On the Crepant Resolution Conjecture for Donaldson-Thomas Invariants

TL;DR

This paper proves a comparison formula for Donaldson-Thomas invariants in the context of the crepant resolution conjecture, establishing new results for point classes and conditional results for general curve classes in Calabi-Yau orbifolds.

Contribution

It provides the first proof of the crepant resolution conjecture for point classes and a conditional proof for general classes, advancing understanding of curve-counting invariants in orbifold resolutions.

Findings

Proved the conjecture for point classes.

Provided a conditional proof for general curve classes.

Identified the image of the standard heart under Bridgeland-King-Reid equivalence.

Abstract

We prove a comparison formula for curve-counting invariants in the setting of the McKay correspondence, related to the crepant resolution conjecture for Donaldson-Thomas invariants. The conjecture is concerned with comparing the invariants of a (hard Lefschetz) Calabi-Yau orbifold of dimension three with those of a specific crepant resolution of its coarse moduli space. We prove the conjecture for point classes and give a conditional proof for general curve classes. We also prove a variant of the formula for curve classes. Along the way we identify the image of the standard heart of the orbifold under the Bridgeland-King-Reid equivalence.