Singularity categories via higher McKay quivers with potential

In 2018, Kalck and Yang showed that the singularity categories associated with $3$-dimensional Gorenstein quotient singularities are triangle equivalent (up to direct summands) to small cluster categories associated with McKay quivers with potential. We introduce higher McKay quivers with potential and generalize Kalck and Yang’s theorem to arbitrary dimensions. The singularity categories we consider occur as the stable categories of categories of maximal Cohen-Macaulay modules. We refine our description of the singularity categories by showing that these categories of maximal Cohen-Macaulay modules are equivalent to Higgs categories in the sense of Wu. Moreover, we describe the singularity categories in the non-Gorenstein case.