  The ability to engineer cavity-mediated interactions has emerged as a
powerful tool for the generation of non-local correlations and the
investigation of non-equilibrium phenomena in many-body systems. Levitated
optomechanical systems have recently entered the multi-particle regime, with
promise for using arrays of massive strongly coupled oscillators for exploring
complex interacting systems and sensing. Here, by combining advances in
multi-particle optical levitation and cavity-based quantum control, we
demonstrate, for the first time, programmable cavity-mediated interactions
between nanoparticles in vacuum. The interaction is mediated by photons
scattered by spatially separated particles in a cavity, resulting in strong
coupling ($G_\text{zz}/\Omega_\text{z} = 0.238\pm0.005$) that does not decay
with distance within the cavity mode volume. We investigate the scaling of the
interaction strength with cavity detuning and inter-particle separation, and
demonstrate the tunability of interactions between different mechanical modes.
Our work paves the way towards exploring many-body effects in nanoparticle
arrays with programmable cavity-mediated interactions, generating entanglement
of motion, and using interacting particle arrays for optomechanical sensing.
