Fermion zero modes and the black-hole hypermultiplet with rigid supersymmetry.

The gravitini zero modes riding on top of the extreme Reissner-Nordstrom black-hole solution of N = 2 supergravity are shown to be normalizable. The gravitini and dilatini zero modes of axiondilaton extreme black-hole solutions of N = 4 supergravity are also given and found to have finite norms. These norms are duality invariant. The finiteness and positivity of the norms in both cases are found to be correlated with the Witten-Israel-Nester construction; however, we have replaced the Witten condition by the pure-spin—constraint on the gravitini. We compare our calculation of the norms with the calculations which provide the moduli space metric for extreme black holes. The action of the N = 2 hypermultiplet with an ofF-shell central charge describes the solitons of N = 2 supergravity. This action, in the Majumdar-Papapetrou multi-black-hole background, is shown to be N = 2 rigidly supersymmetric.