The three-dimensional G-Hilbert scheme (0,2)-McKay counting conjecture

Let $G$ be a finite group acting on $\mathbb{C}^3$, and let $X$ be its three-dimensional $G$-Hilbert scheme, a crepant resolution when the construction applies. The $(0,2)$)-deformations of an $N=(2,2)$ theory give deformation states whose number can be compared with the orbifold conformal field theory count. Three-dimensional G-Hilbert counting conjecture. The counting of states on a three-dimensional $G$-Hilbert scheme corresponding to $(0,2)$-deformations of an $N=(2,2)$ theory matches the conformal field theory orbifold count. The paper proves the correspondence in dimension two and gives three-dimensional examples supporting this claim, but does not establish it in general.