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

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.

Rational curves in a quadric threefold via an SL(2, C)-representation

“…The authors think that a description of its moduli space is just not written down in the literature because it is well-known and easy to experts. However, it seems necessary to record global geometry and cohomological ring in order to serve as concrete examples required by enumerative geometry ([Cao19, CMT18,CLW21]). Now let us introduce the main contents of this paper.…”