D0-D6 states counting and GW invariants

TL;DR

This paper establishes a new correspondence linking the count of certain torsion free sheaves on Calabi-Yau 3-folds to Gromov-Witten invariants of orbifold blowups of weighted projective planes, expanding the GW/quiver correspondence.

Contribution

It introduces a novel variation of the GW/quiver correspondence by changing blowup centers and orders, connecting sheaf counts to rational curve invariants in orbifold settings.

Findings

Established a correspondence between sheaf counts and GW invariants.

Extended the GW/quiver correspondence to new orbifold blowup configurations.

Built on wall-crossing and tropical vertex group theories.

Abstract

We describe a correspondence between the virtual number of torsion free sheaves locally free in codimension 3 on a Calabi-Yau 3-fold and the Gromov-Witten invariants counting rational curves in a family of orbifold blowups of the weighted projective plane $\mathbb{P}(-ch_3, ch_0, 1)$ (with a tangency condition of order $gcd(-ch_3, ch_0)$). This result is a variation of the GW/quiver representations correspondence found by Gross-Pandharipande, when one changes the centres and orders of the blowups. We build on a small part of the theories developed by Joyce-Song and Kontsevich-Soibelman for wall-crossing formulae and by Gross-Siebert-Pandharipande for factorisations in the tropical vertex group.