Dilaton black holes near the horizon

Generic ${\mathrm{U}\left(1\right)\mathrm{}}^2$ four-dimensional (4D) black holes with unbroken $N=1\mathrm{}$ supersymmetry are shown to tend to a Robinson-Bertotti-type geometry with a linear dilaton and doubling of unbroken super-symmetries near the horizon. Purely magnetic dilatonic black holes, which have unbroken $N=2\mathrm{}$ supersymmetry, behave near the horizon as a 2D linear dilaton vacuum $\otimes S^2$. This geometry is invariant under 8 supersymmetries; i.e., half of the original $N=4\mathrm{}$ supersymmetries are unbroken. The supersymmetric positivity bound, which requires the mass of the 4D dilaton black holes to be greater than or equal to the central charge, corresponds to positivity of mass for a class of stringy 2D black holes.