  We study the $m$-graded quiver theories associated to CY $(m+2)$-folds and
their order $(m+1)$ dualities. We investigate how monodromies give rise to
mutation invariants, which in turn can be formulated as Diophantine equations
characterizing the space of dual theories associated to a given geometry. We
discuss these ideas in general and illustrate them in the case of orbifold
theories. Interestingly, we observe that even in this simple context the
corresponding Diophantine equations may admit an infinite number of seeds for
$m\geq 2$, which translates into an infinite number of disconnected duality
webs. Finally, we comment on the possible generalization of duality cascades
beyond $m=1$.
