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Given a negatively graded Calabi-Yau algebra, we regard it as a DG algebrawith vanishing differentials and study its cluster category. We show that thisDG algebra is sign-twisted Calabi-Yau, and realize its cluster category as atriangulated hull of an orbit category of a derived category, and as thesingularity category of a finite dimensional Iwanaga-Gorenstein algebra. Alongthe way, we give two results which stand on their own. First, we show that thederived category of coherent sheaves over a Calabi-Yau algebra has a naturalcluster tilting subcategory whose dimension is determined by the Calabi-Yaudimension and the $a$-invariant of the algebra. Secondly, we prove that two DGorbit categories obtained from a DG endofunctor and its homotopy inverse arequasi-equivalent. As an application, we show that the higher cluster categoryof a higher representation infinite algebra is triangle equivalent to thesingularity category of an Iwanaga-Gorenstein algebra which is explicitlydescribed. Also, we demonstrate that our results generalize the context ofKeller--Murfet--Van den Bergh on the derived orbit category involving a squareroot of the AR translation.

Cluster categories of formal DG algebras and singularity categories