RF system challenges for future  ${e}^+ {e}^-$ circular colliders

The RF system is the centrepiece of any future circular lepton collider. In particular, the system is required to support the high intensity beams needed for pushing the luminosity at the lower energy regimes of future energy-frontier circular lepton colliders (e.g. for operation in the Z peak and at the WW threshold). Capturing, storing the beam and replacing energy losses from synchrotron radiation demand low frequency, low shunt resistance cavities, low number of cells and high RF power per cell. Controlling the beam both transversely and longitudinally requires sophisticated beam control and timing systems. Additional RF systems are used to ensure transverse stability (feedback systems) and to increase the luminosity (crab cavities). Operation at high energies (such as the ZH and ${\mathrm{t}{\overline{\mathrm{t}}}}$ threshold) requires a very large accelerating voltage, since synchrotron radiation leads to significantly higher energy losses per turn which must be compensated. Since the RF system is to be optimised in size and energy efficiency for varying demands for the different operational modes, the spectrum of R&D; challenges covers a wide range of technologies.