Bryan–Gholampour quantum McKay correspondence for surface singularities

Let $G$ be a finite subgroup of $SU(2)$, let $\widehat{\mathbb{C}^{2}/G}$ be the crepant resolution of $\mathbb{C}^{2}/G$, and let $[\mathbb{C}^{2}/G]$ be the corresponding orbifold. Denote by $F_{0}^{\widehat{\mathbb{C}^{2}/G}}(y_{0},y_{1},\ldots,y_{n};\mathbf{q})$ and $F_{0}^{[\mathbb{C}^{2}/G]}(x_{0},x_{1},\ldots,x_{n})$ their $\mathbb{C}^{*}$-equivariant genus-zero orbifold Gromov–Witten potentials. Bryan–Gholampour's quantum McKay correspondence conjecture. After the change of variables

$$y_{0}=x_{0},\qquad y_{R}=\frac{1}{|G|}\sum_{g\in G}\sqrt{2-\chi_{\rho_{1}}(g)}\,\overline{\chi}_{R}(g)x_{\llbracket g\rrbracket},\qquad q_{R}=\exp\left(\frac{2\pi\sqrt{-1}\dim R}{|G|}\right),$$

we have

$$F_{0}^{\widehat{\mathbb{C}^{2}/G}}=F_{0}^{[\mathbb{C}^{2}/G]}.$$

This is the genus-zero quantum form of the McKay correspondence, identifying the Gromov–Witten theory of the crepant resolution with that of the orbifold after analytic continuation and the stated change of variables. The source uses it as the starting point for higher-genus and ancestor extensions; its resolution status is not specified here.