  From biology to social science, the functioning of a wide range of systems is
the result of elementary interactions which involve more than two constituents,
so that their description has unavoidably to go beyond simple
pairwise-relationships. Simplicial complexes are therefore the mathematical
objects providing a faithful representation of such systems. We here present a
complete theory of synchronization of $D$-dimensional oscillators obeying an
extended Kuramoto model, and interacting by means of 1- and 2- simplices. Not
only our theory fully describes and unveils the intimate reasons and mechanisms
for what was observed so far with pairwise interactions, but it also offers
predictions for a series of rich and novel behaviors in simplicial structures,
which include: a) a discontinuous de-synchronization transition at positive
values of the coupling strength for all dimensions, b) an extra discontinuous
transition at zero coupling for all odd dimensions, and c) the occurrence of
partially synchronized states at $D=2$ (and all odd $D$) even for negative
values of the coupling strength, a feature which is inherently prohibited with
pairwise-interactions. Furthermore, our theory untangles several aspects of the
emergent behavior: the system can never fully synchronize from disorder, and is
characterized by an extreme multi-stability, in that the asymptotic stationary
synchronized states depend always on the initial conditions. All our
theoretical predictions are fully corroborated by extensive numerical
simulations. Our results elucidate the dramatic and novel effects that
higher-order interactions may induce in the collective dynamics of ensembles of
coupled $D$-dimensional oscillators, and can therefore be of value and interest
for the understanding of many phenomena observed in nature, like for instance
the swarming and/or flocking processes unfolding in three or more dimensions.
