The local categorical DT/PT correspondence

TL;DR

This paper establishes a categorical wall-crossing formula for specific quivers related to DT/PT moduli spaces on Calabi-Yau 3-folds, providing a categorical analogue of the numerical DT/PT correspondence with applications to sheaves on plane curves.

Contribution

It proves a categorical wall-crossing formula for DT/PT quivers, extending the numerical DT/PT correspondence to a categorical setting with applications to sheaf theory.

Findings

Proved a semiorthogonal decomposition involving quasi-BPS categories.

Established a categorical DT/PT correspondence for sheaves on reduced plane curves.

Extended the understanding of wall-crossing phenomena in derived categories.

Abstract

In this paper, we prove the categorical wall-crossing formula for certain quivers containing the three loop quiver, which we call DT/PT quivers. These quivers appear as Ext-quivers for the wall-crossing of DT/PT moduli spaces on Calabi-Yau 3-folds. The resulting formula is a semiorthogonal decomposition which involves quasi-BPS categories studied in our previous papers, and we regard it as a categorical analogue of the numerical DT/PT correspondence. As an application, we prove a categorical DT/PT correspondence for sheaves supported on reduced plane curves in the affine three dimensional space.