### 6.6 Entropy matching

We will now argue that the temperatures and obtained in Section 6 combined with the
analysis at extremality in Section 4 lead to a (several) microscopic counting(s) of the black hole entropy of
the Kerr, Reissner–Nordström and Kerr–Newman black holes.
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

which is valid when , . As already mentioned in Section 4.4 and argued in [156],
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,

We obtained the value at extremality in Section 4.3 and checked that Cardy’s formula reproduces the
extremal black-hole entropy. One way to uniquely fix the value away from extremality
would consist in matching Cardy’s formula (232) with the Kerr–Newman black-hole entropy
using (233) and the values for the temperatures derived in Section 6.4. Therefore, the matching of
black-hole entropy is true by construction, which is clearly unsatisfactory. It would be more satisfactory to
have an independent computation of away from extremality, but such a computation is currently not
available.
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.