Topology, entropy, and Witten index of dilaton black holes

We have found that for extreme dilaton black holes an inner boundary must be introduced in addition to the outer boundary to give an integer value to the Euler number. The resulting manifolds have (if one identifies imaginary time) a topology ${\mathit{S}}^1$×R×${\mathit{S}}^2$ and Euler number χ=0 in contrast with the nonextreme case with χ=2. The entropy of extreme U(1) dilaton black holes is already known to be zero. We include a review of some recent ideas due to Hawking on the Reissner-Nordström case. By regarding all extreme black holes as having an inner boundary, we conclude that the entropy of all extreme black holes, including [U(1)${]}^2$ black holes, vanishes. We discuss the relevance of this to the vanishing of quantum corrections and the idea that the functional integral for extreme holes gives a Witten index. We have studied also the topology of ‘‘moduli space’’ of multi-black-holes. The quantum mechanics on black hole moduli spaces is expected to be supersymmetric despite the fact that they are not hyper-Kähler since the corresponding geometry has a torsion unlike the BPS monopole case. Finally, we describe the possibility of extreme black hole fission for states with an energy gap. The energy released, as a proportion of the initial rest mass, during the decay of an electromagnetic black hole is 300 times greater than that released by the fission of a ${}_{}{}^{235}{}_{}{}^{}\mathrm{U}$ nucleus.