  Given a negatively graded Calabi-Yau algebra, we regard it as a DG algebra
with vanishing differentials and study its cluster category. We show that this
DG algebra is sign-twisted Calabi-Yau, and realize its cluster category as a
triangulated hull of an orbit category of a derived category, and as the
singularity category of a finite dimensional Iwanaga-Gorenstein algebra. Along
the way, we give two results which stand on their own. First, we show that the
derived category of coherent sheaves over a Calabi-Yau algebra has a natural
cluster tilting subcategory whose dimension is determined by the Calabi-Yau
dimension and the $a$-invariant of the algebra. Secondly, we prove that two DG
orbit categories obtained from a DG endofunctor and its homotopy inverse are
quasi-equivalent. As an application, we show that the higher cluster category
of a higher representation infinite algebra is triangle equivalent to the
singularity category of an Iwanaga-Gorenstein algebra which is explicitly
described. Also, we demonstrate that our results generalize the context of
Keller--Murfet--Van den Bergh on the derived orbit category involving a square
root of the AR translation.
