5.9 Giant gravitons and superstars
It was mentioned in Section 5.7 that angular momentum can stabilise an expanded brane carrying the
same quantum numbers as a lower dimensional brane. I will now review an example of such phenomena,
involving supersymmetric expanding branes in AdS, the so called giant gravitons [386]. In this case, a
rotating pointlike graviton in AdS expands into a rotating brane due to the RR flux supporting the AdS
supergravity solution [395
]. Its angular momentum prevents the collapse of the expanding brane and it can
actually make it supersymmetric [264
, 290].
Consider type IIB string theory in AdS5 × S5. 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
Similar considerations apply in different AdSp+1 realisations of this phenomena [264
, 368].
The field theory interpretation of these states was given in [35] in terms of subdeterminant
operators.
Let us construct these configurations in AdS5 × S5. The bosonic background has a constant dilaton
and non-trivial metric and RR 4-form potential given by
where
stands for the volume form of the 3-sphere in S5 and it is
understood
is made self-dual to satisfy the type IIB equations of
motion.
Giant gravitons consist of D3-branes wrapping such 3-spheres and rotating in the
direction to carry
R-charge from the dual CFT perspective. Thus, one considers the bosonic ansatz
The D3-brane Lagrangian density evaluated on this ansatz and integrating over the 3-sphere world volume
is [264
]
Since
is a Killing vector, the conjugate momentum
is conserved
where the constant
was defined. Computing the Hamiltonian density,
allows us to identify the stable configurations by extremising Eq. (416). Focusing on finite size
configurations, one finds
Notice the latter equality saturates the BPS bound,
, as expected from supersymmetry
considerations.
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 AdS5 × S5 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
Thus, giant gravitons preserve half of the spacetime supersymmetry. Furthermore, they preserve the same
supercharges as a pointlike graviton rotating in the
direction.
General supersymmetric giant graviton construction:
There exist more general giant gravitons charged
under the full
Cartan subalgebra of the full R-symmetry group
. The general construction
of such supersymmetric probes was done in [392
]. The main idea is to embed the bulk 5-sphere into an
auxiliary embedding
space with complex coordinates
and AdS5 into
. In the
probe calculation, the
become dynamical scalar fields subject to the defining quadric constraint
. To prove these configurations are supersymmetric one can use the well known isomorphism
between geometric Killing spinors on both the 5-sphere and AdS5 and parallel spinors in
and
,
respectively. This is briefly reviewed in Appendix B. The conclusion of such analysis is that any
holomorphic function
gives rise to a supersymmetric giant graviton configuration [392]
defined
as the intersection of the 5-sphere with a holomorphic hypersurface properly evolved in world volume time.
The latter involves rotations in each of the
planes in
at the speed of light (in
units), which
is a consequence of supersymmetry and a generalisation of the condition explicitly found in
Eq. (417).
Geometric quantisation and BPS counting:
The above construction is classical and applies
to backgrounds of the form AdS
. In [54], the classical moduli space of holomorphic
functions mentioned above was originally quantised and some of its BPS spectrum matched
against the spectrum of chiral operators in
SYM. Later, in [104, 369], the full
partition function was derived and seen to agree with that of
noninteracting bosons in a 3d
harmonic potential. Similar work and results were obtained for the moduli space of dual giant
gravitons
when
is an Einstein–Sasaki manifold [374]. The BPS partition functions derived from
these geometric quantisation schemes agree with purely gauge theory considerations [69, 341
]
and with the more algebraic approach to counting chiral operators followed in the plethystics
program [67, 210].
Related work:
There exists an extensive amount of work constructing world volume configurations
describing giant gravitons in different backgrounds to the ones mentioned above. This includes
non-supersymmetric giant gravitons with NS-NS fields [131], M-theory giants with 3-form potential
field [132], giants in deformed backgrounds [422] or electric/magnetic field deformed giants in
Melvin geometries [310]. For discussions on supersymmetric D3, fractional D5 and D7-brane
probes in AdS5 × Labc, see [135]. There is also interesting work on bound states of giant
gravitons [430] and on the effective field theory description of many such giants (a non-abelian world
volume description) with the inclusion of higher moment couplings responsible for their physical
properties [317, 318].
5.9.1 Giant gravitons as black-hole constituents
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 AdS5 black holes. These are
solutions to
gauged supergravity with
gauge symmetry [56, 57] properly embedded
into type IIB [157]. Their metric is
with the different scalar functions defined as
All these metrics have a naked singularity at the center of AdS that extends into the 5-sphere. Depending
on the number of charges turned on, the rate at which curvature invariants diverge changes with the
5-sphere direction. Besides a constant dilaton, these BPS configurations also have a non-trivial RR self-dual
5-form field strength
with
with
being volume 3-form of the unit 3-sphere.
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 AdS5 by
one can enclose a point on the brane at
with a small five-sphere in the
directions. The
relevant five-form component is
and by integrating the latter over the smeared direction
and the 3-sphere, one infers the density of
giants at a point
[397
]
If this is correct, the total number of giant gravitons carried by the superstar is
The matching is achieved by comparing the microscopic momentum carried by a single giant at the location
,
, with the total mass of the superstar
Indeed, by supersymmetry, the latter should equal the total momentum of the distribution
which establishes the physical correspondence. There exist extensions of these considerations when more
than a single R-charge is turned on, i.e., when
. See [397] for the specific details, though the
conclusion remains the same.
1/2 BPS superstar and smooth configurations:
Just as supertubes have smooth supergravity
descriptions [205] with U-dual interpretations in terms of chiral states in dual CFTs [361] when some of the
dimensions are compact, one may wonder whether a similar picture is available for chiral operators in
SYM corresponding to collections of giant gravitons. For 1/2 BPS states, the supergravity
analysis was done in [355
]. The classical moduli space of smooth configurations was determined: it is
characterised in terms of a single scalar function satisfying a Laplace equation. When the latter
satisfies certain boundary conditions on its boundary, the entire supergravity solution is smooth.
Interestingly, such boundary could be interpreted as the phase space of a single fermion in
a 1d harmonic oscillator potential, whereas the boundary conditions correspond to exciting
coherent states on it. This matches the gauge theory description in terms of the eigenvalues of the
adjoint matrices describing the gauge invariant operators in this 1/2 BPS sector of the full
theory [150, 68]. Moreover, geometric quantisation applied on the subspace of these 1/2 BPS supergravity
configurations also agreed with the picture of
free fermions in a 1d harmonic oscillator
potential [251, 371]. The singular superstar was interpreted as a coarse-grained description of the typical
quantum state in that sector [37], providing a bridge between quantum mechanics and classical
geometry through the coarse-graining of quantum mechanical information. In some philosophically
vague sense, these supergravity considerations provide some heuristic realisation of Wheeler’s
ideas [492, 493, 39]. Some partial progress was also achieved for similar M-theory configurations [355]. In
this case, the quantum moduli space of BPS gauge theory configurations was identified in [450]
and some steps to identify the dictionary between these and the supergravity geometries were
described in [184]. Notice this set-up is also in agreement with the general framework illustrated in
Figure 7.
Less supersymmetric superstars:
Given the robustness of the results concerning the partition functions of
1/4 and 1/8 chiral BPS operators in
SYM and their description in terms of BPS giant graviton
excitations, it is natural to study whether there exist smooth supergravity configurations preserving this
amount of supersymmetry and the appropriate bosonic isometries to be interpreted as these chiral states.
The classical moduli space of these configurations was given in [142
], extending previous work [182, 181].
The equations describing these moduli spaces are far more complicated than its 1/2 BPS sector cousin,
- 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.