Fix a Calabi-Yau 3-fold X. Its DT invariants count stable bundles and sheaves on X. The generalised DT invariants of Joyce-Song count semistable bundles and sheaves on X. I will describe work with Soheyla Feyzbakhsh showing these generalised DT invariants in any rank r can be written in terms of rank 1 invariants.

By the MNOP conjecture these rank 1 “abelian” invariants are determined by the GW invariants of X. Along the way we also express rank r DT invariants in terms of invariants counting D4-D2-D0 branes: rank 0 sheaves supported on surfaces in X. These invariants are predicted by S-duality to be governed by (vector-valued, mock) modular forms.