  We derive general bounds on the linear response energy absorption rates of
periodically driven many-body systems of spins or fermions on a lattice. We
show that for systems with local interactions, energy absorption rate decays
exponentially as a function of driving frequency in any number of spatial
dimensions. These results imply that topological many-body states in
periodically driven systems, although generally metastable, can have very long
lifetimes. We discuss applications to other problems, including decay of highly
energetic excitations in cold atomic and solid-state systems.
