  We develop a Luttinger liquid theory of the Coulomb drag of persistent
currents flowing in concentric mesoscopic rings, by incorporating non-linear
corrections to the electron dispersion relation. We demonstrate that at low
temperatures, interactions between electrons in different rings generate an
additional phase and thus alter the period of Aharonov-Bohm oscillations. The
resulting nondissipative drag depends strongly on the relative parity of the
electron numbers. We also show that interactions set a new temperature scale
below which the linear response theory does not apply at certain values of
external flux.
