Hall algebras in the derived category and higher rank DT invariants

TL;DR

This paper demonstrates that combining recent works confirms a local description of moduli stacks of semi-Schur objects in derived categories, enabling new proofs of higher rank DT/PT correspondence and rationality in Calabi-Yau 3-folds.

Contribution

It proves the conjectured local description of moduli stacks, facilitating the extension of DT/PT correspondence and rationality results to higher ranks.

Findings

Confirmed the local description of moduli stacks of semi-Schur objects.

Extended DT/PT correspondence to higher rank cases.

Established rationality results for higher rank DT invariants.

Abstract

We remark that the combination of the works of Ben-Bassat-Brav-Bussi-Joyce and Alper-Hall-Rydh imply the conjectured local description of the moduli stacks of semi-Schur objects in the derived category of coherent sheaves on projective Calabi-Yau 3-folds. This result was assumed in the author's previous papers to apply wall-crossing formulas of DT type invariants in the derived category, e.g. DT/PT correspondence, rationality, etc. We also show that the above result is applied to prove the higher rank version of DT/PT correspondence and rationality.