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arXiv:1005.1287v1 [hep-th] 7 May 2010
Preprint typeset in JHEP style - HYPER VERSION
TIFR/TH/
ITFA10-13
Small Hairy Black Holes in AdS5× S5
Sayantani Bhattacharyyaa∗, Shiraz Minwallaa†and Kyriakos Papadodimasb‡
aTata Institute of Fundamental Research,
Homi Bhabha Rd, Mumbai 400005.
bInstitute for Theoretical Physics,
Valckenierstraat 65, 1018 XE Amsterdam,
The Netherlands
Abstract: We study small hairy black holes in a consistent truncation of N = 8 gauged
supergravity that consists of a single charged scalar field interacting with the metric and a
U(1) gauge field. Small very near extremal RNAdS black holes in this system are unstable
to decay by superradiant emission. The end point of this instability is a small hairy
black hole that we construct analytically in a perturbative expansion in its charge. Unlike
their RNAdS counterparts, hairy black hole solutions exist all the way down to the BPS
bound, demonstrating that N = 4 Yang Mills theory has an O(N2) entropy at all energies
above supersymmetry. At the BPS bound these black holes reduce to previously discussed
regular, supersymmetric horizon free solitons. We use numerical methods to continue the
construction of these solitons to large charges and find that the line of soliton solutions
terminates at a singular solution S at a finite charge. We conjecture that a one parameter
family of singular supersymmetric solutions, which emerges out of S, constitutes the BPS
limit of hairy black holes at larger values of the charge. We analytically determine the near
singularity behaviour of S, demonstrate that both the regular and singular solutions exhibit
an infinite set of damped ‘self similar’ oscillations around S and analytically compute the
frequency of these oscillations. At leading order in their charge, the thermodynamics of
the small hairy black holes constructed in this paper turns out to be correctly reproduced
by modeling these objects as a non interacting mix of an RNAdS black hole and the
supersymmetric soliton in thermal equilibrium. Assuming that a similar non interacting
model continues to apply upon turning on angular momentum, we also predict a rich
family of rotating hairy black holes, including new hairy supersymmetric black holes. This
analysis suggests interesting structure for the space of (yet to be constructed) hairy charged
rotating black holes in AdS5× S5, particularly in the near BPS limit.
Keywords: .
∗sayanta@theory.tifr.res.in
†minwalla@theory.tifr.res.in
‡k.papadodimas@uva.nl
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Contents
1. Introduction2
2. A Consistent Truncation and its Equations of Motion
2.1 A Consistent Truncation of Gauged Supergravity
2.2 Equations of Motion
2.3 RNAdS Black Holes
10
10
12
14
3.The Supersymmetric Soliton in Perturbation Theory
3.1Setting up the perturbative expansion
3.2 The Soliton up to O(ǫ9)
14
15
16
4.The Hairy Black Hole in Perturbation Theory
4.1 Basic Perturbative strategy
4.2 Perturbation Theory at O(ǫ)
4.2.1 Far Field Region (r ≫ R)
4.2.2 Intermediate Field Region r ≪ 1 and (r − R) ≫ R3
4.2.3Near Field Region (r − R) ≪ R or (y − 1) ≪ 1
4.3 Perturbation theory at O(ǫ2)
4.3.1 Far Field Region, r ≫ R
4.3.2 Intermediate field region, r ≪ 1 and (r − R) ≫ R3
4.3.3 Near Field Region (r − R) ≪ R or (y − 1) ≪ 1
17
17
18
19
20
23
24
24
25
28
5.All Spherically Symmetric Supersymmetric Configurations
5.1The Equations of Supersymmetry
5.2Classification of Supersymmetric Solutions
5.2.1h(ρ) ≈ ρ−2
5.2.2h(ρ) ≈ ho+ O(ρ2)
5.2.3h(ρ) ≈a
5.2.4The generic solution, α = 2
5.3 ’Regular’ supersymmetric Solutions
5.3.1Solitons
5.3.2Solutions with α = 1
5.3.3 An analytic solution at large charge
5.4 Phase Structure of ‘regular’ supersymmetric solutions
30
31
32
33
35
37
38
39
39
42
44
44
3
ρ
6. Thermodynamics in the Micro Canonical Ensemble
6.1 RNAdS Black Hole
6.2 Supersymmetric Soliton
6.3 A non interacting mix of the black hole and soliton
6.4Hairy Black Hole
46
47
48
48
49
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7. Hairy Rotating Black Holes
7.1Thermodynamics of small Kerr RNAdS Black Holes
7.2 Hairy Rotating Black Holes as a non interacting mix
51
51
53
8. Discussion 55
A. Results for the Perturbative Expansion of Hairy Black Holes
A.1 Far Field Solution
A.2 Intermediate Solution
A.3 Near Field Solution
57
57
58
59
B. Supersymmetric solitons in AdS5× S5and the planar limit
B.1 One-charge solitons
B.2 Three-charge solitons
60
60
62
C. Some numerical results62
1. Introduction
Black hole solutions of IIB theory on AdS5× S5constitute the thermodynamic saddle
points of N = 4 Yang Mills theory on S3via the AdS/CFT correspondence. A complete
understanding of the space of stationary black hole solutions in AdS5×S5is consequently
essential for a satisfactory understanding of the state space of N = 4 Yang Mills theory
at energies of order N2. While the Kerr RNAdS black hole solutions are well known
[1, 2, 3, 4, 5, 6, 7], it seems likely that several additional yet to be determined families of
black hole solutions will play an important role in the dynamics and thermodynamics of
N = 4 Yang Mills theory.
In this paper we will construct a new class of asymptotically AdS5× S5black hole
solutions. The black holes we construct are small, charged, and have parametrically low
temperatures; our construction is perturbative in the black hole charge. Our solutions
are hairy, in the sense that they include condensates of charged scalar fields1. In the
BPS limit these hairy black holes reduce to regular horizon free solitons. We also use
numerical techniques to continue our perturbative construction of these solitons to charges
of order unity, and uncover an intricate self similar behaviour in the space of solitons in
the neighborhood of a finite critical value of the charge. The solutions presented in this
paper suggest a qualitatively new picture of the near BPS spectrum of N = 4 Yang Mills.
Our perturbative construction of hairy black holes is close in spirit and technique to the
constructions presented in the recent paper [14], which may be regarded as an immediate
precursor to the current work. For this reason we first present a brief review of [14] before
1See [8, 9, 10, 11, 12] and references therein for reviews of recent work - sparked by an observation by
Gubser [13] -on hairy black branes in AdS spaces.
– 2 –
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turning to a description and discussions of the new black hole solutions constructed in this
paper.
It was demonstrated in [14] that small charged black holes in global AdS spaces are
sometimes unstable to the condensation of charged matter fields. More precisely any system
governed by the Lagrangian
?
Dµφ = ∂µφ − eiAµφ
d5x√g
?1
2(R + 12) −3
8FµνFµν−3
16
?|Dµφ|2+ ∆(∆ − 4)φφ∗?+ Interactions
?
(1.1)
possesses small RNAdS black holes that are unstable to decay by superradiant discharge of
the scalar field φ whenever e > ∆. The end point of this superradiant tachyon condensation
process is a hairy black hole. The authors of [14] constructed these hairy black hole solutions
(working with a particular toy model Lagrangian of the form (1.1)) in a perturbation
expansion in their mass and charge. At leading order in this expansion, the hairy black
holes of [14] are well approximated by a non interacting mix of a small RNAdS black hole
and a weak static solitonic scalar condensate. In particular, it was shown in [14] that the
leading order thermodynamics of small hairy black holes could be reproduced simply by
modeling them as a non interacting mix of an RNAdS black hole and a regular charged
scalar soliton.
The results of [14] suggest that the density of states of certain field theories with a
gravity dual description might be dominated in certain regimes by previously unexplored
phases consisting of an approximately non interacting mix of a normal charged phase and
a Bose condensate. In order to make definitive statements about the actual behaviour of
N = 4 Yang Mills theory, however, it is necessary to perform the relevant calculations
in IIB supergravity on AdS5× S5rather than a simple toy model Lagrangian; this is the
subject of the current paper. As the IIB theory on AdS5×S5is a very special system, the
reader might anticipate that hairy black holes in this theory have some distinctive special
properties not shared by equivalent objects in the toy model studied in [14] at least at
generic values of parameters. As we will see below, this indeed turns out to be the case.
In order to avoid having to deal with the full complexity of IIB SUGRA, in this paper
we identify2work with a consistent truncation of gauged N = 8 supergravity (itself a
consistent truncation of IIB SUGRA on AdS5× S5). Of the complicated spectrum of
N = 8 supergravity, our truncation retains only a single charged scalar field φ, a gauge
2In unpublished work, S. Gubser, C. Herzog and S. Pufu have independently identified this consistent
truncation, and have numerically investigated hairy black branes in this set up. We thank C. Herzog for
informing us of this.
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field Aµand the metric.3The Lagrangian for our system is given by
S =N2
4π2
?
√g
?1
2(R + 12) −3
8FµνFµν−3
16
?
|Dµφ|2−∂µ(φφ∗)∂µ(φφ∗)
4(4 + φφ∗)
− 4φφ∗
??
Dµφ = ∂µφ − 2iAµφ
Fµν= ∂µAν− ∂νAµ
(1.2)
and is of the general structure (1.1) with e = 2 and ∆ = 2. As e = ∆, small very near
extremal black holes in this system lie at the precipice of the super radiant instability
discussed in [14]. The question of whether they fall over this precipice requires a detailed
calculation. We perform the necessary computation in this paper. Our results demonstrate
that very near extremal small black holes do suffer from the super radiant instability; we
proceed to construct the hairy black hole that constitutes the end point of this instability.
The calculations presented in this paper employ techniques that are similar to those
used in [14]. In particular we work in a perturbation expansion in the scalar amplitude
using very near extremal vacuum RNAdS black holes as the starting point of our expansion.
This expansion is justified by the smallness of the charge of our solutions. We implement
our perturbative procedure by matching solutions in a near (horizon) range, intermediate
range and far field range.This matching procedure is justified by the parametrically
large separation of scales between the horizon radius of the black holes we construct and
AdS curvature radius. We refer the reader to the introduction of [14] for a more detailed
explanation of physical motivation for this perturbative expansion and its formal structure.
Actual details of our calculations may be found in sections 3 and 4 below. In the rest of
this introduction we simply present our results and comment on their significance.
We study black holes of charge Q4(normalized so that each of the three complex Yang
Mills scalars has unit charge) and mass M (normalized to match the scaling dimensions of
dual operators). We find it convenient to deal with the ‘intensive’ mass and charge, m and
q, given by
m =M
N2
As in [14], in this paper we are primarily interested in small black holes for which q ≪ 1
and m ≪ 1. We now recall some facts about RNAdS black holes in this system. First, the
masses of such black holes obey the inequality
q =
Q
N2.
m ≥ mext(q) = 3q + 3q2− 6q3+ O(q4)
Black holes that saturate this inequality are extremal, regular and have finite entropy (see
subsection 6.1 for more details). The chemical potential µext(q) of extremal black holes is
given by
µext(q) = 1 + 2q − 6q2+ O(q3)
3Under the AdS/CFT correspondence, φ is dual to the operator TrX2+TrY2+TrZ2while Aµ is dual
to the conserved current JX¯
µ
+ JY¯Y
µ
+ JZ¯
the N = 4 Lagrangian.
4Q is the charge of the black hole solutions under each of the three diagonal U(1) Cartan’s of SO(6).
The consistent truncation of this paper forces these three U(1) charges to be equal.
XZ
µ . Here X, Y and Z denote the three complex chiral scalars in
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and approaches unity in the limit of small charge. Note also that extremal black holes lie
above the BPS bound
mBPS(q) = 3q
and in particular that mext(q) − mBPS(q) = 3q2+ O(q3)
We will now investigate potential superradiant instabilities (see the introduction of
[14] for an explanation of this term) of these black holes. Recall that a mode of charge e
and energy ω scatters off a black hole of chemical potential µ in a superradiant manner
whenever µe > ω. The various modes of the scalar field φ in (1.2) have energies ω = 2,3,...
and all carry charge e = 2. As the chemical potential of a small near extremal black hole
is approximately unity, it follows that only the ground state of φ (with ω = ∆ = 2) could
possibly scatter of a near extremal RNAdS black hole in a superradiant manner. This mode
barely satisfies the condition for superradiant scattering; as a consequence we will show
in this paper that small RNAdS black holes in (1.2) suffer from a superradiant instability
into this ground state mode only very near to extremality, i.e. when
m − mext(q) ≤ 6q3+ O(q4).
Unstable black holes eventually settle down into a new branch of stable hairy black hole
solutions. We have constructed these hairy black holes in a perturbative expansion in their
charge in Section 4; we now proceed to present a qualitative description of these solutions
and their thermodynamics.
Recall that the zero mode of the scalar field φ obeys the BPS bound (and so is super-
symmetric) at linear order in an expansion about global AdS5. It has been demonstrated
in [15, 16, 17] (and we reconfirm in section 3 below) that this linearized BPS solution
continues into a nonlinear BPS solution upon increasing its amplitude. In this paper we
will refer to this regular solution as the supersymmetric soliton. The hairy black holes of
this paper may approximately be thought of as a small, very near extremal RNAdS black
hole located in the center of one of these solitons. Although the soliton is supersymmetric,
the black hole at its center is not, and so hairy black holes are not BPS in general. These
solutions exist in the mass range
3q ≤ m ≤ 3q + 3q2+ O(q4) (1.3)
At the lower bound of this range (7.14) hairy black holes reduce to the supersymmetric
soliton. At the upper bound (which is also the instability curve for RNAdS black holes)
they reduce to RNAdS black holes.
In Fig. 1 below we have plotted the near extremal micro canonical ‘phase diagram’ for
our system. As is apparent from Fig. 1 our system undergoes a phase transition from an
RNAdS phase to a hairy black hole phase upon lowering the energy at fixed charge. This
phase transition occurs at the upper end of the range (1.3). Note that the phase diagram
of Fig. 1 has several similarities with the phase diagram depicted in Fig. 1 in [14]; however
there is also one important difference. The temperature of the hairy black holes of this
paper decreases with decreasing mass at fixed charge, and reaches the value zero at the
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q
m
Figure 1: Phase diagram as a function of charge q (x axis) and mass m (y axis) at small q. The
solid blue line at the bottom is the BPS bound along which the soliton lives. Hairy black holes
exist - and are the dominant phase - in the shaded region. RNAdS black holes are the only known
solutions (so in particular the dominant phase) in the unshaded region above the solid red curve at
the top. RNAdS black holes also exist (but are dynamically unstable and thermodynamically sub
dominant) between the solid red curve and the dashed curve. The solid red curve is described by
m = 3q +3q2+O(q4), while the blue curve by m = 3q. The dashed curve corresponds to extremal
RNAdS black holes and is given by m = 3q + 3q2− 6q3+ O(q4). The curves have not been drawn
to scale to make the diagram more readable.
BPS bound. In contrast the temperature of the hairy black holes of [14] increases with
decreasing mass (at fixed charge), approaching infinity in the vicinity of the lower bound.
As we have emphasized, the phase diagram depicted in Fig. 1 applies only in the limit
of small charges and masses. We would now like to inquire as to how this phase diagram
continues to large charges and masses. In order to address this question we first focus
on solitonic solutions. These solutions may be determined much more simply than the
generic hairy solution, as they obey the constraints of supersymmetry rather than simply
the equations of motion. It turns out that spherically symmetric supersymmetric solutions
are given as solutions to a single nonlinear, second order ordinary differential equation
[15, 16, 17]. The solitons constitute the unique one parameter set of regular solutions
to this equation. It is easy to continue our perturbative construction of the solitonic
solutions to large charges by solving this equation numerically: in fact this exercise was
already carried out in [16]. This numerical solution reveals that the solitonic branch of
solutions terminates at a finite charge qc= 0.2613. For q > qcthere are no supersymmetric
spherically symmetric solutions to the equations of motion of (1.2)5.
Recall that solitons constitute the lower edge of the space of hairy black hole solutions
of Fig. 1. The non existence of regular supersymmetric solutions for q > qcmight, at first,
suggest that at these charges the space of hairy black hole solutions terminates at a mass
greater than 3q (i.e. does not extend all the way down to supersymmetry). While this is
a logical possibility, we think it is likely that the truth lies elsewhere. As we will explain
5To be more precise, there are smooth solitonic solutions up to a slightly higher value qm = 0.2643, but
in a sense that will be explained in section 5, the point qc marks the boundary between regular solitonic
solutions and singular ones.
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in section 5 the solitonic branch of supersymmetric solutions terminates in a distinguished
singular solution S. It turns out that S is also the end point (or origin) of a one parameter
set of supersymmetric solutions that are all singular at the origin. The charges of these
solutions increase without bound (indeed we have found an explicit analytic solution for
the singular supersymmetric solution in the limit of arbitrarily large charge). The two one
parameter families of solutions, regular and singular ones, are joined at the special solution
S. We conjecture that smooth hairy black hole solutions exist in our system at every q
and for m > 3q. Upon taking the limit m → 3q, these smooth solutions reduce to the
smooth soliton for q < qcbut reduce to the singular supersymmetric solutions described
above when q > qc. In summary, we conjecture that the phase diagram of our system takes
the form displayed in Fig. 2 below.
?
q
m
Figure 2: Conjectured phase diagram as a function of charge q (x axis) and mass m (y axis) for
all values of q. The blue line at the bottom is the BPS bound along which the regular soliton lives
(straight part) and the singular supersymmetric solutions (wiggly part). The solid red curve at the
top marks the phase transition between the regime of RNAdS black holes (above the line) and that
of smooth hairy black holes (below). The black curve indicates a phase transition between different
types of hairy black holes. This curve originates from the BPS line at the black dot which is close
to the point q = qcand could end either in the bulk of the hairy black hole region or could extend
all the way up to the red line.
The distinguished solution S clearly plays a special role in the space of spherically
symmetric supersymmetric solutions. In subsection 5.2 we analytically determine the near
singularity behaviour of this solution. Viewing the 2nd order differential equation that
determines supersymmetric solutions as a dynamical system in the ‘time’ variable lnr, we
demonstrate that the solution S is a stable fixed point of this system, and analytically
compute the eigenvalues that characterize the approach to this fixed point. This eigen-
value has an imaginary part (which damps fluctuations) and a real part (that results in
oscillations). Solitonic - and singular - solutions in the neighborhood of S may be thought
of as configurations that that flow to S at large lnr. The oscillations6referred to above
result in the following phenomenon: the system develops a multiplicity of supersymmetric
6We are extremely grateful to M. Rangamani for suggesting that we look for this ‘self similar’ structure
in the space of solitons in the neighborhood of q = qc. The results reported in this paragraph are the
outcome of investigations that were spurred directly by this suggestion.
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solitonic (or singular) at charges q when q comes near enough to qc. The number of solu-
tions diverges as q → qc. The space of solitonic and singular supersymmetric solutions are
usefully plotted as a curve on a plane parametrized by the charge q and the expectation
value of the operator dual to the scalar φ. On this plane supersymmetric solitons spiral
into the point S, while the singular solutions spiral out of the same point (see Fig. 15);
the two spirals are non self intersecting7. We find this extremely intricate structure of
supersymmetric solutions quite fascinating, and feel that its implications for N = 4 Yang
Mills physics certainly merits further investigation.
In this paper we have so far only considered charged black holes with vanishing angular
momentum. Such solutions are spherically symmetric; i.e. they preserve the SO(4) =
SU(2) × SU(2) rotational isometry group. As all known supersymmetric black holes in
AdS5× S5possess angular momentum [18, 19, 20], it is of interest to generalize the study
of this paper to black holes with angular momentum. Let us first consider a spinning
Kerr RNAdS black hole that preserves only U(1) × U(1) ∈ SU(2) × SU(2). Perturbations
about such a solution are functions of an angle and a radius and are given by solutions
to partial rather than ordinary differential equations. However there exist RNAdS black
holes with self dual angular momentum. The angular momentum of such a black hole lie
entirely within one of the two SU(2) above, and so preserve a U(1)×SU(2) subgroup of the
rotation group. Perturbations around these solutions may be organized in representations
of SU(2) and so obey ordinary rather than partial differential equations.8Consequently,
the generalization of the hairy black hole solutions determined in this paper to solutions
with self dual spin appears to be a plausibly tractable project; which we however leave to
future work.
Even though we do not embark on a serious analysis of charged rotating black holes
in this paper, we do present a speculative appetizer for this problem. In more detail we
present a guess (or a prediction) for the leading order thermodynamics of these spinning
hairy solutions. Our guess is based on the observation that the thermodynamics of the hairy
solutions constructed in this paper can be reproduced, at leading order, by modeling the
hairy black hole as a non interacting mix of a RNAdS black hole and the soliton. In section 7
we simply assume that a similar model works for charged spinning black holes, and use this
model to compute the thermodynamics of a certain class of spinning hairy black holes. The
most interesting aspect of our results concern the BPS limit. Our non interacting model
predicts that extremal hairy black holes are BPS at every value of the angular momentum
and charge. This is in stark contrast with Kerr RNAdS black holes that are BPS only
on a co dimension one surface of the space of extremal black holes. According to our non
interacting model, BPS hairy black holes are a non interacting mix of Gutowski Reall black
holes [18, 19, 20] and supersymmetric solitons. Such a mix is thermally equlibrated at all
values of charge and angular momentum because of an important property of Gutowski
Reall black holes; their chemical potential is exactly unity9. We find this result both
intriguing and puzzling (see e.g. [21]). We emphasize that our prediction is based on
7We are very grateful to V. Hubeny for suggesting the possibility of a spiral structure for these curves.
8We thank A. Strominger for pointing this out to us.
9We are very grateful to S. Kim for explaining this to us.
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the non interacting superposition model, which may or may apply to actual black hole
solutions. We leave further investigation of this extremely interesting issue to future work.
This paper has been devoted largely to the study of small very near extremal charged
black holes in AdS5that are smeared over S5. As uncharged small smeared black holes are
well known to suffer from Gregory-Laflamme type instabilities [22], the reader may wonder
whether the black holes studied in this paper might suffer from similar instabilities. We
believe that this is not the case. Recall that the likely end point of a Gregory Laflamme
type instability is a small black hole of proper horizon radius rH localized on the S5. In
order that this black hole be near extremal, it has to zip around the S5at near the speed of
light, i.e. at v = 1−δ with δ ≪ 1. The AdS5charge of such a black hole is given by q ∝
while its energy above extremality of such a black hole is given by m − 3q ∝ r7
may now solve for rHand δ as a function of q and m−3q. In the near BPS limit of interest
to this paper m − 3q ∼ q2and we find r7
of such a localized black hole ∝ r8
of the black holes studied in this paper. As the the black holes studied in this paper have
higher entropy than S5localized black holes with the same charge, there seems no reason
to expect them to suffer from Gregory- Laflamme type instabilities11. Another pointer to
the same conclusion is the fact it was very important for the analysis of [22] that the black
holes they studied had negative specific heat. The charged black holes at the center of the
hairy solutions here all have positive specific heat12.
r7
H
√δ
H
√δ.10We
H∝ q
3
2×8
3
2 and
√δ ∝√q. It follows that the entropy
7, and so is smaller than the entropy, (∝ q
H∝ q
3
2),
We also note that the Gubser Mitra instability [23, 24] afflicts three equal charge black
holes only when the black holes in question have large enough charge. It follows that the
small black holes primarily studied in this paper do not suffer from Gubser Mitra type
instabilities13.
It is conceivable that the solutions presented in this paper might suffer from further
superradiant instabilities, once embedded in IIB theory on AdS5× S5. In order to see
why this might be the case, let us recall once again why the field φ - dual to the chiral
Yang Mills operator TrX2+TrY2+TrZ2condensed in the presence of very near extremal
charged RNAdS black hole. The reason is simply that the energy ∆ = 2 of this field is
equal to its charge e = 2. As a consequence the Boltzmann suppression factor, e−β(∆−e)
of this mode exceeds unity when µ > 1 causing this mode to Bose condense. However
exactly the same reasoning applies to, for instance, the field φndual to the chiral operator
10To see this let the sphere be given by equations |z1|2+ |z2|2+|z3|2= 1 where zi are the three complex
embedding coordinates. A black hole we study is located at |z1| = |z2| = |z3| =
(1−δ)
√3
momentum in each plane, q = r × p, is given by
energy of the black hole is mpγ.
11Were we interested in black holes with small q and m−3q ∼ O(q) then we would have found δ ∼ 1 and
r7
increase their entropy by condensing, they presumably do suffer from Gregory Laflamme type instabilities.
12We thank V. Hubeny and M. Rangamani for discussions on this point.
13We thank M. Rangamani for a discussion on this point.
1
√3, and moves with speed
H. Its angular
√
in each of the three orthogonal planes. Let the proper mass of the black hole be mp ∝ r7
1
√3×
mpγ(1−δ)
√3
=
mpγ(1−δ)
3
where γ =
1
1−(1−δ)2. The
H∼ q. The entropy of the localized black hole would then have been ∝ q
8
7 > q
3
2. As such black holes can
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Page 11
TrXn+ TrYn+ TrZnall of which have ∆ = e14. It seems likely that there exist other
hairy solutions in which some linear combination of φn(rather than simply φ2) condense
15. It is important to know whether any solution of this form has higher entropy than the
black holes with pure φ2condensate presented in this paper. If this is the case then the
hairy black holes of our paper would likely suffer from superradiant instabilities towards the
condensation to the entropically dominant black hole. On the other hand the black holes of
this paper, with φ2, the lightest chiral scalar operator that preserves all discrete symmetries
of the problem, as the only condensate, are quite special. It seems quite plausible to us that
the solution presented in this paper has the largest entropy of all the hairy solutions with
ρncondensates. If this is indeed the case then the hairy black hole solutions presented in
this paper constitute the thermodynamically dominant saddle point of N = 4 Yang Mills
very near to supersymmetry; and the entropy of N = 4 Yang Mills very near to the BPS
bound is given the formula (6.17) below.
To end this introduction we would like to emphasize that the black hole solutions of
this paper give a qualitatively different picture of the density of states of N = 4 Yang Mills
theory at finite charge compared to a picture suggested by RNAdS black holes. As we have
seen above, there exist no RNAdS black holes with masses between mBPS(q) and mext(q),
a fact had previously been taken to suggest that, for some mysterious reason, there are less
than O(N2) states in Yang Mills theory between mBPS(q) and mext(q). The new black
hole solutions of this paper establish, on the other hand, that N = 4 Yang Mills theory has
O(N2) states all the way down to the BPS bound at least at small charge, and plausibly at
all values of the charge (see Fig. 2).16The saddle point that governs near BPS behaviour is
a mix of a charged Bose condensate and a normal charged fluid. It would be fascinating to
find some (even qualitative) confirmation of this picture from a direct field theory analysis.
2. A Consistent Truncation and its Equations of Motion
2.1 A Consistent Truncation of Gauged Supergravity
N = 8 gauged supergravity constitutes a consistent truncation of IIB theory on AdS5×S5.
In addition to the metric, the bosonic spectrum of this theory consists of 42 scalar fields,
15 gauge fields and 12 two form fields. The scalars transform in the 20 +10c+1 +1 of
SO(6), the gauge fields transform in the 15 dimensional adjoint representation, while the
two form fields transform in the 6crepresentation of SO(6).
It has been shown [26] that N = 8 gauged supergravity admits a further consistent
truncation that retains only the scalars in the 20 and the vector fields in the 15 together
14This statement is true more generally of every operator in the N = 1 chiral ring of the theory.
15In the BPS limit any linear combination of ρns can condense and we have an infinite dimensional moduli
space of solutions (see [25, 17]). We expect the introduction of a black hole to lift this moduli space, to a
discrete set of solutions.
16This difference is starkest in the limit of large charge, i.e. in the Poincare Patch limit. The energy
density, ρE, of RNAdS black branes is bounded from below by cρ
, on the other hand predicts that the energy density of a charged black brane can be arbitrarily small at
any given value of the charge density.
4
3
Qwhere ρQ is the charge density. Fig. 2
– 10 –
Page 12
with the metric, setting all other fields to zero. The action for this consistent truncation
is given by [26]
S =
1
16πG5
−1
?
√g
?
Fi1i2Fi3i4Ai5i6− Fi1i2Ai3i4Ai5jAji6+2
R −1
4T−1
ij(DµTjk)T−1
kl(DµTli) −1
8T−1
ikT−1
jlFij
µν(Fkl)µν− V
48ǫi1···i6
?
5Ai1i2Ai3jAji4Ai5kAki6
?? (2.1)
where
V =1
2
?2TijTij− (Tii)2?
Fij=dAij+ Aik∧ Akj
DµTij=∂µTij+ Aik
π
2N2
µTkj+ Ajk
µTik
G5=
(2.2)
Here (i,j,··· ) denote the SO(6) vector indices and (µ,ν,···) are the space time indices.
Tij are symmetric unimodular (i.e. Tij is a matrix of unit determinant) SO(6) tensors.
Further N is the rank of the gauge group of the dual N = 4 Yang Mills theory, and we
work in units in which the AdS5with unit radius solves (2.1).
We will now describe a further consistent truncation of (2.1). For this purpose we find
it useful to move to a complex basis for the SO(6) vector indices that appear summed in
(2.1). Let (xj
j = 1,··· ,6) denote SO(6) Cartesian directions. We define the complex
coordinates
x2j−1+ ix2j= zj, x2j−1− ix2j= ¯ zj j = 1,··· ,3
We will now argue that the restriction
Tz1z1= Tz2z2= Tz3z3=φ
T¯ z1¯ z1= T¯ z2¯ z2= T¯ z3¯ z3=φ∗
4
4
Tz1¯ z1= Tz2¯ z2= Tz3¯ z3=
√4 + φφ∗
4
= 2iAµ
Az1¯ z1
µ
= Az2¯ z2
µ
= Az3¯ z3
µ
All Others =0
(2.3)
constitutes a consistent truncation of (2.1). To see this is the case note that the permu-
tations of labels (1,2,3), as also separate rotations by π in the z1, z2and z3planes, can
each be generated by separate SO(6) gauge transformations. It follows that these discrete
transformations are symmetries of (2.1). Now it is easy to convince oneself that (2.3) is
the most general field configuration of (2.1) that is invariant separately under each of these
four discrete symmetries. It follows that (2.3) is a consistent truncation of the system (2.1).
– 11 –
Page 13
The consistent truncation (2.3) is governed by the Lagrangian
S =
1
8πG5
=N2
4π2
?
√g
?1
2(R + 12) −3
2(R + 12) −3
8FµνFµν−3
8FµνFµν−3
16
?
?
|Dµφ|2−∂µ(φφ∗)∂µ(φφ∗)
4(4 + φφ∗)
|Dµφ|2−∂µ(φφ∗)∂µ(φφ∗)
4(4 + φφ∗)
− 4φφ∗
??
?
√g
?1
16
− 4φφ∗
??
Dµφ = ∂µφ − 2iAµφ
Fµν= ∂µAν− ∂νAµ
(2.4)
Note that φ has charge 2 and m2= −4. Under the AdS/CFT dictionary this field maps
to an operator of dimension ∆ = 2. Note also that the kinetic term of the gauge field has
the factor prefactor3
8rather than the (more usual)1
4as employed, for instance, in [14].17
2.2 Equations of Motion
We now list the equations of motion that follow from varying (2.4). We find the Einstein
equation
Rµν−1
2gµνR − 6gµν= −3
2TEM
µν
+3
8Tmat
µν
(2.5)
where
TEM
µν
=FµσFσν−1
=1
4gµνFασFσα
Tmat
µν
2[Dµφ(Dνφ)∗+ Dνφ(Dµφ)∗] −1
1
4(4 + φφ∗)
2gµν|Dσφ|2+ 2φφ∗gµν
−
?
∂µ(φφ∗)∂ν(φφ∗) −1
2gµν[∂σ(φφ∗)]2
?
(2.6)
the Maxwell equation
∇σFµσ=i
4[φ(Dµφ)∗− φ∗Dµφ] (2.7)
and the scalar equation
DµDµφ + φ
?[∂σ(φφ∗)]2
4(4 + φφ∗)2−
∇2(φφ∗)
2(4 + φφ∗)+ 4
?
= 0.
(2.8)
17Consequently gauge fields and chemical potentials in this paper and [14] are related by
Ahere=
?
2
3Athere,µhere=
?
2
3µthere
Note also that G5 was set to to unity in [14], while G5 =
π
2N2in this paper. It follows that
Mhere
N2
=2
πMthere,
Shere
N2
=2
πSthere,
Qhere
N2
=2
π
?
1
6Qthere
The factor of
3µheredQhere as is required on on physical grounds.
?
1
6above ensures that TdSthere = dMthere− µtheredQthere implies TdShere = dMhere−
– 12 –
Page 14
In this paper we study static spherically symmetric configurations of the system (2.4)
We adopt a Schwarzschild like gauge and set
ds2= −f(r)dt2+ g(r)dr2+ r2dΩ2
At= A(r)
3
Ar= Ai= 0
φ = φ∗= φ(r)
(2.9)
The four unknown functions f(r), g(r), A(r) and φ(r) are constrained by Einstein’s
equations, the Maxwell equations and the scalar equations. It is possible to demonstrate
that f,g,A,φ are solutions to the equations of motion if and only if
E1=g(r)
?
?
−3?A(r)2+ f(r)?φ(r)2
4f(r)
φ′(r)2
−3
r2− 6
?
+3
4
−
φ(r)2+ 4+A′(r)2+2f′(r)
r
f(r)
+4
r2
?
= 0
E2=g(r)2
f(r)
−g(r)φ′(r)2
?
−A(r)2φ(r)2−A′(r)2
?4
g(r)
?
+
?2rg′(r) − 4g(r)
g(r)2= 0
r2
?
φ(r)2+ 4+r2+ 8 + φ(r)2
?f′(r)
∇2φ(r) +
?
E3=2A(r)g(r)φ(r)2+ A′(r)
f(r)+g′(r)
1 +A(r)2
f(r)
g(r)−6
r
?
φ′(r)2
− 2A′′(r) = 0
E4=
?
1
4 + φ(r)2
??
−
g(r)[φ(r)2+ 4]2
?
φ(r) = 0
(2.10)
where
∇2φ(r) =
g(r)
??f′(r)
f(r)+6
r
?
φ′(r) + 2φ′′(r)
?
− g′(r)φ′(r)
2g(r)2
The equations E1and E2are derived from the rr and tt components of the Einstein
equations, E3is the t component of the Maxwell equation and E4is the equation of the
scalar field.
As in [14] the equations (2.10) contain only first derivatives of f and g, but depend on
derivatives up to the second order for φ and A. It follows that (2.10) admit a 6 parameter
set of solutions. One of these solutions is empty AdS5 space, given by f(r) = r2+ 1,
g(r) =
1+r2, A(r) = φ(r) = 0. We are interested in those solutions to (2.10) that asymptote
to AdS space time, i.e. solutions whose large r behaviour is given by
1
f(r) = r2+ 1 + O(1/r2)
g(r) =
1 + r2+ O(1/r6)
A(r) = O(1) + O(1/r2)
φ(r) = O(1/r2)
1
(2.11)
– 13 –
Page 15
As in [14] it turns out that these conditions effectively impose two conditions on the solu-
tions of (2.10), so that the system of equations admits a four parameter set of asymptot-
ically AdS solutions. We usually also be interested only in solutions that are regular (in
a suitable sense) in the interior. This requirement will usually cut down solution space to
distinct classes of two parameter space of solutions; the parameters may be thought of as
the mass and charge of the solutions.
2.3 RNAdS Black Holes
The AdS-Reissner-Nordstrom black holes constitute a very well known two parameter set
of solutions to the equations (2.10). These solutions are given by
f(r) =µ2R4
r4
−
?R2+ µ2+ 1?R2
?r2− R2??r4+ r2(R2+ 1) − µ2R2?
f(r)
?
φ(r) = 0
r2
+ r2+ 1
=1
r4
g(r) =
1
A(r) =µ1 −R2
r2
?
(2.12)
where µ is the chemical potential of the RNAdS black hole. The function V (r) in (2.12)
vanishes at r = R and consequently this solution has a horizon at r = R. In fact, it can
be shown that R is the outer event horizon provided
µ2≤ (1 + 2R2). (2.13)
As explained in [14] and in the introduction, (2.12) is unstable to superradiant decay
provided in the presence of field of charge e and minimum energy ∆ provided eµ > ∆. Now
our field φ has ∆ = 2 and e = 2. Moreover, in the limit R → 0, RNAdS black holes have
µ ≤ 1 (this inequality is saturated at extremality). It follows that small extremal black
holes lie at the edge of instability, as mentioned in the introduction. We show below that
very near extremal RNAdS black holes do in fact suffer from super radiant instabilities.
3. The Supersymmetric Soliton in Perturbation Theory
In this section we will construct the analogue of the ground state soliton in [14]. The
new feature in here is that the soliton turns out to be supersymmetric (this is obvious at
linearized order).
In this section we generate the solitonic solution in perturbation theory. We use only
the equations of motion without imposing the constraints of supersymmetry, but check
that our final solution is supersymmetric (by verifying the BPS bound order by order in
perturbation theory). This method has the advantage that it generalizes in a straightfor-
ward manner to the construction of non supersymmetric hairy black holes in subsequent
sections.
– 14 –
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