The categorical DT/PT correspondence and quasi-BPS categories for local surfaces

TL;DR

This paper develops a categorical framework linking Donaldson-Thomas and Pandharipande-Thomas theories for local surfaces, providing new semiorthogonal decompositions and wall-crossing formulas.

Contribution

It introduces a categorical analogue of the DT/PT correspondence for local surfaces, including semiorthogonal decompositions and wall-crossing formulas for DT/PT quivers.

Findings

Constructed semiorthogonal decompositions of DT categories for local surfaces.

Proved a categorical wall-crossing formula for DT/PT quivers.

Proposed conjectural K-theory computations for quasi-BPS categories.

Abstract

We construct semiorthogonal decompositions of Donaldson-Thomas (DT) categories for reduced curve classes on local surfaces into products of quasi-BPS categories and Pandharipande-Thomas (PT) categories, giving a categorical analogue of the numerical DT/PT correspondence for Calabi-Yau 3-folds. The main ingredient is a categorical wall-crossing formula for DT/PT quivers (which appear as Ext-quivers in the DT/PT wall-crossing) proved in our previous paper. We also study quasi-BPS categories of points on local surfaces and propose conjectural computations of their K-theory analogous to formulas already known for the three dimensional affine space.