# Derived categories of Thaddeus pair moduli spaces via d-critical flips

## TL;DR

This paper demonstrates that moduli spaces of Thaddeus pairs are connected by d-critical flips and establishes fully-faithful functors between their derived categories, providing evidence for a d-critical analogue of D/K equivalence and categorifying wall-crossing formulas.

## Contribution

It introduces the relation of moduli spaces via d-critical flips and constructs fully-faithful functors between their derived categories, advancing the understanding of derived equivalences in this context.

## Key findings

- Moduli spaces of Thaddeus pairs are related by d-critical flips.
- Existence of fully-faithful functors between derived categories of these moduli spaces.
- Provides evidence for a d-critical analogue of D/K equivalence and categorifies wall-crossing formulas.

## Abstract

We show that the moduli spaces of Thaddeus pairs on smooth projective curves and those of dual pairs are related by d-critical flips, which are virtual birational transformations introduced by the second author. We then prove the existence of fully-faithful functors between derived categories of coherent sheaves on these moduli spaces. Our result gives an evidence of a d-critical analogue of Bondal-Orlov, Kawamata's D/K equivalence conjecture, and also a categorification of wall-crossing formula of Donaldson-Thomas type invariants on ADHM sheaves introduced by Diaconescu.