Correspondence of Donaldson-Thomas and Gopakumar-Vafa invariants on local Calabi-Yau 4-folds over V_5 and V_22

TL;DR

This paper computes Gromov-Witten and Donaldson-Thomas invariants for local Calabi-Yau 4-folds over specific Fano 3-folds, verifying conjectured correspondences with Gopakumar-Vafa invariants and exploring genus 1 invariants.

Contribution

It provides explicit calculations of GW and DT invariants for local CY 4-folds over V_5 and V_22, confirming conjectured DT-GV correspondences in genus 0.

Findings

Verification of DT-GV correspondence in genus 0

Genus 1 GV invariants are zero, consistent with absence of elliptic curves

Explicit invariants computed up to degree 3

Abstract

We compute Gromov-Witten (GW) and Donaldson-Thomas (DT) invariants (and also descendant invariants) for local CY 4-folds over Fano 3-folds, V_5 and V_22 up to degree 3. We use torus localization for GW invariants computation, and use classical results for Hilbert schemes on V_5 and V_22 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 V_5 and V_22 up to degree 3.