Fix a Calabi-Yau 3-fold
X
X
satisfying the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as the quintic 3-fold.
We express Joyce’s generalised DT invariants counting Gieseker semistable sheaves of any rank
r
r
on
X
X
in terms of those counting sheaves of rank 1. By the MNOP conjecture they are therefore determined by the Gromov-Witten invariants of
X
X
.