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 7View Image, 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).
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Figure 7: Relation between the quantisation of the classical moduli space of certain supersymmetric probe configurations, their supergravity realisations and their possible interpretation as black hole constituents.

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 [22Jump To The Next Citation Point], 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 [456Jump To The Next Citation Point]. 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 [174Jump To The Next Citation Point]. 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 𝒩 = 4 d = 4 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”41. 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 objects42 was tested for the standard D0-D4 black hole in [174Jump To The Next Citation Point]. The black hole was deconstructed in terms of D6 and -- D6 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 [219Jump To The Next Citation Point], 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 [395Jump To The Next Citation Point], 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 [237Jump To The Next Citation Point, 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 [238Jump To The Next Citation Point]. 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.

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