### 5.5 Microscopic accounting of superradiance

We mentioned in Section 2.1 that extremal spinning black holes that do not admit a globally-defined
timelike Killing vector spontaneously emit quanta in the range of frequencies (11). This quantum effect is
related by detailed balance to the classical effect of superradiant wave emission, which occur in the same
range of frequencies.
It has been argued that the bound (11) essentially follows from Fermi–Dirac statistics of the fermionic
spin-carrying degrees of freedom in the dual two-dimensional CFT [121] (see also [132]). These arguments
were made for specific black holes in string theory but one expects that they can be applied to
generic extremal spinning black holes, at least qualitatively. Let us review these arguments
here.

One starts with the assumption that extremal spinning black holes are modeled by a CFT, where
the left and right sectors are coupled only very weakly. Therefore, the total energy and entropy are
approximately the sum of the left and right energies and entropies. The state corresponding to
an extremal spinning black hole is modeled as a filled Fermi sea on the right sector with zero
entropy and a thermal state on the left sector, which accounts for the black-hole entropy. The
right-moving fermions form a condensate of aligned spins , which accounts for the
macroscopic angular momentum. It is expected from details of emission rates in several parametric
regimes that fermions are only present on the right sector, while bosons are present in both
sectors [105, 106].

Superradiant spontaneous emission is then modeled as the emission of quanta resulting from interaction
of a left and a right-moving mode. Using details of the model such as the fact that the Fermi energy should
be proportional to the angular velocity , one can derive the bound (11). We refer the reader to [132]
for further details. It would be interesting to better compare these arguments to the present setup, and to
see how these arguments could be generalized to the description of the bound (12) for static
extremal rotating black holes. Let us finally argue that the existence of a qualitative process of
superradiant emission in these models further supports the conjecture that the dual theory to
extremal black holes is a chiral limit of a CFT instead of a chiral CFT with no right-moving
sector.