Consider type IIB string theory in AdS_{5} × S^{5}. It is well known that this theory has BPS graviton
excitations rotating on the sphere at the speed of light. In the dual SYM theory, these
states correspond to single trace operators belonging to the chiral ring [18, 150, 68]. When
their momentum becomes of order , it is energetically favourable for these gravitons to
expand into rotating spherical D3-branes, i.e., giant gravitons. The scaling is easy to argue
for: the conformal dimension must be proportional to the D3-brane tension times the volume
of the wrapped cycle, which is controlled by the AdS radius of curvature , thus giving

Let us construct these configurations in AdS_{5} × S^{5}. The bosonic background has a constant dilaton
and non-trivial metric and RR 4-form potential given by

To check whether the above configuration indeed preserves some supersymmetry, one must check
whether there exists a subset of target space Killing spinors solving the kappa symmetry preserving
condition (214). The 32 Killing spinors for the maximally-supersymmetric AdS_{5} × S^{5} background were
computed in [359, 264]. They are of the form where is a non-trivial Clifford valued
matrix depending on the bulk point and is an arbitrary constant spinor. It was shown in [264] that
Eq. (214) reduces to

Individual giant gravitons carry conformal dimension of order and according to the discussion above, they exhaust the spectrum of chiral operators in the dual CFT, whereas R-charged AdS black holes carry mass of order . The idea that supersymmetric R-charged AdS black holes could be interpreted as distributions of giant gravitons was first discussed in [397], where these bulk configurations were coined as superstars. The main idea behind this identification comes from two observations:

- The existence of naked singularities in these black holes located where giant gravitons sit in AdS suggests the singularity is due to the presence of an external source.
- Giant gravitons do not carry D3-brane charge, but they do locally couple to the RR 5-form field strength giving rise to some D3-brane dipole charge. This means [397] that a small (five-dimensional) surface enclosing a portion of the giant graviton sphere will carry a net five-form flux proportional to the number of D3-branes enclosed. If this is correct, one should be able to determine the local density of giant gravitons at the singularity by analysing the net RR 5-form flux obtained by considering a surface that is the boundary of a six-dimensional ball, which only intersects the three-sphere of the giant graviton once, at a point very close to the singularity.

To check this interpretation, let us review these supersymmetric R-charged AdS_{5} black holes. These are
solutions to gauged supergravity with gauge symmetry [56, 57] properly embedded
into type IIB [157]. Their metric is

To test the microscopic interpretation for the superstar solutions, consider the single R-charged
configuration with . This should correspond to a collection of giant gravitons rotating along
with a certain distribution of sizes (specified by ). To measure the density of giant
gravitons sitting near a certain , one must integrate over the appropriate surface. Describing the
3-sphere in AdS_{5} by

- 1/4 BPS configurations depend on a 4d Kähler manifold with Kähler potential satisfying a non-linear Monge–Ampère equation [142],
- 1/8 BPS configuration depend on a 6d manifold, whose scalar curvature satisfies a non-linear equation in the scalar curvature itself and the square of the Ricci tensor [338].

Some set of necessary conditions for the smoothness of these configurations was discussed in [142]. A more thorough analysis for the 1/4 BPS configurations was performed in [360], where it was argued that a set of extra consistency conditions were required, the latter constraining the location of the sources responsible for the solutions. Interestingly, these constraints were found to be in perfect agreement with the result of a probe analysis. This reemphasises the usefulness of probe techniques when analysing supergravity matters in certain BPS contexts.

Living Rev. Relativity 15, (2012), 3
http://www.livingreviews.org/lrr-2012-3 |
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