Donaldson-Thomas theory for Calabi-Yau 4-folds

TL;DR

This paper develops Donaldson-Thomas type invariants for Calabi-Yau 4-folds, relating them to existing invariants, and explores their computation, localization, and wall-crossing phenomena.

Contribution

It introduces $DT_{4}$ invariants for Calabi-Yau 4-folds, extending Donaldson-Thomas theory and establishing connections with Gromov-Witten invariants and non-commutative versions.

Findings

Defined $DT_{4}$ invariants for Calabi-Yau 4-folds

Established $DT_{4}/GW$ correspondence in certain cases

Computed invariants for smooth moduli spaces and toric examples

Abstract

Let $X$ be a compact complex Calabi-Yau 4-fold. Under certain assumptions, we define Donaldson-Thomas type deformation invariants ($DT_{4}$ invariants) by studying moduli spaces of solutions to the Donaldson-Thomas equations on $X$. We also study sheaves counting problems on local Calabi-Yau 4-folds. We relate $DT_{4}$ invariants of $K_{Y}$ to the Donaldson-Thomas invariants of the associated Fano 3-fold $Y$. When the Calabi-Yau 4-fold is toric, we adapt the virtual localization formula to define the corresponding equivariant $DT_{4}$ invariants. We also discuss the non-commutative version of $DT_{4}$ invariants for quivers with relations. Finally, we compute $DT_{4}$ invariants for certain Calabi-Yau 4-folds when moduli spaces are smooth and find a $DT_{4}/GW$ correspondence for $X$. Examples of wall-crossing phenomenon in $DT_{4}$ theory are also given.