Chen-Ruan cohomology of ADE singularities

We study Ruan’s $\mathit{\text{cohomological}}\mathit{\text{crepant}}\mathit{\text{resolution}}\mathit{\text{conjecture}}$ for orbifolds with transversal ADE singularities. In the $A_n$-case we compute both the Chen-Ruan cohomology ring $H_{\mathrm{C}\mathrm{R}}^{\ast }([Y])$ and the quantum corrected cohomology ring $H^{\ast }(Z)(q_1,\mathrm{.}\mathrm{.}\mathrm{.},q_n)$. The former is achieved in general, the later up to some additional, technical assumptions. We construct an explicit isomorphism between $H_{\mathrm{C}\mathrm{R}}^{\ast }([Y])$ and $H^{\ast }(Z)(-1)$ in the $A_1$-case, verifying Ruan’s conjecture. In the $A_n$-case, the family $H^{\ast }(Z)(q_1,\mathrm{.}\mathrm{.}\mathrm{.},q_n)$ is not defined for $q_1=\mathrm{.}\mathrm{.}\mathrm{.}=q_n=-1$. This implies that the conjecture should be slightly modified. We propose a new conjecture in the $A_n$-case which we prove in the $A_2$-case by constructing an explicit isomorphism.