Let us assume that there is a two-dimensional CFT () describing the Kerr black hole, a two-dimensional CFT () describing the Reissner–Nordström black hole and a family of two-dimensional CFTs (, ) describing the Kerr–Newman black hole. If these CFTs are dual to the black hole, the entropy is reproduced by Cardy’s formula, the regime , is not a necessary condition for Cardy’s formula to be valid if these CFTs have special properties such as admitting a long string picture, as reviewed in Section 3.3.
Let us discuss the values of the central charges. In a CFT, the difference is proportional to the diffeomorphism anomaly of the CFT [188, 187]. One can then argue from diffeomorphism invariance that the two left and right sectors should have the same value for the central charge,
However, the resulting central charge is, however, non-trivial. For the , we obtain . For , we have and for the , we find . Quite remarkably, these central charges are expressed solely in terms of quantized charges. They do not depend on the mass of the black hole. This is a non-trivial feature that has no explanation so far.
The presence of several CFTs dual to the Kerr–Newman black hole is curious but not inconsistent. Each CFT describes part of the low-energy dynamics of probe scalar fields and multiple CFTs are needed in order to reproduce the full dynamics for arbitrary ratios of the probe angular momentum to probe electric charge. Therefore, each CFT description has therefore a range of applicability away from extremality.
Living Rev. Relativity 15, (2012), 11
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