we can compute genus 1 GV invariants via the conjecture of Cao-Toda, which turned out to be 0. These fit into the fact that there are no smooth elliptic curves in V5 and V22 up to degree 3.

AB - We compute Gromov-Witten (GW) and Donaldson-Thomas (DT) invariants (and also descendant invariants) for local CY 4-folds over Fano 3-folds, V5 and V22 up to degree 3. We use torus localization for GW invariants computation, and use classical results for Hilbert schemes on V5 and V22 for DT invariants computation. From these computations, one can check correspondence between DT and Gopakumar-Vafa (GV) invariants conjectured by Cao-Maulik-Toda in genus 0. Also we can compute genus 1 GV invariants via the conjecture of Cao-Toda, which turned out to be 0. These fit into the fact that there are no smooth elliptic curves in V5 and V22 up to degree 3.

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KW - Fano varieties

KW - Gopakumar-Vafa invariants

KW - Gromov-Witten invariants

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