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

Let $G$ be a finite subgroup of $\mathrm{S}\mathrm{U}(4)$ 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 ${\mathbb{C}}^4\mathrm{/}G$, 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 $[{\mathbb{C}}^4\mathrm{/}{\mathbb{Z}}_r]$, $[{\mathbb{C}}^4\mathrm{/}{\mathbb{Z}}_2\times {\mathbb{Z}}_2]$. Combining these two parts, we formulate a crepant resolution correspondence which relates the above two theories.