A Donaldson-Thomas crepant resolution conjecture on Calabi-Yau 4-folds

TL;DR

This paper explores a conjecture relating Donaldson-Thomas invariants and K-theoretic stable pairs on Calabi-Yau 4-folds, proposing formulas and a crepant resolution correspondence in the context of orbifolds and their resolutions.

Contribution

It introduces new conjectures and formulas for invariants on Calabi-Yau 4-folds and develops a crepant resolution correspondence connecting two theories.

Findings

Conjectured closed formulas for generating series of invariants.

Defined K-theoretic stable pair invariants on crepant resolutions.

Developed a vertex formalism for computing DT invariants in toric cases.

Abstract

Let $G$ be a finite subgroup of $\mathrm{SU}(4)$ whose elements have age not larger than one. In the first part of this paper, we define $K$-theoretic stable pair invariants on the crepant resolution of the affine quotient $\mathbb{C}^4/G$, and conjecture closed formulae for their generating series, expressed in terms of the root system of $G$. In the second part, we define degree zero Donaldson-Thomas invariants of Calabi-Yau 4-orbifolds, develop a vertex formalism that computes the invariants in the toric case and conjecture closed formulae for the quotient stacks $[\mathbb{C}^4/\mathbb{Z}_r]$, $[\mathbb{C}^4/\mathbb{Z}_2\times \mathbb{Z}_2]$. Combining these two parts, we formulate a crepant resolution correspondence which relates the above two theories.