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 [395Jump To The Next Citation Point]. Its angular momentum prevents the collapse of the expanding brane and it can actually make it supersymmetric [264Jump To The Next Citation Point, 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 𝒩 = 4 d = 4 SYM theory, these states correspond to single trace operators belonging to the chiral ring [18, 150Jump To The Next Citation Point, 68Jump To The Next Citation Point]. When their momentum becomes of order N, it is energetically favourable for these gravitons to expand into rotating spherical D3-branes, i.e., giant gravitons. The N 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 L4, thus giving

4 Δ ∝ TD3L4 = N . (411 )
Similar considerations apply in different AdSp+1 realisations of this phenomena [264Jump To The Next Citation Point, 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

( 2) 2 ( ) ds2 = − 1 + r-- dt2 +--dr-2-+ r2d&tidle;Ω2 + L2 dπœƒ2 + cos2πœƒd Ο•2 + sin2 πœƒdΩ2 , 10 L24 1 + rL2 3 4 3 4 4 4 C4 = L4 sin πœƒ dΟ• ∧ ω3, (412 )
where ω3 stands for the volume form of the 3-sphere in S5 and it is understood dC4 is made self-dual to satisfy the type IIB equations of motion39. Giant gravitons consist of D3-branes wrapping such 3-spheres and rotating in the Ο• direction to carry R-charge from the dual CFT perspective. Thus, one considers the bosonic ansatz
σ0 = t, σi = ωi, πœƒ = πœƒ0, Ο• = Ο• (τ), r = 0. (413 )
The D3-brane Lagrangian density evaluated on this ansatz and integrating over the 3-sphere world volume is [264Jump To The Next Citation Point]
[ ∘--------------- ] β„’ = N-- − sin3πœƒ 1 − L2 cos2πœƒ Ο•Λ™2 + L sin4 πœƒΟ•Λ™ . (414 ) L4 4 4
Since k = ∂ Ο• is a Killing vector, the conjugate momentum P Ο• is conserved
⌊ ⌋ L4 sin3πœƒ cos2πœƒΟ•Λ™ P Ο• = N ⌈ ∘----------------+ sin4πœƒ⌉ ≡ N p , (415 ) 1 − L24cos2 πœƒΟ•Λ™2
where the constant p was defined. Computing the Hamiltonian density,
∘----------------------- β„° = P Ο• Λ™Ο• − β„’ = N-- p2 + tan2 πœƒ(p − sin2πœƒ)2, (416 ) L4
allows us to identify the stable configurations by extremising Eq. (416View Equation). Focusing on finite size configurations, one finds
-- sinπœƒ0 = √ p = ⇒ Λ™Ο• = -1- =⇒ β„° = PΟ•-. (417 ) L4 L4
Notice the latter equality saturates the BPS bound, Δ ≡ β„°L4 = P Ο•, 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 (214View Equation). The 32 Killing spinors for the maximally-supersymmetric AdS5 × S5 background were computed in [359, 264Jump To The Next Citation Point]. They are of the form πœ– = M πœ–∞ where M is a non-trivial Clifford valued matrix depending on the bulk point and πœ–∞ is an arbitrary constant spinor. It was shown in [264Jump To The Next Citation Point] that Eq. (214View Equation) reduces to

(Γ tΟ• − 1)πœ–∞ = 0. (418 )
Thus, giant gravitons preserve half of the spacetime supersymmetry. Furthermore, they preserve the same supercharges as a pointlike graviton rotating in the Ο• direction.

General supersymmetric giant graviton construction:
There exist more general giant gravitons charged under the full 3 U (1) Cartan subalgebra of the full R-symmetry group SO (6). The general construction of such supersymmetric probes was done in [392Jump To The Next Citation Point]. The main idea is to embed the bulk 5-sphere into an auxiliary embedding β„‚3 space with complex coordinates zi i = 1,2,3 and AdS5 into β„‚1,2. In the probe calculation, the Zi become dynamical scalar fields subject to the defining quadric constraint ∑ |Zi|2 = 1 i. 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 3 β„‚ and 1,2 β„‚, respectively. This is briefly reviewed in Appendix B. The conclusion of such analysis is that any holomorphic function F (Z1, Z2,Z3 ) gives rise to a supersymmetric giant graviton configuration [392] defined

|Z |2 + |Z |2 + |Z |2 = 1, 1 2 3 F (e−itβˆ•L4Z1, e−itβˆ•L4Z2,e− itβˆ•L4Z3) = 0, (419 )
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 β„‚ planes in 3 β„‚ at the speed of light (in 1βˆ•L4 units), which is a consequence of supersymmetry and a generalisation of the condition explicitly found in Eq. (417View Equation).

Geometric quantisation and BPS counting:
The above construction is classical and applies to backgrounds of the form AdS5 × β„³5. 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 𝒩 = 4 d = 4 SYM. Later, in [104, 369], the full partition function was derived and seen to agree with that of N noninteracting bosons in a 3d harmonic potential. Similar work and results were obtained for the moduli space of dual giant gravitons40 when β„³5 is an Einstein–Sasaki manifold [374]. The BPS partition functions derived from these geometric quantisation schemes agree with purely gauge theory considerations [69, 341Jump To The Next Citation Point] 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 N 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 N 2. The idea that supersymmetric R-charged AdS black holes could be interpreted as distributions of giant gravitons was first discussed in [397Jump To The Next Citation Point], where these bulk configurations were coined as superstars. The main idea behind this identification comes from two observations:

  1. 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.
  2. 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 [397Jump To The Next Citation Point] 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 𝒩 = 2 d = 5 gauged supergravity with U (1)3 gauge symmetry [56, 57] properly embedded into type IIB [157]. Their metric is

2 √-- [ − 1 2 −1 2 2 2] ds10 = Δ − (H1H2H3 ) fdt + (f dr + r dΩ 3) 1 ∑3 ( ) + √--- Hi L2d μ2i + μ2i[L4dΟ•i + (H −i1− 1)dt]2 , (420 ) Δ i=1
with the different scalar functions defined as
r2 qi f = 1 + --2H1H2H3 with Hi = 1 + -2 , L 4 r ∑3 μ2i ∑3 2 Δ = H1H2H3 ---, with μi = 1 . (421 ) i=1Hi i=1
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 R5 = dC4 + ∗dC4 with
4 3 C = − -r-Δ dt ∧ ω − L ∑ qμ2 (L dΟ• − dt) ∧ ω , (422 ) 4 L4 3 4 i=1 i i 4 i 3
with ω3 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 q2 = q3 = 0. This should correspond to a collection of giant gravitons rotating along Ο• 1 with a certain distribution of sizes (specified by μ = cosπœƒ 1 1). To measure the density of giant gravitons sitting near a certain πœƒ1, one must integrate R5 over the appropriate surface. Describing the 3-sphere in AdS5 by

dΩ2 = d α2+ sin2 α (dα2 + sin2α dα2), (423 ) 3 1 1 2 2 3
one can enclose a point on the brane at πœƒ1 with a small five-sphere in the {r, πœƒ1,Ο•1αi} directions. The relevant five-form component is
(R ) = 2q L2 sin πœƒ cos πœƒ sin2 α sin α , (424 ) 5 πœƒ1Ο•1α1α2α3 1 4 1 1 1 2
and by integrating the latter over the smeared direction Ο•1 and the 3-sphere, one infers the density of giants at a point πœƒ1 [397Jump To The Next Citation Point]
∫ dn1- -N---- 3 -q1 dπœƒ1 = 4π3L4 (R5)πœƒ1Ο•1α1α2α3dΟ•1d α = N L2 sin2 πœƒ1. (425 ) 4 4
If this is correct, the total number of giant gravitons carried by the superstar is
∫ πβˆ•2 dn1 q1 n1 = 0 dπœƒ1 dπœƒ--= N L2. (426 ) 1 4
The matching is achieved by comparing the microscopic momentum carried by a single giant at the location πœƒ1, Pmicro = N sin2 πœƒ1, with the total mass of the superstar
N-2 q1- M = 2 L3 . (427 ) 4
Indeed, by supersymmetry, the latter should equal the total momentum of the distribution
∫ 2 M = P-Ο•1= πβˆ•2 dπœƒ dn1-Pmicro = N-- q1-, (428 ) L4 0 1d πœƒ1 L4 2 L34
which establishes the physical correspondence. There exist extensions of these considerations when more than a single R-charge is turned on, i.e., when q2,q3 ⁄= 0. 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 𝒩 = 4 d = 4 SYM corresponding to collections of giant gravitons. For 1/2 BPS states, the supergravity analysis was done in [355Jump To The Next Citation Point]. 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 N 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 7View Image.

Less supersymmetric superstars:
Given the robustness of the results concerning the partition functions of 1/4 and 1/8 chiral BPS operators in 𝒩 = 4 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 [142Jump To The Next Citation Point], extending previous work [182, 181]. The equations describing these moduli spaces are far more complicated than its 1/2 BPS sector cousin,

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.

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