Rank $r$ DT theory from rank $1$

TL;DR

This paper expresses higher-rank Donaldson-Thomas invariants for Calabi-Yau 3-folds in terms of rank 1 invariants, linking them to Gromov-Witten invariants via the MNOP conjecture.

Contribution

It provides a formula to compute rank $r$ DT invariants from rank 1 invariants, assuming the Bogomolov-Gieseker conjecture.

Findings

Rank $r$ DT invariants are determined by rank 1 invariants.

Connection established between DT invariants and Gromov-Witten invariants.

Results apply to Calabi-Yau 3-folds satisfying the conjecture.

Abstract

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