One-loop supergravity corrections to black hole entropy and residual supersymmetry

We study the one-loop corrections to the effective on-shell action of $N=2\mathrm{}$ supergravity in the background of the Reissner-Nordström black hole. In the extreme case the contributions from the graviton, gravitino, and photon to the one-loop corrections to the entropy are shown to cancel. This gives the first explicit example of the supersymmetric nonrenormalization theorem for the on-shell action (entropy) for BPS configurations which admit Killing spinors. We display the residual supersymmetry of the perturbations of a general supersymmetric theory in a bosonic BPS background.