4 Microscopic Entropy of Extremal Black Holes

We discussed that near-horizon geometries of compact extremal black holes are isolated systems with universal properties and we reviewed that in all analyzed cases they have no local bulk dynamics. Given the non-trivial thermodynamic properties of these systems even at extremality, one can suspect that some non-trivial dynamics are left. It turns out that such non-trivial dynamics appears at the boundary of the near-horizon geometry. We now show that near-horizon geometries can be extended to a large class describing extremal boundary excitations. The set of all near-horizon geometries will admit additional symmetries at their boundary – asymptotic symmetries – which will turn out to be given by one copy of the Virasoro algebra. We will then argue that these near-horizon geometries are described by chiral limits of two-dimensional CFTs, which we will use to microscopically derive the entropy of any charged or spinning extremal black hole.

 4.1 Boundary conditions and asymptotic symmetry algebra
 4.2 Absence of SL (2,ℝ) asymptotic symmetries
 4.3 Virasoro algebra and central charge
 4.4 Microscopic counting of the entropy

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