5.10 Deconstructing black holes
Both supertubes and giant gravitons are examples of supersymmetric states realised as classical solitons
in brane effective actions and interpreted as the microscopic constituents of small black holes. The bulk
entropy is matched after geometric quantisation of their respective classical moduli spaces. This
framework, which is summarised in Figure 7, suggests the idea of deconstructing the black hole
into zero-entropy, minimally-charged bits, reinterpreting the initial black-hole entropy as the
ground-state degeneracy of the quantum mechanics on the moduli space of such deconstructions
(bits).
In this subsection, I briefly mention some work in this direction concerning large supersymmetric
AdS5 × S5 black holes, deconstructions of supersymmetric asymptotically-flat black holes in terms of
constituent excitations living at the horizon of these black holes and constituent models for extremal static
non-BPS black holes.
Large supersymmetric AdS5 black holes:
Large supersymmetric AdS5 × S5 black holes require the addition
of angular momentum in AdS5, besides the presence of R-charges, to achieve a regular macroscopic
horizon while preserving a generic 1/16 of the vacuum supersymmetries. The first examples were
reported in [280]. Subsequent work involving more general (non-)BPS black holes can be found
in [279, 143, 350].
Given the success in identifying the degrees of freedom for R-charged black holes, it is natural to analyse
whether the inclusion of angular momentum in AdS5 can be accomplished by more general (dual) giant
graviton configurations carrying the same charges as the black hole. This task was initiated in [339]. Even
though their work was concerned with configurations preserving 1/8 of the supersymmetry, the importance
of a non-trivial Poynting vector on the D3-brane world volume to generate angular momentum was already
pointed out, extending the mechanism used already for supertubes. In [340], the first extension of these
results to 1/16 world volume configurations was considered. The equations satisfied for the most
general 1/16 dual giant D3-brane probe in AdS5 × S5 were described in [22
], whereas explicit
supersymmetric electromagnetic waves on (dual) giants were constructed in [23]. Similar interesting work
describing giant gravitons in the pp-wave background with non-trivial electric fields was reported
in [15].
All these configurations have interest on their own, given their supersymmetry and the conserved
charges they carry, but further evidence is required to interpret them as bulk black hole constituents. This
task was undertaken in [456
]. Instead of working in the vacuum, these authors studied the spectrum of
classical supersymmetric (dual) giant gravitons in the near horizon geometries of these black holes in [457],
following similar reasonings for asymptotically-flat black holes [174
]. The partial quantisation of this
classical moduli space [456] is potentially consistent with the identification of dual giants as the
constituents of these black holes, but this remains an open question. In the same spirit, [22] quantised the
moduli space of the wobbling dual giants, 1/8 BPS configurations with two angular momentum in AdS5
and one in S5 and agreement was found with the gauge theory index calculations carried out
in [341].
There have also been more purely field theoretical approaches to this problem. In [250], cohomological
methods were used to count operators preserving 1/16 of the supersymmetries in
SYM,
whereas in [97] explicit operators were written down, based on Fermi surface filling fermions models and
working in the limit of large angular momentum in AdS5. These attempted to identify the pure states
responsible for the entropy of the black hole and their counting agreed, up to order one coefficients, with the
Hawking–Bekenstein classical entropy.
Large asymptotically-flat BPS black holes:
There exists a large literature on the construction of
supersymmetric configurations with the same asymptotics and charges as a given large BPS
black hole, but having the latter carried by different constituent charges located at different
“centers”.
The center locations are non-trivially determined by solving a set of constraint equations, called the bubble
equations. The latter is believed to ensure the global smoothness and lack of horizon of the configuration.
These constraints do reflect the intrinsic bound state nature of these configurations. The identification of a
subset of 1/2 BPS centers as the fundamental constituents for large black holes was further developed
in [38].
One of the new features in these deconstructions is that the charges carried by the different constituents
do not have to match the charges carried by the black hole, i.e., a constituent can carry D6-brane charge
even if the black hole does not, provided there exists a second centre with anti-D6-brane charge, cancelling
the latter.
This idea of deconstructing a given black hole in terms of maximally entropic configurations of constituent
objects
was tested for the standard D0-D4 black hole in [174
]. The black hole was deconstructed in terms of
and
branes with world volume fluxes turned on, inducing further D4-D2-D0 charges, and a large set of
D0-branes. Working in a regime of charges where the distance between centres scales to zero, i.e., the scaling
solution, all D0-branes become equidistant to the D6-branes, forming some sort of accretion disk and
the geometry deep inside this ring of D0-branes becomes that of global AdS3 × S2, when
lifting the configuration to M-theory. Using the microscopic picture developed in [219
], where
it was argued that the entropy of this black hole came from the degeneracy of states due to
non-abelian D0-branes that expand into D2-branes due to the Myers’ effect [395
], the authors
in [174] manage to extend the near horizon wrapping M2-branes found in [455] to M2-branes
wrapping supersymmetric cycles of the full geometry. It was then argued that the same counting
done [219], based on the degeneracy of the lowest Landau level quantum mechanics problem
emerging from the effective magnetic field on the transverse Calabi–Yau due to the coupling
of the D2-D0 bound states to the background RR 4-form field strength, would apply in this
case.
The same kind of construction and logic was applied to black rings [206, 199] in [239]. Further
work on stable brane configurations in the near horizon on brane backgrounds can be found
in [130].
Extremal non-BPS deconstructions:
These ideas are also applicable to non-supersymmetric systems, though
one expects to have less control there. For the subset of static extremal non-BPS black holes in the
STU model [155, 194, 58], these methods turned out to be successful. The most general static
black-hole solution, including non-trivial moduli at infinity, was found in [237
, 358]. It was pointed
out in [237] that the mass of these black holes equals the sum of four mutually local 1/2 BPS
constituents for any value of the background moduli fields and in any U-duality frame. Using probe
calculations, it was shown that such constituents do not feel any force in the presence of these
black holes [238
]. This suggested that extra quanta could be added to the system and located
anywhere. This is consistent with the multi-center extremal non-BPS solutions found in [218]: their
centres are completely arbitrary but the charge vectors carried by each centre are constrained to
be the ones of the constituents identified in [238] (or their linear combinations). This model
identifies the same constituents as the ones used to account for the entropy of neutral black
holes in [204] and extends it to the presence of fluxes. No further dynamical understanding of
the open string degrees of freedom is available in terms of non-supersymmetric quiver gauge
theories.
As soon as angular momentum is added to the system, while keeping extremality, the location
of the deconstructed constituents gets constrained according to non-linear bubble equations
that ensure the global smoothness of the full supergravity solution [61, 62]. These are fairly
recent developments and one expects further progress to be achieved in this direction in the
future. For example, very recently, an analysis of stable, metastable and non-stable supertubes in
smooth geometries being candidates for the microstates of black holes and black rings was
presented in [63]. This includes configurations that would also be valid for non-extremal black
holes.