  We construct a system of $N$ interacting particles on the unit sphere
$S^{d-1}$ in $d$-dimensional space, which has $d$-body interactions only. The
equations have a gradient formulation derived from a rotationally-invariant
potential of a determinantal form summed over all nodes, with antisymmetric
coefficients. For $d=3$, for example, all trajectories lie on the 2-sphere and
the potential is constructed from the triple scalar product summed over all
oriented 2-simplices. We investigate the cases $d=3,4,5$ in detail, and find
that the system synchronizes from generic initial values, for both positive and
negative coupling coefficients, to a static final configuration in which the
particles lie equally spaced on $S^{d-1}$. Completely synchronized
configurations also exist, but are unstable under the $d$-body interactions. We
compare the relative effect of 2-body and $d$-body forces by adding the
well-studied 2-body interactions to the potential, and find that higher-order
interactions enhance the synchronization of the system, specifically,
synchronization to a final configuration consisting of equally spaced particles
occurs for all $d$-body and 2-body coupling constants of any sign, unless the
attractive 2-body forces are sufficiently strong relative to the $d$-body
forces. In this case the system completely synchronizes as the 2-body coupling
constant increases through a positive critical value, with either a continuous
transition for $d=3$, or discontinuously for $d=5$. Synchronization also occurs
if the nodes have distributed natural frequencies of oscillation, provided that
the frequencies are not too large in amplitude, even in the presence of
repulsive 2-body interactions which by themselves would result in asynchronous
behaviour.
