  We work over a perfect field. Recent work of the third-named author
established a Derived Auslander Correspondence that relates finite-dimensional
self-injective algebras that are twisted $3$-periodic to algebraic triangulated
categories of finite type. Moreover, the aforementioned work also shows that
the latter triangulated categories admit a unique differential graded
enhancement. In this article we prove a higher-dimensional version of this
result that, given an integer $d\geq1$, relates twisted $(d+2)$-periodic
algebras to algebraic triangulated categories with a $d\mathbb{Z}$-cluster
tilting object. We also show that the latter triangulated categories admit a
unique differential graded enhancement. Our result yields recognition theorems
for interesting algebraic triangulated categories, such as the Amiot cluster
category of a self-injective quiver with potential in the sense of Herschend
and Iyama and, more generally, the Amiot-Guo-Keller cluster category associated
with a $d$-representation finite algebra in the sense of Iyama and Oppermann.
As an application of our result, we obtain infinitely many triangulated
categories with a unique differential graded enhancement that is not strongly
unique. In the appendix, B. Keller explains how -- combined with crucial
results of August and Hua-Keller -- our main result yields the last key
ingredient to prove the Donovan-Wemyss Conjecture in the context of the
Homological Minimal Model Program for threefolds.
