On window theorem for categorical Donaldson-Thomas theories on local surfaces and its applications

TL;DR

This paper establishes a window theorem for categorical Donaldson-Thomas theories on local surfaces, enabling new insights into wall-crossing phenomena and functorial relations between different categorical DT theories.

Contribution

It introduces a novel window theorem for categorical DT theories on local surfaces and applies it to prove wall-crossing equivalences and functorial embeddings between DT categories.

Findings

Proved wall-crossing equivalences for DT categories of stable sheaves.

Established fully-faithful functors from PT to MNOP categorical theories.

Indicated categorifications of numerical DT invariants and connections to D/K conjecture.

Abstract

In this paper, we prove a window theorem for categorical Donaldson-Thomas theories on local surfaces as an analogue of window theorem for GIT quotient stacks. We give two applications of our main result. The first one is a proof of wall-crossing equivalences of DT categories for one dimensional stable sheave on local surfaces, under some technical condition on strictly semistable sheaves. The second one is to show the existence of fully-faithful functors from categorical PT theories to categorical MNOP theories, when the curve class is reduced. These results indicate categorifications of wall-crossing formulas of numerical DT invariants, and also regarded as d-critical analogue of D/K conjecture in birational geometry.