Semiorthogonal decompositions for categorical Donaldson-Thomas theory via $\Theta$-stratifications

TL;DR

This paper develops semiorthogonal decompositions for Donaldson-Thomas categories using $ heta$-stratifications, providing new tools for understanding categorical DT theory and its applications to d-critical flips and flops.

Contribution

It introduces a semiorthogonal decomposition framework for categorical DT theory via $ heta$-stratifications, extending window theorems and applications to wall-crossing phenomena.

Findings

Established semiorthogonal decompositions for DT categories.

Proved fully-faithful functors exist under d-critical flips and flops.

Applied results to DT categories of Pandharipande-Thomas moduli spaces.

Abstract

We show the existence of semiorthogonal decompositions of Donaldson-Thomas categories for $(-1)$-shifted cotangent derived stacks associated with $\Theta$-stratifications on them. Our main result gives an analogue of window theorem for categorical DT theory, which has applications to d-critical analogue of D/K equivalence conjecture, i.e. existence of fully-faithful functors (equivalences) under d-critical flips (flops). As an example of applications, we show the existence of fully-faithful functors of DT categories for Pandharipande-Thomas stable pair moduli spaces under wall-crossing at super-rigid rational curves for any relative reduced curve classes.