D0-D6 states counting and GW invariants

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 $\mathbb{P}(-c{h}_3,c{h}_0,1)$ (with a tangency condition of order $gcd(-c{h}_3,c{h}_0)$). 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.