Page 1
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
Page 2
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
– 1 –
Page 3
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 –
Page 4
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.
– 3 –
Page 5
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
– 4 –
Page 6
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
– 5 –
Page 7
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.
– 6 –
Page 8
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.
– 7 –
Page 9
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.
– 8 –
Page 10
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
– 9 –
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 –
Page 16
In Section 5 we will revisit this solitonic solution; we will rederive it by imposing the
constraints of supersymmetry from the start. That method has the advantage that it
permits a relatively simple extrapolation of supersymmetric solutions to large charge.
3.1 Setting up the perturbative expansion
We now turn to the description of our perturbative construction. To initiate the perturba-
tive construction of the supersymmetric soliton we set
f(r) = 1 + r2+
?
∞
?
ǫ2nA2n(r)
n
ǫ2nf2n(r)
g(r) =
1
1 + r2+
n=1
ǫ2ng2n(r)
A(r) = 1 +
∞
?
n=1
φ(r) =
ǫ
1 + r2+
∞
?
n=1
φ2n+1(r)ǫ2n+1
(3.1)
and plug these expansions into (2.10). We then expand out and solve these equations order
by order in ǫ. All equations are automatically solved up to O(ǫ). At order ǫ2nthe last
equation in (2.10) is trivial while the first three take the form
d
dr
?
r2(1 + r2)2g2n(r)
?
?
?
= P(g)
2n(r)
d
dr
?
?f2n(r)
r3dA2n(r)
1 + r2
=2(1 + 2r2)
r
g2n(r) + P(f)
2n(r)
d
dr dr
= P(A)
2n(r).
(3.2)
On the other hand, at order ǫ2n+1the first three equations in (2.10) is trivial while the last
equation reduces to
d
dr
?
r3
1 + r2
d
dr
?(1 + r2)φ2n+1(r)??
= P(φ)
2n+1(r)(3.3)
Here the source terms P(g)
by the solution to lower orders in perturbation theory, and so should be thought of as
known functions, in terms of which we wish to determine the unknowns f2n, g2n, A2nand
φ2n+1.
The equations (3.2) are all easily integrated. It also turns out that all the integration
constants in these equations are uniquely determined by the requirements of regularity,
2n(r), P(f)
2n(r), P(A)
2n(r) and P(φ)
2n+1(r) are completely determined
– 15 –
Page 17
normalisability and our definition of ǫ, exactly as in [14]. The solution is given by
g2n(r) =
1
r2(1 + r2)2
??r
?∞
??s
??∞
0
P(g)
2n(s)ds
?
f2n(r) = − (1 + r2)
r
?2(1 + 2s2)
P(A)
s
g2n(s) + P(f)
2n(s)
?
ds
A2n(r) = −
?∞
r
ds
s3
0
2n(s′)ds′
?
???s
φ2n+1(r) = −
1
1 + r2
r
ds
?1 + s2
s3
0
P(φ)
2n+1(s′)ds′
??
(3.4)
3.2 The Soliton up to O(ǫ9)
The perturbative procedure outlined in this subsection is very easily implemented to arbi-
trary order in perturbation theory. In fact, by automating the procedure described above,
we have implemented this perturbative series to 17th order in a Mathematica programme.
In the rest of this subsection we content ourselves with a presentation of our results to
O(ǫ9).
g2(r) = 0
f2(r) = −
1
4(1 + r2)
1
8(1 + r2)
1
8(1 + r2)3
A2(r) = −
φ3(r) =
(3.5)
g4(r) =
r4
192(1 + r2)5
f4(r) = −
A4(r) = −r4+ 3r2+ 3
384(r2+ 1)3
φ5(r) =6r4+ 4r2+ 55
2304(r2+ 1)5
r4
192(1 + r2)5
(3.6)
g6(r) =r4?4r4+ 15r2+ 20?
f6(r) = −12r8+ 45r6+ 60r4+ 20r2+ 5
23040(r2+ 1)5
A6(r) = −6r8+ 30r6+ 60r4+ 55r2+ 25
23040(r2+ 1)5
φ7(r) =120r8+ 460r6+ 1095r4+ 558r2+ 2368
460800(r2+ 1)7
7680(r2+ 1)7
(3.7)
– 16 –
Page 18
g8(r) =r4?169r8+ 1024r6+ 2640r4+ 3320r2+ 2180?
?5?169r8+ 1024r6+ 2580r4+ 3344r2+ 2288?r2+ 3096?r2+ 516
?5??169?r4+ 7r2+ 21?r2+ 5819?r2+ 5543?r2+ 14721?r2+ 4191
?5?1014r8+ 6124r6+ 18257r4+ 30484r2+ 36676?r2+ 75784?r2+ 155759
2211840(r2+ 1)9
f8(r) = −
11059200(r2+ 1)7
A8(r) = −
22118400(r2+ 1)7
φ9(r) =
132710400(r2+ 1)9
(3.8)
The soliton obeys the BPS relation m = 3q to the order to which we have carried out
our computation (we present more details of the thermodynamics in Section 6).
4. The Hairy Black Hole in Perturbation Theory
4.1 Basic Perturbative strategy
We will now present our perturbative construction of hairy black hole solutions. In order
to set up the perturbative expansion we expand the metric gauge field and the scalar fields
as
f(r) =
∞
?
∞
?
∞
?
∞
?
n=0
ǫ2nf2n(r)
g(r) =
n=0
ǫ2ng2n(r)
A(r) =
n=0
ǫ2nA2n(r)
φ(r) =
n=0
ǫ2n+1φ2n+1(r)
(4.1)
where the unperturbed solution is taken to be the RNAdS black hole
f0(r,R) = V (r), g0(r,R) =
1
V (r)
A0(r,R) = µ0(1 −R2
r2)
?
V (r) = 1 + r2
1 −R2µ2
0+ R2+ R4
r4
+R4µ2
0
r6
?
(4.2)
– 17 –
Page 19
The chemical potential of our final solution will be given by an expression of the form
µ = µ(ǫ,R) =
?
n=0
ǫ2nµ2n(R)
µ2n(R) =
∞
?
k=0
µ(2n,2k)R2k
µ(0,0)= 1
(4.3)
Note that, at the leading order in the perturbative expansion, µ = 1.
Our basic strategy is to plug the expansion (4.1) into the equations of motion and
then to recursively solve the later in a power series in ǫ. We expand our equations in a
power series in ǫ. At each order in ǫ we have a set of linear differential equations (see below
for the explicit form of the equations), which we solve subject to the requirements of the
normalisability of φ(r) and f(r) at infinity together with the regularity of φ(r) and the
metric at the horizon. These four physical requirements turn out to automatically imply
that A(r = R) = 0 i.e. the gauge field vanishes at the horizon, as we would expect of a
stationary solution. These four physical requirements determine 4 of the six integration
constants in the differential equation, yielding a two parameter set of solutions. We fix
the remaining two integration constants by adopting the following conventions to label
our solutions: we require that φ(r) fall off at infinity like
the horizon area of our solution is 2π2R3(definition of R). This procedure completely
determines our solution as a function of R and ǫ. We can then read of the value of µ in
(4.3) on our solution from the value of the gauge field at infinity.
ǫ
r2 (definition of ǫ) and that
As in [14], the linear differential equations that arise in perturbation theory are difficult
to solve exactly, but are easily solved in a power series expansion in R, by matching near
field, intermediate field and far field solutions. At every order in ǫ we thus have a solution
as an expansion in R. Our final solutions are, then presented in a double power series
expansion in ǫ and R.
4.2 Perturbation Theory at O(ǫ)
In this section we present a detailed description of the implementation of our perturba-
tive expansion at O(ǫ). The procedure described in this subsection applies, with minor
modifications, to the perturbative construction at O(ǫ2m+1) for all m.
Of course all equations are automatically obeyed at O(ǫ0). The only nontrivial equation
at O(ǫ) is D2φ = 0 where Dµ= ∇µ−2iAµis the linearized gauge covariantised derivative
about the background (4.2). We will now solve this equation subject to the constraints of
normalisability at infinity, regularity at the horizon, and the requirement that
φ(r) ∼
ǫ
r2+ O(1/r4)
at large r.
– 18 –
Page 20
4.2.1 Far Field Region (r ≫ R)
Let us first focus on the region r ≫ R. In this region the black hole (4.2)
f0(r) =µ2
r4
=1
r4
1
f(r)
?
∞
?
µ(0,0)=1
0R4
−
?R2+ µ2+ 1?R2
?r2− R2??r4+ r2(R2+ 1) − µ2
r2
+ r2+ 1
0R2?
g0(r) =
A0(r) =µ0
1 −R2
r2
?
µ0=
k=0
R2kµ(0,2k)
(4.4)
is a small perturbation about global AdS space. For this reason we expand
φout
1(r) =
∞
?
k=0
R2kφout
(1,2k)(r), (4.5)
where the superscript out emphasises that this expansion is good at large r. In the limit
R → 0, (4.4) reduces to global AdS space time with At = 1. A stationary linearised
fluctuation about this background is gauge equivalent to a linearised fluctuation with time
dependence e−itabout global AdS space with At= 0 (Atis the temporal component of the
gauge field). The required solution is simply the ground state excitation of an m2= −4
minimally coupled scalar field about global AdS and is given by
φout
(1,0)(r) =
1
1 + r2
(4.6)
The overall normalisation of the mode is set by our definition of ǫ which implies
φout
(1,0)(r) =1
r2+ O(1/r4).
We now plug (4.5) into the equations of motion D2φ = 0 and expand to O(R2) to solve
for φout
(1,2). Here D2is the gauge covariant Laplacian about the background (4.4). Now
(D2)out= (D2
0)out+ R2(D2
2)out+ ...
where (D2
ground gauge field At= 1. It follows that, at O(R2),
0)outis the gauge covariant Laplacian about global AdS space time with back-
(D2
0)outφout
(1,2)= −(D2
2)outφout
(1,0)= −(D2
2)out
?
1
1 + r2
?
This equation is easily integrated and we find
φout
(1,2)(r) = −
?
1
1 + r2
??µ(0,2)
r2
− 2?µ(0,2)− 2?log(r) +?µ(0,2)− 2?log?r2+ 1?+
2
r2+ 1
?
(4.7)
– 19 –
Page 21
We could iterate this process to generate φout
tions we need to solve, at order O(R2k), takes the form
d
dr1 + r2
(1,2k)upto any desired order k. The equa-
?
r3
d
dr
?
(1 + r2)φout
(1,2k)(r)
??
= Pout
(1,2k)(r)(4.8)
where Pout
sion at lower orders in perturbation theory.
As in (4.7), it turns out that the expressions φout
0. In fact it may be shown that the most singular piece of φout
logarithmic corrections. In other words the expansion of φoutin powers of R2is really an
expansion inR2
r2 (upto log corrections) and breaks down at r ∼ R.
To end this subsection we summarize our results to O(R2). We have
φout
1
=
1 + r2+
?
+ O(R4
(1,2k)(r) is a source function, whose form is determined by the results of the expan-
(1,2k)are increasingly singular as r →
(1,2k)scales like
1
r2k, upto
1
+ R2
−
?
1
1 + r2
??µ(0,2)
r2
− 2?µ(0,2)− 2?log(r) +?µ(0,2)− 2?log?r2+ 1?+
2
r2+ 1
??
r4)
(4.9)
Expanding φout
1(r) in a Taylor series about r = 0 we find
φout
1 (r) =?1 − r2+ O(r4)?
+ R2?µ(0,2)
+ O(R4
r2
+ (µ(0,2)− 2)(2log(r) + 1) + O(r2)
?
r4)(1 + O(r2)) + ...
(4.10)
Note that this result depends on the as yet unknown parameter µ(0,2). This quantity
will be determined below by matching with the intermediate field solution of the next
subsection.
4.2.2 Intermediate Field Region r ≪ 1 and (r − R) ≫ R3
Let us now turn to intermediate region R3≪ r−R ≪ 1. Over these length scales the small
black hole is far from a small perturbation about AdS5space. Instead the simplification
in this region stems first from the fact that we focus on radial distances of order R (
r ∼ R ≪ 1). Over these small length scales the background gauge field, which is of order
unity, is negligible compared to the mass scale set by the horizon radius
A second simplification results from the fact that we insist that (r − R) ≫ R3, i.e.
we do not let our length scales become too small. At these distances the back hole that
we perturb around are effectively extremal (rather than slightly non extremal) at leading
order. Moreover the black hole may also be thought of (at leading order) as a small black
hole in flat rather than global AdS space.18
1
R.
18As we will see below, deviations of the black hole from extremality (and deviations of the form of its
metric from the metric of a flat space black hole) are crucial to dynamics at r − R ∼ R3, but are small
perturbations on dynamics when (r − R) ≫ R3.
– 20 –
Page 22
In this region it is convenient to work in a rescaled radial coordinate y =
rescaled time coordinate τ =
R. Note that the near field region consists of space time
points with y of order unity (but not too near to unity). Points with y of order
larger) and y − 1 of order O(R2) (or smaller) are excluded from the considerations of this
subsection.
The metric and the gauge field of the background black hole take the form
r
Rand a
t
1
R(or
ds2=R2
?
−V (y)dτ2+
1 −1
y2
?
∞
?
µ(0,0)=1
dy2
V (y)+ y2dΩ2
1 −µ2
1 −1
y2
3
?
?
V (y) =
?
??
0
y2+ R2(1 + y2)
Aτ(y) =Rµ0
?
µ0=
k=0
R2kµ(0,2k)
(4.11)
As in the previous subsection we expand
φmid
1
(y) =
∞
?
k=0
R2kφmid
(1,2k)(y)
(4.12)
To determine the unknown functions in this expansion, we must solve the equation
D2φmid= 0, where D2is the gauge covariant Laplacian about the background (4.11). Our
solutions must match with the far field expansion of the previous subsection, and the near
field expansion of the next subsection, but are subject to no intrinsic boundary regularity
requirements.
At O(R2k) our equations take the form
1
y3
dydy
d
?
y3V0(y)d
?
φmid
(1,2k)(y) = Pmid
(1,2k)(y) (4.13)
where
V0(y) =
?
1 −1
y2
?2
and Pmid
Ignoring the requirements of matching, for a moment, the solution to this equation is
determined only upto two integration constants at every order. It turns out that φmid
grows like y2k(upto possible logarithmic corrections) at large y and grows like
approaches unity. It follows that the expansion (4.13) is good only when
(1,2k)(y) is a source term determined (recursively) by the perturbative procedure.
(1,2k)(y)
1
(y−1)kas y
R2≪ (y − 1) ≪1
R
We now work out the explicit solutions at low orders. Pmid
solution for φmid
(1,0)(y) is particularly simple, and takes the form
(1,0)(y) = 0 vanishes, so the
φmid
(1,0)(y) = c1+
c2
y2− 1
– 21 –
Page 23
c1and c2are the two constants. It is easy to check that the matching of φmid
φout
(1,0)(r) sets c1= 1. It follows on general grounds that matching with the (as yet unde-
termined) near field solution forces c2to vanish. This is because, were c2to be nonzero,
it would match onto a near field solution of order O?1
R → 0 limit.
We can now iterate the procedure of this subsection to solve to order in R2in the
intermediate field region. We find
(1,0)(y) with
R2
?in the near field region (see the
next subsection for details), violating the requirement that that our solution has a smooth
φmid
(1,0)(y) =1
φmid
(1,2)(y) = − y2− 2log(y2− 1) + c3+
c4
y2− 1
(4.14)
so that
Here c3and c4are the two integration constants. c3may immediately be determined
by matching with the far field solution; it turns out that this procedure also determines
µ(0,2)= 0, the chemical potential that was left undetermined in the previous subsection.
In order to perform this matching we expand φmid
1
(y) about large y
φmid
1
(y) = 1 + R2
?
−y2+ c3− 4log(y) + O
?1
y2
??
+ O(R4y4) (4.15)
The strategy is now to substitute y =r
course one should only compare those terms that are reliable in both expansions. Terms of
order R2mr2nare reliably computed from (4.15) only when m + n ≤ 1. Terms of the same
form are reliably computed from (4.10) only when m ≤ 1. Consequently, the only terms
that one may reliably compare are those of the form O(R0r0), O(R0r2), O(R2r0) together
with logarithmic corrections. The difference between the sum of the corresponding terms
(in (4.15) and (4.10)) is given by
Rin (4.15) and then to compare with (4.10). Of
Difference = R2?µ(0,2)
and vanishes provided µ(0,2)= 0 and c3= −2(1 + 2logR) so that
r2
+ 2µ(0,2)log(r) + µ(0,2)− 2 − 4log(R) − c3
?
φmid
1
(y) = 1 + R2
?
−y2− 2log(y2− 1) − 2(1 + 2logR) +
c4
y2− 1
?
+ O(R4) (4.16)
c4will be determined below by matching to the near field region. To facilitate this matching
in the next subsection, we present the expansion of φmid
1
(y) expanded around y = 1.
φmid
1
(y) = 1 + R2
?
c4
2(y − 1)−
?c4
4+ 3 + 2log2 + 4logR
?
− 2log(y − 1) + O(y − 1)
?
+O(
R4
(y − 1)2)
(4.17)
– 22 –
Page 24
4.2.3 Near Field Region (r − R) ≪ R or (y − 1) ≪ 1
In this subsection we will determine the scalar field in the near field region r − R ≪ R.
More particularly, we will work in terms of a further rescaled radial coordinate z =y−1
Note that the black hole horizon occurs at z = 0. Note points at finite z are located at
r − R ∼ R3or y − 1 ∼ R2. It is also convenient to work with the new time coordinate
T = Rt = R2τ As in the previous subsection, the background gauge field makes a small
direct contribution to dynamics in this region. However deviation of the black hole metric
from extremality (and the difference between an AdS and flat space black hole metric) are
all important in this region, and have to be dealt with exactly rather than perturbatively.
In the new coordinates, the metric and gauge field take the form
R2.
ds2
R2=V (z)dT2+
AT(y) =µ0
R
dz2
V (z)+ (1 + R2z)2dΩ2
1
(1 + R2z)2
3
?
1 −
?
= 2µ0Rz2+ O(R3z4)
µ0=
∞
?
k=0
R2kµ(0,2k)
µ(0,0)=1, µ(0,2)= 0
V (z) =1
R4×
= = 4z(1 + z) + O(R2)
?
1 +
µ2
0
(R2z + 1)4−µ2
0+ 1 + R2
(R2z + 1)2+ R2?1 + R2z?2?
(4.18)
As in previous subsections we expand the field φ(z) as
φin
1(z) =
∞
?
k=0
R2kφin
(1,2k)(z)
(4.19)
and plug this expansion into the equations of motion. At O(R2k) the equations take the
form
d
dz dz
?
4z(1 + z)d
?
φin
(1,2k)(z) = Pin
(1,2k)(z) (4.20)
where, as usual, Pin
perturbation theory at lower orders. We solve the equation (4.20) subject to the require-
ment of regularity at z = 0. It is possible to argue that the solution to φin
like zk−1(upto logarithmic corrections) at large z.
At lowest order (k = 0) Pin
(1,0)(y) = 0 vanishes, and the unique regular solution for
φin
(1,0)(y) is the constant. Matching determines the value of the constant to be unity.
At next order, (i.e. O(R2)) the solution - after imposing the requirement of regularity
- is given by
(1,2k)(z) is a source term whose form is determined from the results of
(1,2k)(z) behaves
φin
(1,2)(z) = 1
(1,2)(z) = α −1
φin
2log2(z + 1) − 2log(z + 1) − Li2(−z)
(4.21)
– 23 –
Page 25
where α is the constant which we will now determine by matching with the intermediate
field solution. Expanding φin
1(z) around z = ∞ we find
?
We now substitute z =
R2 in (4.22) and then compare with (4.17). We find a perfect
match provided α and c4are chosen to be the following
φin(z) = 1 + R2
α +π2
6
− 2logz + O
?1
z
??
(4.22)
y−1
c4= 0 and α = −
?π2
6
+ 3 + 2log 2 + 8logR
?
4.3 Perturbation theory at O(ǫ2)
We now briefly outline the procedure used to evaluate the solution at O(ǫ2). We proceed
in close imitation to the previous subsection. The main difference is that at this (and all
even orders) in the ǫ expansion, perturbation theory serves to determine the corrections to
the functions f, g and A rather than the function φ. The procedure described here applies,
with minor modifications, to the perturbative construction at O(ǫ2m) for all m.
4.3.1 Far Field Region, r ≫ R
When r ≫ R we expand
∞
?
gout
2 (r) =
?
Aout
2(r) =
?
where
fout
2
(r) =
k=0
∞
R2kfout
(2,2k)(r)
k=0
∞
R2kgout
(2,2k)(r)
k=0
R2kAout
(2,2k)(r)
(4.23)
fout
0
(r) =V (r), gout
0 (r) =
1
V (r), Aout
0(r) = µ0
?
1 −R2
r2
?
µ0=
∞
?
k=0
R2kµ(0,2k)
µ(0,0)=1,µ(0,2)= 0
(4.24)
As in the previous subsection, we plug this expansion into the equations of motion and
solve the resultant equations recursively. The equations take the form
d
dr
?
r2(1 + r2)2gout
(2,2k)(r)
?
= Source
d
dr
?fout
(2,2k)(r)
1 + r2
?
−2(1 + 2r2)
r
gout
(2,2k)(r) = Source
d
dr
?
r3dAout
(2,2k)(r)
dr
?
= Source.
(4.25)
– 24 –
Page 26
and may be thought of as the equations governing sourced linearized fluctuations about
empty global AdS space with At= 1.
The equations (4.25) are easily solved by integration. One of the integration constant
in the first equation is fixed by the requirement that fout
The remaining three integration constants (one in the first equation and two in the last)
will be fixed by matching with the intermediate field solution below.
The constraints of matching are particularly simple at O(R0); they require that the
solutions for gout
(2,0)are all regular at r = 0. This is because a far field
solution of the form
rkwould match onto an intermediate solution of the form
this contradicts our basic assumption that our solutions have a smooth R → 0 limit.
It follows that at O(R0)19all our functions obey the same equations - and boundary
conditions - for the 2nd order fluctuations about the supersymmetric soliton and we obtain
the same (unique) solution
(2,2k)(r) is normalizable (see (2.11)).
(2,0), fout
(2,0)and Aout
1
1
ykRk. But
gout
(2,0)(r) = 0
fout
(2,0)(r) = −
1
4(1 + r2)
1
8(1 + r2)
Aout
(2,0)(r) = −
(4.26)
At order O(R2) we find
gout
(2,2)(r) = −
1
4r2(r2+ 1)3−
3 + 5r2
4r2(1 + r2)2+
1 + 2r2
r2(1 + r2)2+
k
r2(1 + r2)2
fout
(2,2)(r) =
1
1 + r2log
2
1 + r2log
?1 + r2
?1 + r2
r2
?
+k
r2
Aout
(2,2)(r) =
r2
?
+ h1+h2
r2
(4.27)
Here k, h1 and h2 are the three undetermined constants, which will be determined by
matching with the intermediate field solution. To facilitate this determination below we
end this subsection by presenting an expansion of (4.27) about r = 0
gout
2 (r) = R2
?
−k +1
4
r2
+ O(r0)
?
?
?4h2+ 1
+ O(R4)
fout
2
(r) =
?
?
−1
4+ O(r2)
?
?
+ R2
k +3
r2
4
+ O(r0)
?
+ O(R4)
Aout
2 (r) =
−1
8+ O(r2)+ R2
4r2
+ O(r0)
?
+ O(R4)
(4.28)
4.3.2 Intermediate field region, r ≪ 1 and (r − R) ≫ R3
As in the previous section, we find it convenient to work with the variables y =
τ =
Rin the intermediate field region. Recall also that, in these coordinates, the leading
r
Rand
t
19At higher orders the same reasoning does not forbid singularities, but determines them in terms of the
known intermediate field behaviour at one order lower.
– 25 –
Page 27
order metric has an overall factor of R2. The metric variables that obey simple equations
have this factor of R2stripped from them. For that reason we define Here
fmid(y) =gττ
R2= gtt and gmid(y) =gyy
R2= grr
(here gµν are metric components. In a similar fashion we define Amid=Aτ
these definitions we expand
R= At. With
fmid
2
(y) =
∞
?
∞
?
∞
?
k=0
R2kfmid
(2,2k)(y)
gmid
2
(y) =
k=0
R2kgmid
(2,2k)(y)
Amid
2
(y) =
k=0
R2kAmid
(2,2k)(y)
(4.29)
where
fmid
0
(y) =V (y), gmid
0
(y) =
1
V (y), Amid
0
(y) = µ0
?
1 −1
y2
?
µ0=
∞
?
k=0
R2kµ(0,2k)
µ(0,0)=1,µ(0,2)= 0
(4.30)
The equations are slightly simpler when rewritten in terms of a new function
K(2,2k)(y) = V0(y)gmid
(2,2k)(y) +
fmid
(2,2k)(y)
V0(y)
where
V0(y) =
?
1 −1
y2
?2
In terms of this function the final set of equations take the form
dK(2,2k)(y)
dy
= Source
d
dy
?
y3dAmid
(2,2k)(y)
dy
?
−
?dK(2,2k)(y)
?dAmid
dy
dy
?
?
= Source
d
dy
?
y2fmid
(2,2k)(y)
?
− 2yK(2,2k)(y) + 2
(2,2k)(y)
= Source
(4.31)
These equations are all easily solved by integration, upto four undetermined integration
constants (one each from the first and third equation, and two for the second). It will
turn out that two of these constants are determined by matching with the far field solution
while the other two are determined by matching with the near field solution. As in the
– 26 –
Page 28
previous section we will find that kthorder solutions scale like y2kat large y, but scale like
1
(z−1)kat small z.
The solution at leading order, R0, is given by
fmid
(2,0)(y) =α1
?
1 −1
α1y4
(y2− 1)3−
y2
?
−2α2
(y2− 1)4+(2α2− α4)y6
y2+2α3
2α3y4
y4+α4
y2
gmid
(2,0)(y) = −
Amid
(y2− 1)4
(2,0)(y) =α2−α3
y2
(4.32)
Here α1, α2, α3 and α4are the four integration constants to be determined by matching.
Expanding (4.32) around y = ∞ one finds
(y) =α1+(α4− α1− 2α2)
y2
(y) =(2α2− α1− α4)
y2
(y) =α2−α3
fmid
2
+ O
?1
?1
?
y4
?
+ O(R2)
gmid
2
+ O
y4
+ O(R2)
Amid
2
y2+ O(R2)
(4.33)
As usual, we substitute as y =r
determines
Rand then match relevant terms of (4.33) and (4.28). This
α1= −1
α2= −1
4
8
?
h2= −
k =α4−1
α3+1
4
?
4
(4.34)
To facilitate matching with the near field region in the next subsection we expand fmid
about y = 1
2
(y), gmid
2
(y) and Amid
2
(y)
fmid
2
(y) =
??1
+ O?(y − 1)3??
?
+ O
y − 1
(y) = −
4+ 2α3+ α4
?
+ O(R2)
−1 + 8α3+ 8α4
32(y − 1)3
+ O(R2)
− (1 + 8α3+ 2α4)(y − 1) +
?3
2+ 20α3+ 3α4
?
(y − 1)2
gmid
2
(y) =
−1 + 8α3+ 4α4
64(y − 1)4
?
?1
−1 + 8α3+ 44α4
128(y − 1)2
1
??
?
Amid
2
8+ α3
+ 2α3(y − 1) + O?(y − 1)2?+ O(R2)
(4.35)
– 27 –
Page 29
4.3.3 Near Field Region (r − R) ≪ R or (y − 1) ≪ 1
As in the previous section we work with the shifted and rescaled radial coordinate z =y−1
In this coordinate the black hole horizon is at z = 0. As we have seen in the previous section,
gTT and grrhave an overall factor of R2even at leading order. For this reason the natural
dynamical variables in the problem are
R2.
gTT
R2=gtt
R4
and
gzz
R2= R4grr
(here gµνis the metric). For easy of matching with the intermediate field solution however,
we will continue to use the notation
fin= gtt= R4×gTT
R2
gin= grr=
1
R4×gzz
R2
Ain= At= R2×AT
R
And so our perturbative expansion takes the form (note the lower limits of the sums)
fin
2(z) =
∞
?
∞
?
∞
?
k=2
R2kfin
(2,2k)(z)
gin
2(z) =
k=−2
R2kgin
(2,2k)(z)
Ain
2(z) =
k=1
R2kAin
(2,2k)(z)
(4.36)
We now come to an important subtlety of our expansion procedure. First recall that
the radial coordinate r employed in this paper has geometrical significance; it parametrizes
the volume of the S3at that point. For this reason reparametrizations of r do not form
a symmetry of the equations in this paper, in general. At leading order in the near field
region, however, the metric metric takes the form
ds2
R2= −4z(1 + z)dT2+
0
dz2
4z(1 + z)+ dΩ2
3
(4.37)
Note in particular that the size of three sphere (at leading order) is a constant independent
of z. For this reason the leading order metric equations in the near field region admit
a whole functions worth (instead of 4 numbers worth) of solutions, parametrized by any
O(ǫ2R0) redefinition of z coordinate. So without even doing any calculations, we have
deduced that one linear combination of the three functions is undetermined at leading
order.
Now let us move to the next order, O(R2). As the homogeneous part of the equations
are same at every order, the same linear combination of second order fluctuations disappears
from (i.e. is undetermined by) the second order equations. However the 0 order ‘gauge
– 28 –
Page 30
transformation’ is now not a symmetry of the O(R2) equations (because, at this order, we
see the fact that the size of the sphere is not really constant). So the zero order ‘gauge
transformation function’ shows up in the second order equations. As this term comes with
an explicit R2(without this factor the equations cannot distinguish it from pure gauge)
it cannot multiply any of the 2nd order unknowns, and so appears as a genuine unknown
all by itself. The net upshot of all this is that at every order other than the leading, we
actually do have as many equations as variables. The variables, however, consist of two
unknown functions at that order coupled with the one unknown ‘gauge transformation’ at
the previous order!
We will now say all of this more precisely. Our equations can be simplified by perform-
ing the following redefinition of the functions (W is essentially the ‘gauge transformation’)
fin
(2,2k)(z) =d
dz[4z(1 + z)]W(2,2k)(z)
(2,2k)(z) =ζ(2,2k)(z) +d
gin
dz
?
1
4z(1 + z)
?
W(2,2k)(z) +
?
1
2z(1 + z)
?
d
dz
?W(2,2k)(z)?
Ain
(2,2k)(z) =χ(2,2k)(z) + 2W(2,2k)(z)
(4.38)
In terms of these functions the equations at order R2ktake the form
d
dz
?z(1 + z)(1 + 2z)2ζ(2,2k)(z)?= Source
?χ(2,2k)(z)?− 4z(1 + z)ζ(2,2k)(z) = Source
d2
dz2
d
dz
?W(2,2k−2)(z)?= Source
(4.39)
As we anticipated above, W(2,2k)(z) does not appear in the homogeneous equations at
O(R2k) as at this order it is pure gauge. But it appears in the homogeneous equations of
O(R2k+2). Therefore to completely determine the metric and gauge field (upto integration
constants) at any given order R2k, one has to solve one more equation at the order R2k+2
along with all the equations at order R2k.
The equations (4.39) are completely well posed, and may easily be integrated to solve
for ζ(2,2k)(z), χ(2,2k)(z) and W(2,2k−2)(z) in terms of four integration constants. Two of
these constants are determined by the requirement that fin
the horizon z = 0. The remaining two constants are determined by matching with the
intermediate range solution.
Solving the first two equations at O(R0) and the third equation at O(R2) one can find
ζ(2,0)(z), χ(2,0)(z) and W(2,0)(z) respectively. The solution is the following.
(2,2k)(z) and Ain
(2,2k)(z) vanish at
ζ(2,0)(z) =
Λ1
z(1 + z)(1 + 2z)2
χ(2,0)(z) =4Λ1z
1 + 2z+ Λ2
W(2,0)(z) = −log[8(1 + z)]
8
+
Λ1
1 + 2z+ zβ1+ β2
(4.40)
– 29 –
Page 31
Regularity at the horizon implies that
Λ1= β2+3log2
8
, Λ2= 0
After imposing the regularity at z = 0 the solution at the leading order
fin
(2,4)(z) = −1
gin
(2,−4)(z) =
(2,2)(z) = −1
2(1 + 2z)log(1 + z) − 3(log2)z + 4β1z(1 + 2z) + 8β2z
1
32z2(1 + z)2[(1 + 2z)log(1 + z) + 2z(3log 2 − 1)] +
β1− 2β2
4z(1 + z)2
Ain
4log(1 + z) + 2zβ1
(4.41)
Here β1and β2are the two constants which are to be determined by matching. Expanding
around z = ∞ one finds
fin
gin
Ain
2(z) =R4?8β1z2+ O(z)?+ O(R6)
2(z) =R2?2β1z + O(z0)?+ O(R4)
R2 this expansion will match with (4.35) provided one chooses the
constants in the following way
2(z) =O(R−2)
(4.42)
After substituting z =y−1
α3= −1
α4=0
β1= −1
8
8
(4.43)
In the whole solution at this order there are two constants left undetermined. The first is
β2in the near field solution and the second is h1in the far field solution. In the expansion
of fin
in expansion of fmid
2
(y). Therefore to compute this constant one needs the solution upto
O(R2) in the intermediate region. Upon determining this solution to O(R2) one can solve
for β2(as well as all the new constants appearing in the O(R2) intermediate solution) in
terms of h1. It turns out that
2(z) the constant β2appears at O(R4z) which is equivalent to a term of O?R2(y − 1)?
β2=
1
16+38log2 −h1
2
So at the end the full solution to O(R2) is determined in terms of a single constant, h1,
which in turn is determined only by the ǫ3order scalar field analysis. It turns out that
h1= 0
5. All Spherically Symmetric Supersymmetric Configurations
In this section we will analyze the set of spherically symmetric supersymmetric solutions
of (1.2). The configurations we will find will include the solitons of section 3 (determined
– 30 –
Page 32
more simply than in that section), but will also include several solutions that are singular
at the origin. In particular (as we have explained in the introduction) we will identify
a one parameter set of singular supersymmetric solutions which we will conjecture to be
physical; we will conjecture that these singular configurations may be obtained as the limit
of nonsingular nonextremal solutions.
5.1 The Equations of Supersymmetry
The action (1.2) is a consistent truncation of N = 8 gauged supergravity. Hence any
solution of the equations of motion (2.10), which saturates the BPS bound m = 3q, cor-
responds to a supersymmetric solution of N = 8 gauged supergravity and consequently of
IIB SUGRA on AdS5× S5. Here we present a more direct analysis of the supersymmetry
equations for the consistent truncation.20
The supersymmetry conditions in theories of this type have been analyzed in [15, 16,
17, 27]. In these works supersymmetric solutions were found for a more general truncation
of N = 8 supergravity to U(1)3gauged supergravity coupled to 3 hypermultiplets. Our
theory is a special case of theirs where all three U(1) charges and three hyperscalars are
taken to be equal. Specializing their results to our theory we find that spherically symmetric
supersymmetric configurations can be written as follows. The metric, gauge field and scalar
are21
ds2= −1 + ρ2h3
h2
dt2+
h
1 + ρ2h3dρ2+ ρ2hdΩ2
3
A = h−1dt,φ = 2
?
(h + ρh′/2)2− 1
(5.1)
The entire solution is then determined by the single function h(ρ) which is constrained to
satisfy the following ordinary differential equation
(1 + ρ2h3)?3h′+ ρh′′?= ρ
?
4 −?2h + ρh′?2?
h2
(5.2)
Notice that prime denotes differentiation with respect to the variable ρ.
This parametrization of the metric is somewhat different from the one that we used in
the previous section, so we explain how the two are related. Comparing the coefficient of
dΩ2
3in the metric (5.1) to that of (2.9) we see that the two radial coordinates are related
by
r2= ρ2h(ρ) (5.3)
20Let us first briefly describe how one could supersymmetrize the action (1.2). The bosonic field content
of our theory is that of minimal gauged supergravity (i.e. the graviton and the U(1) gauge field) coupled
to matter (the charged scalar φ). The scalar field can be thought of as a member of a hypermultiplet. A
complete hypermultiplet would contain 2 complex scalar fields. Therefore, to supersymmetrize the action
(1.2) we have to add one more scalar field, besides the fermions, and the resulting theory is minimal gauged
supergravity coupled to a hypermultiplet. However, in the set of solutions that we are interested in, the
additional scalar field can be consistently set to zero so we can ignore it in what follows.
21Let us explain our notation in relation to the notation of [16]. We have: rthere= ρhere,
(H2)there= (H3)there= hhere,Athere= −Ahere,
(H1)there=
2sinh(φthere) = φhere.
– 31 –
Page 33
Comparing the other coefficients of the two metrics we find
f(r) =1 + ρ2h3
h2
,g(r) =
4ρ2h2
(2ρh + ρ2h′)2(1 + ρ2h3)
(5.4)
With these identifications it is a matter of algebra to verify that equation (5.7) is sufficient
for the equations of motion (2.10) to be satisfied.
In summary the most general spherically symmetric supersymmetric solutions to the
equations of motion (2.10) is given by the configuration
g(r) =
4ρ2h2
(2ρh + ρ2h′)2(1 + ρ2h3)
f(r) =1 + ρ2h3
h2
1
h(r)
?
A(r) =
φ(r) = 2
(h + ρh′/2)2− 1
(5.5)
with
r2= ρ2h(ρ)(5.6)
and h(ρ) any function that obeys (5.7).
5.2 Classification of Supersymmetric Solutions
As we have explained in the previous subsection, supersymmetric solutions to the equations
of motion are given by solutions to the second order differential equation
(1 + ρ2h3)?3h′+ ρh′′?= ρ
?
4 −?2h + ρh′?2?
h2
(5.7)
In this paper we are only interested in regular normalizable solutions to these equations. It
is of crucial importance to this section that the condition of normalizability is automatically
met; an analysis of (5.7) at large ρ immediately reveals that all solutions to this equation
behave at large ρ like
h(ρ) = 1 +2q
ρ2+ ...
22ensuring normalizability for all physical fields23. The importance of this observation is
the following; one may study the small ρ behaviour of supersymmetric solutions in a purely
local manner, without having to worry about when the solutions we study have acceptable
22Using (5.3) and (5.1) we find that this implies the large r behavior of the gauge field
A(r) = 1 −2q
r2+ ...
so the constant q may be identified with the electric charge of the solution, in the conventions of previous
sections.
23This fact has a natural explanation from the viewpoint of the dual N = 4 Yang Mills field theory;
a deformation of the Lagrangian of that theory by only mass term TrX2+ TrY2+ TrZ2preserves no
supersymmetry.
– 32 –
Page 34
large ρ behaviour, as that is always guaranteed. This fact allows us, in this section, to
use local analysis to present a simple classification of normalizable supersymmetric solu-
tions. Relatedly, supersymmetric solutions may be obtained by solving (5.7) as an initial
value problem with initial conditions set at small ρ. This is numerically and conceptually
simpler than the boundary value problem we would have to solve by shooting methods off
supersymmetry.
It remains to impose the condition of ‘regularity’. Let us first explain what we mean
by this term. We call a supersymmetric configuration ’regular’ if it can be regarded as
the limit of a one parameter set of smooth nonsupersymmetric (and so non extremal)
solutions to the equations of motion (2.10). While every smooth supersymmetric solution
is automatically ‘regular’, a singular susy solution may also be ‘regular’, if its singularity
can be removed upon heating the solution up infinitesimally.
Solutions to (5.7) can develop singularities only at ρ = 0. In this subsection we will
classify all possible behaviours of solutions to (5.7) near ρ = 0. In a later subsection we
will then go on to present conjectures about which of these solutions are ‘regular’.
In order to investigate possible behaviours of solutions to (5.7) at small ρ we plug in
the ansatz h(ρ) =
ρα into the equation. It is easy to check that the only values of α that
solve the equation near ρ = 0 are α = 0,2
3,1,2. It is also possible to demonstrate (see
below) that there is a unique solution with α =2
and α = 1 both appear in a one parameter family. Finally, the generic solution to the
differential equation has α = 2; solutions with α = 2 appear in a 2 parameter family.24
A
3. On the other hand solutions with α = 0
5.2.1 h(ρ) ≈ ρ−2
As we have mentioned above, there is a unique solution with α =2
expanded at small ρ as follows
3
3. This solution may be
h(ρ) = ρ−2/3+9
26ρ2/3−
243
20956ρ2+ O(ρ8/3) (5.8)
We now present a crude estimate for validity domain of the expansion (5.8). Note that the
formal procedure that generates the series expansion (5.8) treats the term proportional to
4 (in the RHS of (5.7) ) as subleading to the term proportional to h. This procedure is
valid whenever ρ ≪ 1; as a consequence we expect the expansion (5.8) to be valid whenever
ρ ≪ 1 but to break down at larger values of ρ.
For ρ ≪ 1 the metric, gauge field and scalar corresponding to this solution take the
following form
ds2≈ −2r2dt2+9
A(r) ≈ rdt
φ(r) ≈
3r
8dr2+ r2dΩ2
3
4
(5.9)
24One special exact solution of (5.7) is h(ρ) = 1 +
α = 2. For this solution we notice that the scalar field is not turned on (φ = 0). So in a sense it is
qualitatively different from the “hairy” configurations of interest to us in this paper.
q
ρ2, the so-called “superstar” [28]. This solution has
– 33 –
Page 35
where we used the relations (5.5) and (5.6) to bring the solution in the form of (2.9).
We will denote the distinguished singular solution of this subsection by S. We will
now explain that there is a sense in which S is a fixed point of the equation (5.7) viewed as
a flow equation in the variable logρ = x (see [29, 30, 31] for similar discussions in distinct
but similar contexts). For this purpose we redefine
h(ρ) = e−2
3xf(x)
The differential equation (5.7) becomes
9f′′(1 + f3) + 3f′(2 + 3f2f′+ 10f3) + 8(f4− f) − 36e4x/3f2= 0
For very small ρ (that is for x → −∞) the last term in the equation can be ignored, and the
equation becomes approximately time translation invariant (or an autonomous equation,
in the language of dynamical systems). With this approximation the system has an exact
solution f(x) = 1, which is precisely the leading small ρ approximation to the solution S.
We will restrict attention to large negative values of x in the rest of this subsubsection,
and so study the truncated equation
9f′′(1 + f3) + 3f′(2 + 3f2f′+ 10f3) + 8(f4− f) = 0 (5.10)
Let us consider a small perturbation about f = 1, i.e. we set
f(x) = 1 + εg(x). (5.11)
To linear order in ε the (5.10) turns into the linear ODE
3g′′+ 6g′+ 4g = 0(5.12)
The two linearly independent solutions to this equation are
g(x) = eλx
λ = −1 ±
i
√3
(5.13)
Note that the real part of the each of these eigenvalues is negative, which demonstrates
that f = 1 is a stable fixed point of the dynamical system (5.10). Moving back to the
variable ρ, it follows that arbitrary small perturbations around the solution S behaves like
h(ρ) ≈ ρ−2/3
?
1 + ε1
ρcos
?
1
√3logρ + a
??
Note that all perturbations die out for ρ ≫ ǫ (this is a restatement of the fact that f = 1
is a stable fixed point).
– 34 –
Page 36
5.2.2 h(ρ) ≈ ho+ O(ρ2)
We now turn to regular solutions to (5.7), i.e. solutions with α = 0. Such solutions appear
in a one parameter set, labeled by h0= h(0). The small ρ expansion of (5.7) is given by
h(ρ) = h0+1
2(h2
0− h4
0)ρ2+1
6(h3
0− 5h5
0+ 4h7
0)ρ4+ O(ρ6) (5.14)
The solutions of this subsection are simply the solitons studied in section 3. These solutions
were generated perturbatively (i.e. at h0− 1 small) in section 3.
Let us now turn to the opposite limit of large h0.
validity domain of the expansion (5.14). As the term h3ρ2on the LHS of (5.7) is of
order ρ2, the formal process that generates the power series expansion (5.14) treats this
term as subleading compared to unity. This is actually correct only when h3
follows that when h0is large the series expansion (5.14) will break down at the small value
ρ ∼ ρrb∼ h−3
When h0is large, the expansion (5.14) does not apply in the range ρ ≫ h−3
in this range, however, the general arguments presented above guarantee that our solution
behaves like ρ−αfor one of the allowed values of α described above. What solution does the
expansion (5.14) match onto in this range? A clue to the answer to this question is given by
noting that the value of h(ρ), at the point of break down of (5.14) is approximately given
by h0∼ ρ−2
previous subsubsection, at ρ ∼ ρrb. This suggests that the special solution S of the previous
subsubsection is the limit as h0→ ∞ of the solutions of this subsubsection. We now present
numerical evidence that strongly supports this guess. In Fig. (3) we present numerically
generated plots of the regular solution parametrized by h0for successively increasing values
of h0. Note that, as h0increases, the entire profile of the solution approaches a limiting
shape, with a sharp spike near ρ = 0. The spike becomes sharper as we increase h0, while
the solution at larger values of ρ remains almost unchanged. The limiting solutions indeed
appears to be the special solution S of the previous subsubsection (denoted by the solid
line in Fig. 3). Thus it appears that the solution S forms the end point of the family of
regular supersymmetric solitons.
We will now study in more detail how solutions with large h0 approach the special
solution S. We work in the language of the dynamical system (5.10). We are given a
function f(x) that starts out, at large negative values of x (small ρ) as
Let us first inquire as to the
0ρ2∼ 1. It
2
o .
2
o
. If ρ ≪ 1
3
rb. Thus the solution (5.14) could smoothly match onto the special solution S of
f(x) = h0e
2x
3?1 + O(e2x)?. (5.15)
We wish to study how f(x) evolves under (5.10) at later times. We are interested in the
limit in which h0is large; as we have argued above, we expect f(x) to increase to a value
of order unity at a time x0∼ −3
point f = 1. Note that x0≪ −1 (this follows because of our assumption that h0is large) ,
so that f should settle down to very near unity well within the domain of applicability of
the dynamical system (5.10).25
2lnh0, and thereafter stabilize exponentially to the fixed
25In this language, the conjecture of the previous paragraph is equivalent to the assumption that this
fluctuation lies within the domain of attraction of the fixed point f = 1.
– 35 –
Page 37
0.00.2 0.40.6 0.81.0
0
2
4
6
8
ρ
h(ρ)
Figure 3: Convergence of the numerical solutions
for the regular solitons to the special singular so-
lution S as we increase h0= h(0). The black line
corresponds to the solution S with ρ−2/3behavior
near ρ = 0. The blue lines correspond to regular
solitons of h0= 2,3,4,6,8, starting from the low-
est blue curve and going up.
05 1015 20
?0.4
?0.2
0.0
0.2
0.4
logh0
q−qc
2log h0
e−3
Figure 4: The damped oscillations of q around
the critical value qcfor large h0.
Let us now compute f(x1) for some fixed (h0independent) x1that obeys26
x0≪ x1≪ −1.
In order to do this we recall that the equation (5.10) is invariant under translations in x.
Now the judiciously chosen translation
x′= x +3lnh0
2
eliminates the h0dependence of the initial condition (5.15) . Let χ(x) be the (unique, h0
independent) solution to (5.10) that reduces at early (large negative) times to χ(x′) = e
It follows that the solution of interest to us is
2x′
3 .
f(x) = χ(x +3lnh0
2
).
The key assumption of this section, is that the function χ lies within the domain of
attraction of the fixed point f = 1 (as we have seen above there is impressive numerical
evidence for this assumption). If this is the case it follows from (5.11) and (5.13) that at
large x′(i.e. for x ≫ x0;)
χ(x′) = 1 + Ae−x′cos(x′
√3+ δ)
for some unknown, order unity constants A and δ. It follows that
f(x1) ≈ 1 + Ae−3
2logh0+x1cos
?√3
2
log(h0+ x1) + δ
?
(5.16)
Of course the x1dependence of this result may be absorbed into a redefinition of A and δ.
26Recall that (5.10) is valid only for large and negative x1.
– 36 –
Page 38
Let ρ1= ex1. (5.16) gives us a formula for h(ρ1) and h′(ρ1) for the solution of interest;
these values may be used as an ‘initial conditions’ to generate h(ρ) for all ρ > ρ1. The
resultant solution will take the form
h(ρ) ≈ hS(ρ) + δh(ρ)
where hS(ρ) is the special solution S and δh(ρ) is a small fluctuation (of order ∼ O(1
) about this solution. To leading order in this small parameter, the function δh(ρ) obeys
a linear differential equation, and so depends linearly on h(ρ1) and h′(ρ1). It follows that
(5.16) then determines the behaviour of every observable (like the charge) of the solution
that depends only on the behaviour of h(ρ) for ρ of order unity or greater. In particular it
follows that the dependence of the charge of solutions on h0is given approximately by
h
3
2
0
)
q(h0) ≈ qc+ Ae−3
2logh0cos
?√3
2
logh0+ δ
?
(5.17)
for some constants A,δ. While A and δ can only be determined numerically, we have a
sharp analytic prediction for the form (5.17). A similar formula applies for the vacuum
expectation value of the operator dual to φ as a function of h0.
We have verified the prediction (5.17) numerically; in Fig. 4 we present a plot of the
rescaled oscillations of q about qc. This graph displays precisely the damping (reflected in
the h0dependent renormalization of the y axis in Fig. 4) and the oscillations predicted by
(5.17). We will give more details below.
5.2.3 h(ρ) ≈a
Next we consider solutions with α = 1, i.e solutions that behave near ρ = 0 likea
one parameter set of these solutions may be labeled by a. At small ρ our solution takes the
forma
ρP(ρ) where P(ρ) is a regular power series. The first few terms in the power series
expansion are given by
ρ
ρ. The
h(ρ) =1
ρ
?
a +
1
3a2ρ +18a4− 5
36a5
ρ2+−90a4+ 31
270a8
ρ3+ O(ρ4)
?
(5.18)
The formal procedure that generates the power series (5.18) treats the term proportional
to unity (on the LHS of (5.7)) as subleading compared to the term proportional to h3ρ2.
When a ≪ 1, this is justified only when ρ ≪ a3. Consequently we expect the expansion
(5.18) to break down at ρsb∼ a3.
As in the previous subsubsection, the solution presented here must reduce to one of the
other solutions of this section when a3≪ ρ ≪ 1. Noting that, at the point of breakdown
of (5.18), the function h may be estimated by h ∼
solution of this subsubsection tends to the special solution S in the limit of small a. We
now present strong numerical evidence in support of this guess. In figure (5) we present
numerical plots of the solution of this subsection for a range of decreasing values of a. Note
that the solution converges to the solution S (denoted by the solid line in Fig. (5)) at small
a.
1
a2∼ ρ−2
3
sb, it is natural to guess that the
– 37 –
Page 39
0.0 0.20.4 0.60.8 1.0
0
2
4
6
8
10
ρ
h(ρ)
Figure 5: Convergence of the numerical solutions
for singular solitons with ana
special singular solution S as we decrease a. The
black line correspondsto the solution S with ρ−2/3
behavior near ρ = 0. The red lines correspond to
singular solitons of a = 0.35,0.5,1,2.5,5, starting
from the lowest red curve and going up.
ρsingularity to the
?8
?7
?6
?5
?4
?3
?2
?1
?0.4
?0.2
0.0
0.2
0.4
loga
q−qc
e3 log a
Figure 6: The damped oscillations of q around
the critical value q for small a.
For ρ ≪ 1 the metric, gauge field and scalar corresponding to this solution take the
following form
ds2≈ −r2dt2+4r2
A(r) ≈r2
φ(r) ≈a2
r2
a4dr2+ r2dΩ2
3
a2dt
(5.19)
Precisely as in the previous subsubsection, we can analytically study the approach of
the solution with small a to the solution S. Repeating an analysis very similar to that of
the previous subsubsection, we conclude that the dependence of, for instance, the charge
of the solution on a is given by the formula
q(h0) ≈ qc+ Ae3logacos
?√3loga + δ
?
for some constants A,δ which cannot be determined analytically. We have verified this
prediction numerically (see Fig. 6). A similar formula applies for the vacuum expectation
value of the operator dual to the field φ, as a function of h0.
5.2.4 The generic solution, α = 2
Finally we move to the case m = 2. Now we find a two parameter set of solutions, labeled
by two arbitrary constants a,b, which have the form
h(ρ) =
1
ρ2
?
a +1 − b2
2a
ρ4−(b2− 1)(3a(5b2− 1) − 2)
24a4
ρ8+ O(ρ12)
?
(5.20)
The value b = 1 is special; as we have already remarked h = 1+a
hints of this fact are already visible in the expansion (5.20). It follows in particular that
ρ2is an exact solution;
– 38 –
Page 40
the values b = 1 lies outside the basin of attraction of the fixed point S at least when b = 1.
A very rudimentary numerical investigation suggests that this is also true for all values
of a at (for example) b = 2. Although we have not carefully investigated this question,
it seems possible that the solutions with α = 2 are completely disconnected from all the
other solutions studied above.
For the reasons outlined in the previous paragraph, the ‘generic’ solutions of this
subsubsection will make no further appearance in our paper. We suspect that all - or at
least most - of these solutions are genuinely singular, in the sense that they cannot be
regarded as the limit of smooth solutions. We leave a fuller study of these solutions, and
their possible physical significance, to future work.
5.3 ’Regular’ supersymmetric Solutions
In this section we present the results of a numerical analysis of the space of ’regular’
supersymmetric solutions. Let us first describe what we believe the space of these solutions
to be. The smooth supersymmetric solitons (with α = 0) of the previous subsection are
clearly regular. However this space of solutions ends at finite charge (see below) as h0→ ∞.
As we have described above, this line of solutions spirals into (and ends in ) the special
solution S. We have also seen above that another line of solutions - those with α = 1
and small values of a - spiral out of the solution S. As we will see below, the charge
of this new line of solutions increases without bound at large a. It thus seems that the
solutions with α = 0 and α = 1 may be regarded as two different segments of a single
line of supersymmetric solutions. The two segments are joint together (by a very intricate
non intersecting double spiral structure) at the special solution S. At least one member of
this line of solutions exists at every value of the charge, and so constitutes a candidate end
point of the phase diagram Fig. 2. We conjecture that it is indeed the case that hairy black
holes at every value of the charge exist for all energies above the BPS bound. In the BPS
limit, these solutions reduce to some configuration on this special line of solutions; either
to the smooth supersymmetric soliton (at small charges) or the α = 1 solutions (at large
charges). In particular we conjecture that all solutions on the special line described in this
are ‘regular’, where this word is used in the sense specified in the previous subsection. We
now proceed to study these conjecturally regular solutions in more detail.
5.3.1 Solitons
We first present the numerical analysis of regular solutions of (5.7). For this we fix the
initial conditions h(0) = h0 ≥ 127and h′(0) = 0 and integrate the equation outwards.
For each value of h0we compute the solution h(ρ) numerically and we evaluate the charge
q(h0). The results can be seen in Figs 7, 8 and 9, in various magnifications.28
27The condition h0 ≥ 1 is necessary since the scalar field at ρ = 0 is given by φ(0) = 2?h2
to be real by assumption (2.9).
28To partly check the validity of the numerics it is easy to perform a perturbative analysis of equation
(5.7), similar to that of the previous sections i.e. in a small amplitude of h(ρ) − 1. One finds agreement
between numerics and perturbation theory (i.e. convergence of the perturbative solution to the numerical
one, for small enough values of h0− 1). Notice that the case h0 = 1 is precisely empty AdS.
0− 1 and has
– 39 –
Page 41
05 10 15 20
0.00
0.05
0.10
0.15
0.20
0.25
0.30
h0
q
Figure 7: Charge q of spherically symmetric su-
persymmetric regular solitons as a function of the
value h0≡ h(0).
0 1020 304050
0.260
0.261
0.262
0.263
0.264
0.265
h0
q
Figure 8: The same graph with different scales
on the axes, where we can see the maximum
charge.
02468 10
?0.00003
?0.00002
?0.00001
0
0.00001
0.00002
0.00003
logh0
q − qc
Figure 9: Magnification of the previous graph. We wee the next oscillation around qc.
The most striking feature of the numerical analysis is the existence of a maximum
value29
qmax≈ 0.2643
for the charge of regular supersymmetric solitons. This charge is obtained for the value
hqmax≈ 9.821 of the initial conditions at the center. The existence of a maximum charge
for these solitons was also noticed in [16]30. For higher values of h0 the charge of the
solution starts to decrease and asymptotically it approaches the limiting value
(5.21)
qc≈ 0.2613 (5.22)
as h0→ ∞.
A more careful analysis reveals that the convergence of the function q(h0) towards the
critical value qcis not monotonic. Instead, the function q(h0) undergoes slow oscillations
around the critical value, as can be seen in Fig. 9 and in more detail in Fig. 4. These
oscillations are periodic, with damped amplitude, if expressed in terms of x = logh0. As
we explained in section 5.2.2 the asymptotic form of these oscillations can be determined
analytically by matching the regular solution for large h0to the special ρ−2/3solution S
29We have solved the equations numerically using Mathematica.
30Notice that we are working in slightly different conventions from [16], in which qthere= 2qhere. This is
consistent with the maximum charge (qm)there= 0.529 reported in that paper.
– 40 –
Page 42
and we have the following asymptotic formula for large h0
q(h0) ≈ qc+ Ae−γ logh0cos(ω logh0+ δ)(5.23)
In section 5.2.2 we saw that the period of the oscillations and the damping constant can
be determined analytically from the matching procedure to be
γ = 3/2,ω =
√3/2. (5.24)
while A and δ cannot be fixed in this way. If one tries to fit this formula to the numerical
data one finds
γ ≈ 1.50,ω ≈ 0.87,A ≈ −0.19,δ ≈ 0.12 (5.25)
which are in very good agreement with the exact values.
The number of regular solitonic solutions as a function of the charge are as follows: for
small enough charge there is only one solution. As we increase the charge, at some point
we hit the first oscillation around qc, which increases the number of solutions to three.
Increasing q further we encounter the second oscillation and we have five solutions, and
so on. As we approach the critical value qcfrom below the number of solutions is always
an odd integer which goes to infinity. Now let us consider the large charge behavior. For
q > qmaxwe have no regular solitonic solution. As we decrease the charge and we go below
qmaxwe first find two solution. As we decrease further we encounter the first oscillation
above qc, giving us four solutions, then the second oscillation to six solution and so on.
Hence for q > qcwe always have an even number of solutions (possibly zero) which tends
to infinity as we approach qcfrom above.
Notice that since the BPS bound m = 3q is satisfied for all of these solutions, the
figures 7,8 also show the dependence of the mass of the solution on the value of the field
at the center. This qualitative behavior, i.e. the existence of a maximum mass, and of a
slightly lower critical value of the mass which is approached asymptotically for large central
density via a function which undergoes damped oscillations, is typical in related problems
in general relativity [32, 33, 34].31To our knowledge, however, this is the first time such
behaviour has been observed in family of supersymmetric solutions.
Let us now study the expectation value of the scalar operator dual to φ, which we
denote by ?Oφ?, as a function of h0. Since Oφis an operator of dimension ∆ = 2, its
expectation value can be determined from the large r expansion of φ as
φ(r) =?Oφ?
r2
+ ...
31Generally, when a family of gravitational solutions has the property that their mass has a local maximum
for some value of the initial conditions at the center, it is the sign that one of the two branches (to the left
or right of the local maximum) is unstable (under radial perturbations) and thus unphysical. This is the
analogue of the “Chandrasekhar instability”: if we expand the equations of motion around the solution at
the local maximum of the mass they have a zero mode, since the total mass does not change to first order
in the perturbation. Generically this zero mode will be stable on one side and unstable (i.e. tachyonic) on
the other side of the local maximum. This is what happens for example in the case of boson stars [35]. In
our case the solutions are supersymmetric for all values of h0. It would be very interesting to check what
this implies about their stability.
– 41 –
Page 43
We plot the results in figures 16,17,18 in appendix C. The qualitative behavior is similar
to that of the charge q: the expectation value ?Oφ? is an increasing function of h0up to a
maximum value
?Oφ?max≈ 1.8906
which is realized for h0≈ 6.580 and then decreases and approaches the asymptotic value
?Oφ?c≈ 1.8710
as h0→ ∞, while performing small oscillations around it. Again the oscillations can be
determined following the logic of section 5.2.2 and are captured by the formula
(5.26)
(5.27)
?Oφ?(h0) ≈ ?Oφ?c+ Ae−γ logh0cos(ω logh0+ δ)
The analytic prediction is γ =3
2, ω =
data one finds
(5.28)
√3
2. If one tries to fit this formula to the numerical
γ ≈ 1.50,ω ≈ 0.87,A ≈ −0.66,δ ≈ 0.48 (5.29)
which are in good agreement with the exact values.
Before we proceed let us point out that the value of h0at which we have the maximum
charge (hqmax≈ 9.821) differs from the one at which we have the largest expectation value
?Oφ?max, which turns out to be h?Oφ?max≈ 6.580. More generally, and as we will see more
clearly in subsection 5.4, while there are pairs of regular solitonic solutions with the same
charge q or the same expectation value ?Oφ?, there are no such pairs which have the same
q and same ?Oφ? simultaneously.
5.3.2 Solutions with α = 1
We now present the results of a numerical investigation of the other segment of the line of
(conjecturally) ’regular’ supersymmetric solutions: those whose small ρ behaviour is given
by a/ρ for small ρ. We compute the entire solution numerically and calculate the charge
q as a function of a. The results are shown in Figs 10,11,12. As we see in the figures
the charge of this family starts (at small a) precisely at the point q = qc(5.22) where the
family of regular solitons ended, then as we increase a the charge seems to decreases, down
to a minimum value
qmin= 0.2605(5.30)
and then increases all the way to arbitrarily large values.
As before, a more careful analysis shows that the entire radial profile of solutions with
a
ρsingularities converges to the special solution S in the limit a → 0, as shown in Fig. 5.
Again, a closer inspection shows that in the regime between a = 0 and the point where
q = qminthe function q(a) is not monotonically decreasing, but rather is undergoing small
damped oscillations around the value qcas a function of loga. This is shown in Fig. 6.
For small values of a the form of these oscillations can be determined by the matching
procedure discussed in section 5.2.3 and we find the following formula
q(a) ≈ qc+ Aeγ logacos(ω loga + δ) (5.31)
– 42 –
Page 44
0.0 0.51.0
a
1.5 2.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
q
Figure 10: Charge q of spherically symmetric
supersymmetric solitons with a singularity of the
forma
ρat ρ = 0.
0.000.05 0.100.15
a
0.200.25 0.30
0.2600
0.2605
0.2610
0.2615
0.2620
q
Figure 11: Detail of the previous graph with
different scales on the axes, where we can see the
minimum of the charge near a = 0.
?6
?5
?4
?3
?2
?1
?4.?10?6
?2.?10?6
0
2.?10?6
4.?10?6
loga
q − qc
Figure 12: Magnification of the previous graph. We wee the next oscillation around qc.
with the analytically determined values (see 5.2.3) γ = 3, ω =√3. From the numerics we
find
γ ≈ 3.00,
which are in good agreement with the exact values.
We find similar behavior for the expectation value ?Oφ? as shown in Figs 20,21 in
appendix C: the expectation value starts at the point (5.27) where the regular soliton
family ended, it then goes down to
ω ≈ 1.73,A ≈ 0.18,δ ≈ 0.70(5.32)
?Oφ?min≈ 1.8658 (5.33)
and then increases indefinitely.
Finally we have the following oscillatory behavior for small a which is shown in Fig.
23
?Oφ?(a) ≈ ?Oφ?c+ Aeγ logacos(ω loga + δ)
with the exact values γ = 3, ω =√3. From the numerics we find
(5.34)
γ ≈ 2.97,ω ≈ 1.74,A ≈ 0.59,δ ≈ −0.34(5.35)
Let us mention that the numerical results depicted in Fig. 14 agree with the perturba-
tive analysis of section 3 in the regime of small q. According to the results of that section
– 43 –
Page 45
we expect that for small q the expectation value ?Oφ? goes like
?Oφ? = 4√q + ...
One can indeed verify that the small q behavior of the curve in Fig. 14 agrees with this
result.
5.3.3 An analytic solution at large charge
While we have no analytic control of α = 1 solutions in general, we can see from the
numerical analysis that for large a the charge of the solution can be well approximated by
the formula q =a2
4+ subleading. In fact, in the limit of large a one can find an analytic
form of the solution as follows: let us consider the first factor (1 + ρ2h3) on the LHS of
equation (5.7). At small value of ρ the term ρ2h3dominates over the 1 since by assumption
h ∼a
to assume that in the limit of very large a we can make the approximation 1+ρ2h3≈ ρ2h3
for the entire range of ρ. Then the differential equation (5.7) becomes
ρ. At large values of ρ the same is true since h ∼ 1+2q
ρ2. Hence it is not unreasonable
ρh?3h′+ ρh′′?=
?
4 −?2h + ρh′?2?
(5.36)
This equation can be solved exactly32and if we impose the desired behavior near ρ = 0
the solution is
h∞(ρ) =
?
1 +a2
ρ2
(5.37)
It is not hard to check that in the large charge limit the numerical solutions do indeed
converge towards the solution (5.37) in the entire range of ρ, as shown in figure 13. The
solution (5.37) goes likea
ρnear ρ = 0 and like 1 +
that the charge q ∼a2
?Oφ? ∼ a2. So in the limit of large q we have
a2
2ρ2for large ρ. As we said this implies
4. One also finds that in this limit the expectation value goes like
?Oφ? = 4q + ...
which describes the behavior of the red curve in Fig. 14 for large q.
5.4 Phase Structure of ‘regular’ supersymmetric solutions
Let us now put everything together and describe the space of supersymmetric solutions.
In Figs 14,15 we show the expectation value ?Oφ? and the charge q of the family of regular
solitons (blue curve) and those with ana
ρsingularity (red curve). As we explained above
the two families meet at the solution S with ρ−2/3behavior, which is denoted by a black
dot. Near the point S the two curves develop into two intertwined spirals which are
asymptotically described by equations (5.23),(5.28),(5.31),(5.34). We zoom into the point
S in Figs. 24,25 in appendix C.
32The general solution of (5.36) is h(ρ) =
?
1 +c1
ρ2+c2
ρ4.
– 44 –
Page 46
0.00.2 0.40.60.8 1.0
0
1
2
3
4
ρ
h(ρ)
h∞(ρ)
Figure 13: Convergence of the numerical solutions for singular solitons with ana
the family of approximate solutions h∞(ρ)as we increase a. We plot the ratio of the two functions
for various values of a and we see that it converges to 1 as we raise a. The values plotted are
a = 0.1,0.2,0.5,1.5, from top to bottom.
ρsingularity to
0.0 0.1 0.2 0.3
q
0.4 0.50.6
0
1
2
3
4
?Oφ?
Figure 14: Expectation value ?Oφ? vs charge q
for the family of regular solitons (blue) and for the
family of solitons with ana
two curves meet at the point denoted by the black
dot which corresponds to the special solution S
with ρ−2/3behavior.
ρsingularity (red). The
0.250 0.2550.260 0.265 0.2700.275
1.84
1.85
1.86
1.87
1.88
1.89
1.90
q
?Oφ?
Figure 15: Detail of the previous graph around
the point S where the two families meet. The
blue curve is the regular soliton and the red curve
is the soliton with thea
ρsinguality.
From these figures we see that the curves are non intersecting, which means that if
we fix the charge q and the expectation value ?Oφ? there is at most one solution. If we
consider the number of solutions as a function of the charge q leaving ?Oφ? arbitrary we
have the following pattern: for small enough q we have only one regular solitonic solution.
As we increase q we first encounter the point qminwhere two new solutions appear, bringing
the total number of solutions to 3. Increasing q we cross a point where two more regular
solutions are added and the total number of solutions becomes 5. This pattern continues as
we approach the point qcwith an ever-increasing number of total solutions (at each step we
add, alternatively, either two regular or two singular solutions). Notice that this number is
always odd. After we cross the point qcthe pattern is reversed, we successively lose pairs of
solutions until we end up with a single singular solution for q > qmax. Similar statements
hold if we look for solutions with given expectation ?Oφ? instead of given charge. However,
the fact that the curves in these figures are non-intersecting, means that if we specify both
q and ?Oφ? there is always at most one solitonic solution with these values.
– 45 –
Page 47
The phase diagram we have proposed for our gravitational system is depicted in Fig.
2. In this diagram we have included a phase transition curve that meets the BPS line near
q = qc≈ 0.2613; we will now explain our rational for doing so. In this paragraph we assume
that both the regular and the singular supersymmetric solutions found before, may each be
obtained as a limit of non singular non extremal hairy black hole solutions. As we saw above
for any given value of q we may have either one, or a larger odd number of supersymmetric
configurations depending on whether q lies outside or inside the interval (qmin,qmax) =
(0.2605,2643). It follows that there may exist more than one near supersymmetric regular
hairy black hole solutions in the charge range q ∈ (qmin,qmax) approximately centered
around qc, on which we now focus. These configurations differ by the expectation value
?Oφ? of the operator dual to the field φ. Let SR(q,δe) denote the entropy of the hairy black
hole that reduces to the regular soliton with the largest value of ?Oφ? when δe → 0 (here
δe denotes the energy above BPS). Let SS(q,δe) denote the entropy of the hairy black hole
that reduces to the singular supersymmetric solutions with the smallest value of ?Oφ?. We
suspect that
SR(q,δe) > SS(q,δe) when q < qP(δe)
SS(q,δe) > SR(q,δe) when q > qP(δe)
(5.38)
for some qP(δe) such that limδe→0qP(δe) ∈ (qmin,qmax). Moreover we suspect that the
entropies of the plethora of intermediate phases that appear at charges near to qc are
always smaller than either SS(q,δe) or SR(q,δe). In other words we suspect that our
system undergoes the micro canonical analogue of a a single first order phase transition at
q = qP(δe); this is the black curve we have depicted in Fig. 2. The phase transition curve,
which originates at the BPS line, could either extend all the way to the phase transition
curve between RNAdS and hairy black holes, or could terminate somewhere in the bulk
of the hairy black hole phase, at a triple point analogous to the water steam system. Of
course the considerations of this paragraph have been highly speculative. It would be very
interesting to investigate this further.
Before we continue we would like to mention that it would be important to clarify the
stability of these solutions under linearized perturbations. As mentioned in footnote 31
on general grounds one would expect regular solutions past qmaxto be unstable. On the
other hand our solutions are supersymmetric and from this point of view it would seem
more natural to believe that they are stable. This is a confusing issue that deserves further
study.
6. Thermodynamics in the Micro Canonical Ensemble
In this section we present thermodynamical formulae for RNAdS black holes, the super-
symmetric solitons, and hairy black holes, in a small charge and near extremal limit. We
also demonstrate that the leading order thermodynamical formulae for hairy black holes
are reproduced by modeling them by a non interacting mix of a soliton and an RNAdS
black hole with µ = 1.
– 46 –
Page 48
6.1 RNAdS Black Hole
The basic thermodynamics for an RNAdS black hole is summarised by the following for-
mulae
m ≡M
N2=3
Q
N2=µ
S
N2= πR3
4R2?1 + R2+ µ2?
2R2
q ≡
s ≡
T =
1
2πR
?1 + 2R2− µ2?
(6.1)
were Q is the charge, M is the mass of the black hole, S is its entropy, T its temperature
and µ its chemical potential. Note that
µ2≤ (1 + 2R2). (6.2)
(this follows from the requirement that R is the outer rather than the inner event horizon
of the black hole).
In this paper we are interested in small RNAdS black holes - i.e. black holes with
m ≪ 1 and q ≪ 1 that are also very near extremality. The mass of RNAdS black holes
at fixed charge is bounded from below by the mass of the extremal black hole of the same
charge; at small q we have
m ≥ 3?q + q2− 2q3?+ O(q4) (6.3)
For every pair (m,q) that obeys this inequality, there exists a unique black hole solution.
In this paper we are interested in black holes whose mass above extremality of of order
O(q3).33For this reason we define the shifted and rescaled mass variable δ2by
δ2q3= m − 3?q + q2− 2q3?
We are interested in q ≪ 1 but δ of order unity. In this regime the entropy, chemical
potential and temperature of this black hole is given by
√2− 10√3δ + 33√2 −24√3
3− 6 +4√6
δ
(6.4)
s = πq
3
2
?
?
2√2 +
?
2√3δ − 6√2
?
q +
?
3δ2
δ
?
q2+ O?q3??
µ = 1 +2 −
?2
3δ
?
q +
?
δ2
?
q2+ O?q3?
πT = q
1
2
?
δ
√3−
?3√2δ3− 10√3δ2+ 24√3?q
6δ
+ O?q2??
(6.5)
33Note that the mass of an extremal black hole, at charge q, exceeds the mass of a BPS black hole at the
same charge by 3q2−6q3+O(q4). Consequently the deviation of the mass of our black holes from the BPS
bound is given by 3q2+ (δ − 6)q3, and in particular is O(q2) rather than O(q3).
– 47 –
Page 49
34
Although the black holes we study are very small, their temperature is very small (it
scales like√q) because we focus on the near extremal limit35. Moreover the black hole
temperature decreases as we decrease δ, reaching zero at δ = 0 + O(q).
Note also that the chemical potential µ of these black holes is unity at leading order.
The first correction to this leading order result is of order O(q) and is positive when δ2< 6
but negative otherwise. This already suggests that RN-AdS black holes with δ2< 6 are
unstable to super radiant decay; we will see below that this is indeed the case. As we will
see below, the end point of the resultant tachyon condensation process is a hairy black
hole.
Finally note that the radius of the black holes we study is O(√q) so that the entropy
is of order O(q
3
2).
6.2 Supersymmetric Soliton
The mass and charge of the supersymmetric soliton are given by
m =3
4
?ǫ2
?ǫ2
4+
ǫ4
192+
ǫ4
384+
ǫ6
1920+
ǫ6
3840+
169ǫ8
2211840+ O?ǫ10??
169ǫ8
4423680+ O?ǫ10??
q =1
2 8+
(6.6)
Note that
m = 3q
µ = 1
(6.7)
Of course the soliton is dynamically stable as it is supersymmetric. It carries no entropy.
6.3 A non interacting mix of the black hole and soliton
In this subsection we will determine the thermodynamics of a hypothetical non interacting
mixture of the small black holes and the supersymmetric solitons of the previous subsection.
Of the net mass m and charge q of the system, let mass 3qsand and charge qslie in
condensate so that the mass and charge of the black hole are given by
mb= m − 3qs
qb= q − qs
(6.8)
The charge qsis determined by maximising the entropy of the system, which determines
the black hole chemical potential to be unity. This condition gives
mb= 3qb+ 3q2
b
(6.9)
34The reader may worry that the appearance of inverse powers of δ in (6.5) signify that black hole
thermodynamics degenerate in the extremal limit; this, however, is not the case. At extremality δ = δext(q).
The function δext(q) starts out at O(q) and so is small at small charge, but does not identically vanish.
Infact δext(q) may be determined as a power expansion in q by equating the temperature in (6.5) to zero.
Plugging this function into the remaining expressions in (6.5), yields nonsingular, analytic expressions as a
function of q.
35In contrast small black holes in [14] all had a very high temperatures.
– 48 –
Page 50
(this exact formula may also be verified to O(q4) by setting δ2= 6 in (6.5)) plugging (6.8)
into (6.9) yields a quadratic equation for qs. Solving this equation we find
qs=
?
?m − 3q
q −
?m − 3q
3
?
qb=
3
(6.10)
The squared radius of the black hole is given by R2
and temperature of the mixture are given by
b= 2qb. At leading order the entropy
s = πR3= π
?
2
?m − 3q
?1
3
?3
2
T =
√2
π
?m − 3q
3
4
(6.11)
As the m−3q for the mixture is of order q2, it is convenient to define a shifted and rescaled
mass variable
q2ρ =4
3(m − 3q)
in terms of which
s = πq
3
2ρ
3
4
T =√qρ
1
4
π
(6.12)
We will see below that these results correctly reproduce the leading order thermody-
namics of hairy black holes.
6.4 Hairy Black Hole
Once we have our solutions for hairy black holes from Appendix A, the evaluation of their
thermodynamic charges and potentials is a straight forward exercise. At low orders in the
– 49 –
Download full-text