#### A systematic construction of microstate geometries with low angular momentum

HJE
A systematic construction of microstate geometries with low angular momentum
Iosif Bena 0 1
Pierre Heidmann 0 1
Pedro F. Ram rez 0
Gif-sur-Yvette 0
France 0
Instituto de F sica Teorica UAM/CSIC 0
0 C/ Nicolas Cabrera , 13-15, C.U. Cantoblanco, 28049 Madrid , Spain
1 Institut de Physique Theorique, Universite Paris Saclay, CEA , CNRS
We outline a systematic procedure to obtain horizonless microstate geometries that have the same charges as three-charge macroscopically-large horizon area and an arbitrarily-small angular momentum. There are two routes through which such solutions can be constructed: using multi-center GibbonsHawking (GH) spaces or using superstratum technology. So far the only solutions corresponding to microstate geometries for black holes with no angular momentum have been obtained via superstrata [1], and multi-center Gibbons-Hawking spaces have been believed to give rise only to microstate geometries of BMPV black holes with a large angular momentum [2]. We perform a thorough search throughout the parameter space of smooth horizonless solutions with four GH centers and nd that these have an angular momentum that is generally larger than 80% of the cosmic censorship bound. However, we nd that solutions with three GH centers and one supertube (which are smooth in six-dimensional supergravity) can have an arbitrarily-low angular momentum. Our construction thus gives a recipe to build large classes of microstate geometries for zero-angular-momentum black holes without resorting to superstratum technology.
Black Holes in String Theory; String Duality; Superstring Vacua
1 Introduction 2
Supertubes and microstate geometries
Supersymmetric solutions with a Gibbons-Hawking base
Symplectic transformations
Three-supertube scaling BPS solutions in Taub-NUT
Microstate geometries from three-supertube con gurations
B
C
2.1
2.2
2.3
2.4
4.2
4.3
3
Four-GH-center solutions with a hierarchy of scales
3.1
Exploration of the parameter space
3.1.1
3.1.2
Systematic generation of solutions
Main results of the analysis
3.2
A particular solution
3.2.1
Scaling the solution
4
A supertube with three Gibbons-Hawking centers
4.1
Exploration of the parameter space
4.1.1
4.1.2
Systematic generation of solutions
Main results of the analysis
An example of a solution without scale di erences
4.2.1
Scaling solutions
A solution with very small angular momentum
4.3.1
Scaling solutions
A Solving the BPS equations
Mathur in 2003,1 has been reinforced by recent information-theory based fuzzball/ rewall
1See [4{6] for reviews.
{ 1 {
arguments2 that establish that the only way a black hole can release information without
a violation of Quantum Mechanics is if there exists a structure that modi es the physics
at the scale of the horizon.
The only construction of such a structure when gravity is present has been done in
the context of the \microstate geometries programme" that aims to construct horizonless
solutions with black hole charges purely within supergravity. Since supersymmetry
significantly simpli es the equations governing the solutions, most microstate geometries that
have been constructed so far correspond to supersymmetric black holes [1, 2, 9, 10].3
We will focus on the rotating three-charge BPS black hole in ve dimensions, known
HJEP10(27)
as the BMPV black hole, which has two equal angular momenta satisfying the cosmic
censorship bound J1 = J2
pQ1Q2Q3. This solution can be embedded in string theory as
a black hole with three M2 brane charges, corresponding to M2 branes wrapping three
2tori inside a 6-torus. In another duality frame, the three charges correspond to D1 and D5
branes that share a common direction, and momentum, P, along this direction. In the later
duality frame one of the charges gives rise to a nontrivial bration of an internal direction
over spacetime, so the black hole and microstate geometries thereof are asymptotically
example of the way in which most singularities are resolved in String Theory [17{19]. The
resulting solutions are smooth and horizonless.
A convenient choice of four-dimensional base space is given by the Gibbons-Hawking
family of spaces, whose tri-holomorphic U(1) isometry implies that all solutions are
determined by harmonic functions in R3 [20{22]. To obtain singularity-free horizonless solutions
the poles of the harmonic functions must satisfy certain relations [23{25], and the sizes and
positions of the bubbles are also constrained by the absence of closed timelike curves via
the so-called bubble equations [23, 26].
Only a few explicit examples of smooth horizonless solutions which have the same
charges an angular momenta as a BMPV black hole with a macroscopically-large horizon
area are known [1, 2, 9, 10], and this is because most solutions one can construct by putting
uxes on a multi-center Gibbons-Hawking base have an angular momentum larger than
the black hole cosmic censorship (cc) bound. This was rst discovered in [27], where it
was pointed out that smooth multi-center BPS solutions with a GH base with a large
number of centers have angular momenta at and slightly above the cosmic censorship
bound. Furthermore, in [2], a generic recipe was given to construct solutions with four GH
centers that have angular momenta below the c.c. bound; however, when the aspect ratios
of the distances between the centers are of the same order, all these solutions were found to
2See, for instance, [7, 8].
3However, one can also construct microstate geometries corresponding to extremal non-supersymmetric
black holes [11{13] and to non-supersymmetric and non-extremal black holes [14{16].
{ 2 {
have J at 99% of the cc bound. Thus, trying to nd multi-GH-center microstate geometries
with low angular momentum appears to resemble searching for a needle in a haystack.
The rst obstacle is to nd an appropriate class of multi-center solutions with no closed
timelike curves (ctc's). Since all bubbling solutions have charges dissolved in
uxes, and
since these uxes have di erent signs, the most likely outcome of trying to obtain a solution
by putting random values of uxes on various cycles is a solution with regions of positive
and negative charge densities. Such solutions are not supersymmetric, and imposing a
supersymmetric ansatz on them gives in general a solution with ctc's. Furthermore, since
the ux on every cycle interacts with the ux on every other cycle, making sure there are no
regions of negative charge density is a very complicated problem, that has not been solved
yet.4 To bypass this problem, one of the authors proposed a recipe to construct generic
ctcfree solutions with four centers, starting from ctc-free solutions describing three supertubes
in Taub-NUT, going to a scaling limit, and performing a combination of spectral ows and
gauge transformations to transform the supertube centers into smooth GH centers without
introducing ctc's [29]. One can then use the fact that the solutions have a scaling limit to
remove certain constants in the harmonic functions and obtain asymptotically-R4;1 smooth
horizonless solutions with four GH centers and BH charges [2].
This recipe is e cient because it is relatively easy to obtain ctc-free solutions with
three supertubes of di erent kinds and a GH center: unlike solutions with GH centers, the
charges of these solutions come from the supertubes themselves, and hence by ensuring
that the supertube charges are positive one avoids ctc's. Furthermore, since any solution
with four GH centers can be transformed via spectral
ows into a solution with three
supertubes and a GH center, the method of [2] is guaranteed to yield the most generic
ctcfree solutions with four GH centers. Moreover, if one performs this procedure and uses only
two spectral ows, one obtains the most generic ctc-free solution with three GH centers and
a single supertube, which is singular in the M2-M2-M2 ( ve-dimensional) duality frame but
is smooth in the D1-D5-P (six-dimensional) duality frame.
The second obstacle is to implement a lter for solutions with angular momentum at a
nite fraction of the cc bound. Indeed, starting from generic three-supertube solutions will
almost always produce solutions with J slightly below this bound, and hunting for solutions
with a parametrically-lower J is challenging. To do this one has to nd physical quantities
which will discriminate three-supertube solutions that will produce near-maximally
spinning 4-GH-center solutions from those that will produce 4-GH-center solutions with lower
angular momentum. To do this, it is useful to follow the procedure of [2] and introduce
the so-called entropy parameter :
H
bound of the black hole with the same charges.5
4In [28] a strategy to solve this problem will be proposed.
5Of course, microstate geometries have no horizons and their angular momentum can easily be above
black holes with a nite H parameter (typically 0.4 with a maximum around 0.6).6 The key
di erence between the geometries with four GH centers we construct and those of [2], which
have H < 10 2, is that the aspects ratios of the new geometries are parametrically larger
than one. Given that the only other known method for constructing
nite-H
multi-GHcenter microstate geometries, via mergers of clusters of bubbles [9] also produces solutions
with parametrically-large aspect ratios, this appears to be a universal feature of
multi-GHcenter solutions with angular momenta signi cantly below the cc bound. It would be very
interesting to nd a deeper physical reason for this.
Our technology can also be used to produce solutions with three GH centers and a
supertube, that have zero or very small angular momentum. So far, the only method to
obtain such BTZ microstate geometries has been to use superstratum technology [1, 30],
which is technically much more di cult than the construction of solutions with a GH base
space. Furthermore, this technology produces asymptotically-AdS3
S3 geometries [1],
and extending these solutions to obtain asymptotically- at D1-D5-P microstates is quite
nontrivial [31]. In contrast, our technology produce very easily large classes of smooth
asymptotically- at zero-angular-momentum black-hole microstate geometries.
The trade-o
is that the CFT
dual of superstratum
solutions is exactly
known [1, 30, 32, 33] (which makes superstrata amenable to precise holographic
investigations), while the CFT dual of any solution with more than two GH centers is not known.
The multi-center solutions we obtain do have a scaling limit, so they have a throat that
can resemble a black hole throat to arbitrary accuracy; hence one can argue that they are
dual to CFT states that have long e ective strings [9] and therefore live in the same CFT
sector as the states that count the black hole entropy. However, identifying these states
precisely remains a challenging open problem.
The method we employ reveals itself as a very powerful tool to study the spectrum of
four-center microstate geometries. It will be interesting to be able to perform similar studies
for even more general solutions, with an arbitrary number of centers or with the inclusion
of non-Abelian elds [25]. We plan to adress these questions in future work [28, 34].
In section 2 we summarize the structure of the two classes of four-center solutions we
study: solutions with four GH centers or with three GH centers and one supertube, and we
explain how to generate them using generalized spectral ows and gauge transformations
on solutions with three supertubes in Taub-NUT. In section 3 we present an exhaustive
analysis of solutions with four GH centers. We show that imposing a hierarchy of scales
between the inter-center distances is a necessary ingredient to construct solutions with an
angular momentum signi cantly below the cc bound. In section 4 we apply the same kind
of analysis on solutions with three GH centers and one supertube and construct microstate
geometries for black holes with arbitrarily-small angular momentum.
nonetheless because it facilitates the comparison between the microstate geometry and the corresponding
black hole.
has H
0:28.
6The only other known microstate geometry with multiple GH centers and low angular momentum [9]
{ 4 {
2.1
Supersymmetric solutions with a Gibbons-Hawking base
We work in the context of ve dimensional N = 1 Supergravity coupled to two vector
multiplets in the STU model.7 This theory has been shown to be obtained from compacti cation
of eleven dimensional Supergravity on a Calabi-Yau threefold [37].8 Its supersymmetric
solutions with a compact spatial isometry are completely speci ed in terms of a set of 8
harmonic functions in R3, which we take of the form
V = q
1 + X qa
a ra
; KI = k1I + X k
a
I
a ;
ra
; LI = l1I + X l
I
a ; (2.1)
1 and charges a,9 such that
where ra is the Euclidean three-dimensional distance measured from the center with
coordinates ~xa, and with I = 1; 2; 3. It is convenient to introduce a vector with the harmonic
(V; KI ; LI ; M ), which implicitly de nes a set of vectors of asymptotic
con=
1 +
a
ra
:
In this article we are mostly interested in the spacetime metric and its properties, so
we shall focus on this aspect of the solution. We refer the reader to the appendix A for
a description of the complete eld content and the solving of the BPS equations. The
ve-dimensional metric is given by
ds52 =
(Z1Z2Z3) 2=3 (dt + k)2 + (Z1Z2Z3)1=3 ds42;
where ds24 is a four-dimensional ambipolar Gibbons-Hawking space [41, 42]
{ 5 {
ds42 = V 1 (d
The warp factors ZI and the 1-form k are given by
with CIJK = j IJK j and
1
6
?(3)d! = h ; d i
:
=
V 2CIJK KI KJ KK +
V 1KI LI + M ;
1
2
the STU model.
by a truncation [38{40].
9For instance, a = qa; ka1; ka2; ka3; la1; la2; la3; ma .
7Our conventions mostly coincide with those of [5]. See [35, 36] for information about the theory and
8Alternatively, it can be obtained from the compacti cation of Heterotic Supergravity on T 5 followed
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
ned as10
In the last expression h ; i is a symplectic product of vectors A =
A0; AI ; AI ; A0
dehA; Bi
and microstate geometries later, we emphasize here that all physically sensible solutions
need to be free of closed timelike curves and Dirac-Misner strings. The rst condition
requires the positivity of the quartic invariant I4 (see appendix A),
I4
while the second restricts the position of the centers [26],
b
rab
where rab is the distance between the pair of centers located at ~xa and ~xb. These are known
as the bubble equations and impose strong constraints on the space of parameters leading
to physically sensible con gurations. Solving those equations is usually the hardest step
when building multi-center solutions.
2.2
Symplectic transformations
Any vector of harmonic functions de nes a solution, and any linear transformation, 0 = g
with g 2 GL(8; R), maps a solution to another solution. A special subgroup of these
transformations is Sp(8; R), corresponding to linear transformations that preserve the symplectic
product and, therefore, leave the bubble equations invariant. Among all possible Sp(8; R)
transformations, the most attractive are those that also leave the function I4 invariant.
We are interested in two subgroups with these characteristics [43]:
Generalized spectral
ows.
These transformations can be understood as simple
changes of coordinates when the ve dimensional solution is embedded in six
dimensional Supergravity [29], and correspond to a subgroup of the E7(7) duality
transformations from the eleven dimensional perspective [13]. Generalized spectral ows are
generated by three real parameters I
M 0 = M;
KI 0 = KI
Even though they act non-trivially on ZI and , one can check that I4 and the bubble
equations remain invariant under the action of (2.11).
10Another more symmetric convention where the harmonic function M is twice the one we use here is
also used in the literature.
{ 6 {
(2.8)
(2.9)
(2.10)
(2.11)
HJEP10(27)
Gauge transformations. These transformations leave the physical properties of the
solution unchanged and their sole e ect is a gauge transformation of the vector elds.
They are just a re ection of the fact that the construction of solutions in terms of
8 harmonic functions contains redundancies. There are three independent gauge
transformations (one for each vector) parametrized by gI , acting as
V 0 = V;
to get rid of the constant terms appearing after spectral ows in the functions KI ,
since these introduce singularities for ve dimensional asymptotically- at solutions.
Consequently, generalized spectral ows and gauge transformations can be used to
generate smooth horizonless geometries starting from a three-supertube solution in
a Taub-NUT hyper-Kahler space. This plays a central role in our study.
There is one additional subgroup of Sp(8; R) that leaves I4 invariant that involves
rescalings of the harmonic functions, but since we are not going to make use of this type
of transformations we refer the interested reader to [43].
2.3
Three-supertube scaling BPS solutions in Taub-NUT
Our starting point is a system of three two-charge supertubes of di erent species in which
the 4-dimensional hyper-Kahler metric is the Euclidean Taub-NUT solution [44, 45]. This
is a multi-supertube generalization [46] of the con guration constructed in [47]. However,
since the supertubes are of di erent kinds, this con guration is not smooth in the D1-D5-P
duality frame.
are given by
Each supertube carries a dipole charge kI and two electric charges Q(aI) at the centers
a 6= I. Consequently, the 8 harmonic functions that characterize such a eld con guration
(2.13)
(2.14)
(2.15)
(2.16)
V = q
1 +
KI =
q0
r0
;
I + X3 ka I
a=1 ra a ;
a=1
4ra
3
a=0 ra
1
:
M = m1 + X ma
aI ;
{ 7 {
In these expressions ra is the three-dimensional Euclidean distance measured from the ath
center. We consider axisymmetric supertube con gurations. The positions of the supertube
Therefore,
In the analysis performed in [2, 12, 46] it was derived that regularity at the centers and
the absence of asymptotic Dirac-Misner strings requires xing the following parameters
m1 =
achieved imposing the four bubble equations (2.10), which x the positions of the centers,
and the global bound (2.9). The bubble equations can be conveniently written as
2 =
3 = 0 :
centers are given by the distances z1, z2 and z3 on the z-axis of the three-dimensional base
space of the solution in the following order
(2.17)
(2.18)
(2.19)
(2.20)
(2.21)
r12
r12
r23
ab = h a; bi and rab is the distance between the centers a and b. Provided
those conditions are satis ed, we have a family of regular solutions free of ctc's labeled
by eight parameters; kaI, QIa, q
1 and q0. However, one should not expect the whole space
of parameters to be compatible with (2.9) and (2.20). Moreover, experience shows that
nding a set of appropriate parameters can involve a vast exploration.
Among all possible physical solutions, the most interesting correspond to scaling
geometries. These are con gurations in which the distances between the supertubes and
the GH center can be made arbitrarily small while preserving the value of the asymptotic
charges practically constant. If one de nes the aspect ratios dI as zI =
dI with d3 of order
one, this is achieved in practice for con gurations in which the terms on the left-hand side
of (2.20) are almost vanishing when we replace the inter-center distances by the aspect
ratios. Thus, scaling solutions of three supertubes and a GH center must satisfy the scaling
conditions:
12 +
d12
d12
d23
21 +
32 +
13
d13
23
d23
31
d13
8q0
8q0
8q0
m1
d1
m2
d2
m3
d3
0
0
0;
{ 8 {
with zIJ =
dIJ .
When these relations are satis ed, the limit
1 in the bubble
equations is well-de ned at rst order in . By summing the three equations (2.21) we see
that m1, m2 and m3 cannot be all of the same sign. Since all Q(I) are taken positive to
J
avoid ctc's one of the dipole charges k1, k2 and k3 must have di erent sign from the other
two. The warp factors of the solution (2.5) have a term quadratic in the dipole charges,
and when the kI have opposite signs this term is negative and can be problematic. However
we avoid this by choosing the kI to be smaller than the square roots of the charges.
As one approaches the scaling limit, the AdS2
S3 throat of the solution becomes
longer, and the solution resembles more and more the near-horizon geometry of an extremal
black hole. Therefore after the application of spectral ow transformations that render the
metric smooth at the centers we will construct a completely regular, horizonless solution
with near-horizon-like throat of large but nite depth that caps o smoothly.
2.4
Microstate geometries from three-supertube con gurations
Several classes of smooth BPS solutions can be generated through the application of two or
three generalized spectral ows and gauge transformations on a system of three-supertubes
in a Taub-NUT space, see for instance [2, 29, 46]. In this manner we can investigate large
classes of regular supersymmetric solutions with multiple Gibbons-Hawking centers, which
are usually di cult to generate otherwise. As we explained in section 2.2, no closed timelike
curves or Dirac-Misner strings are generated in this process.
However, this method to generate smooth microstate geometries presents a drawback.
It has been recently argued that generalized spectral ows result in a signi cant increase
of the angular momentum, at least when the inter-center distances are of the same order
of magnitude for four-center solutions [2]. Actually, when all the distances between the
centers are of the same order the solutions are near-maximally spinning. In particular,
while spectral ows do not modify the quartic invariant, it seems that they simultaneously
increase the value of the two terms in its de ning expression (2.9).
To be more precise, recall that we de ned the entropy parameter H in (1.1) as:
H
where QI and J are the asymptotic charges and angular momentum. For classical black
holes the entropy is proportional to horizon area, given by the square root of the numerator.
The numerator can also be read o from the coe cient of the 1=r4 in the quartic invariant.
If the numerator is negative, the black hole solution will be singular. Thus, H is 0 when
rotation is maximal, while it is 1 when there is no rotation at all.
Within this construction scheme, there are two possible strategies that we are going
to explore in order to avoid angular momenta near the cosmic censorship bound. The rst
possibility is to look for con gurations in which there is a hierarchy in the distances between
centers. The second option simply consists in applying two spectral ows instead of three.
Both approaches involve a large exploration of the parameter space, since there is no way,
in principle, to know how the input parameters should be chosen to produce a high value
{ 9 {
1. Four smooth Gibbons-Hawking centers.
of the entropy parameter. On the bright side, the procedures to generate smooth solutions
can be systematized, as we will brie y explain, and therefore such an exploration is feasible.
If one applies the three possible generalized spectral ow transformations to an initial
solution with three supertubes in Taub-NUT, one obtains a four-GH-center con guration
described by a set of harmonic functions with
laI =
ma =
1
This guarantees that the resulting solution is horizonless and smooth [5, 23, 24].
2. One supertube and three smooth Gibbons-Hawking centers.
Let us consider the application of two types of generalized spectral ows (2.11) to an initial
system of three supertubes in Taub-NUT. For instance let us denote by the index J the
spectral ow transformation which is not applied, so
J = 0. Then it is straightforward
to check that the set of four-center harmonic functions obtained satisfy
laI =
ma =
kJI = 0
lJJ = 0 ;
mJ =
2 CIKL kaK kaL
1
qa
12 CIKL kaIkaK kaL
1
q2
a
4 CJKL lJKkJlJL ;
1
J
a 6= J ;
a 6= J ;
I 6= J ;
where the notation is that of (2.1).
This con guration describes a supertube in the
presence of three smooth Gibbons-Hawking centers. Much like vanilla two-charge
supertubes [48{52], these solutions are not smooth in the M2-M2-M2 duality frame where they
are described by
ve-dimensional supergravity. However, they become smooth once one
dualizes them to a D1-D5-P duality frame where the supertube charges correspond to D1
and D5 branes, and the solution can be described by a six-dimensional supergravity [53].
3
Four-GH-center solutions with a hierarchy of scales
In this section we explore the possibility of constructing smooth geometries with four
Gibbons-Hawking centers with an angular momentum far below the cosmic censorship
bound. We will see that when the analysis technique of [2] is applied to solutions in which
the inter-center distances have a hierarchic structure, it is possible to build solutions with
small angular momentum. The particular
ve-center solution found in [9] provides the
motivation to study this type of con gurations in detail, since it is characterized by a
hierarchic distribution of the centers and its entropy parameter is H = 0:28.
(2.22)
HJEP10(27)
(2.23)
The details of the numerical analysis we perform are contained in appendix B. Here we
give a qualitative description of the procedure pioneered in [2] and explain our results.
The program is based on the automation of the method to build four-center microstate
geometries described in the previous section. This allows us to scan the space of parameters
and look for the maximization of the entropy parameter H.
Systematic generation of solutions
Let us discuss how the solution generating technique is automatized. Before proceeding, we
point out some generalities about this construction scheme. In rst place, we are interested
in con gurations that present a hierarchy of scales, which at the beginning we take to be
HJEP10(27)
z1
z2
z2
z3
Q(2)
Q(2) and Q(3)
3
1 3
The Gibbons-Hawking metric (2.4) is fully determined by the function V . Although it
might seem that this metric becomes singular at the centers, it can be easily checked that
this is not true as long as the coe cients qa are integer numbers.11 Then, we need to
take care of that fact and impose that all Gibbons-Hawking charges are integer numbers.
Moreover their sum has to be necessarily 1, since we want this space to asymptote to R4.
On the other hand, the application of spectral ow transformations to a system of
supertubes does not guarantee a good asymptotic behaviour. In particular we are interested
in asymptotically- at 5-dimensional spacetimes. However this type of con gurations will
never be obtained directly if we start from a system of supertubes in a Taub-NUT base
space, since one cannot eliminate simultaneously all constant factors in the functions V
and KI . The best one can do is to perform three gauge transformations (2.12) to eliminate
the integration constants in KI , and remove by hand the constant in V afterwards, hoping
that this does not generate ctc's.
The initial system of supertubes is speci ed by seven parameters: k1, k2, k3, q0, QQ((213)) ,
1
Q(21) . The solution also depends on q , but the value of this parameter is not
1
essential when looking for scaling solutions. Our recipe is the following:
1. Choose a value for the seven degrees of freedom of the three-supertube solution, k1,
k2, k3, q0, QQ((213)) , Q(2)
3
1 1
Q(2) ,
Q(3)
1
the k's has a sign di erent from the other two. We also give a non-vanishing value
to q , which is necessary in order to be able to cancel the constant terms of all KI
in a later step. Therefore, the base space is Taub-NUT.
Q(21) . Recall that we can only obtain scaling solutions if one of
3
11This fact becomes evident performing a local coordinate transformation ra = 4
describes the orbifold space R4=Zjqaj, which is harmless in the context of string theory.
a . The local metric
2. Using (3.1) as an equality, we impose the scaling condition (2.21) as three exact
equations from which we obtain the precise value of all the Q(aI) parameters.
Afterwards, we round these values to some close rational numbers and solve the bubble
equations (2.20) to determine the positions of the centers z1, z2 and z3. Thus, (3.1)
and (2.21) cease to be equalities and become approximations, as intended. This step
ensures that we construct a scaling three-supertube solution free of ctc's.
3. We perform three generalized spectral ows and three gauge transformations. We x
the values of the spectral ow parameters I by imposing some particular, integer
values of the Gibbons-Hawking charges q1, q2 and q3 such that P qa = 1. The values
of the gauge parameters gI are found requiring that the constant terms in all the
At this stage, we have a BPS scaling solution with four Gibbons-Hawking centers.
However, there are still two problems that need to be solved. First, the harmonic
function V still has a constant term. This means that the four-dimensional base space
of the solution is asymptotically R
3
S1 instead of at R4. Second, because all the
parameters of the transformations I and gI are xed by polynomial equations, the
resulting charges and dipole charges of the solution are general real numbers. Since
those are expected to be quantized when interpreted in the full context of string
theory, it is desirable that they take integer values.
For the numerical analysis of the entropy parameter, we do not apply the next three
steps because they do not signi cantly change the value of the charges and the
angular momentum. They are just technical steps to build proper
asymptotically-5dimensional solutions.
4. It is not possible to remove the constant of V using transformations that preserve the
bubble equations. Thus we remove it by hand. The impact of this removal on the
solution takes place mainly on the bubble equations. Changing the right hand side of
the bubble equations (2.10) necessarily results in a change of the inter-center distances
in the left hand side. In the scaling limit, when all these distances are very small, one
may think that a change of constant terms can be compensated by an in nitesimally
small change of distances. However, this it is not necessarily true for axisymmetric
con gurations [
54
]. In our construction we will carefully select the solutions for
which it is possible to perform this truncation preserving the axisymmetry of the
center con guration.
5. Since we want the monopole and dipole charges to be integer numbers, we proceed
in two steps. The rst step consists in obtaining solutions whose harmonic functions
have rational poles. For that purpose, we round the values of the parameters kaI to
be rational and obtain all the other charges laI and ma using (2.22). Since one can
nd rational numbers arbitrarily close to any irrational number, this procedure is
guaranteed not to change signi cantly the properties of the solution. Hence, we have
a fair bit of freedom in rounding the irrational numbers to rational ones, and we can
use it to obtain kaI that have the same denominator. This rounding does not leave
the bubble equations invariant, and we need to solve them again and check again the
absence of ctc's. The second step is to obtain solutions whose harmonic functions
have integer poles. To do this we use the following transformations parametrized by
any real numbers fs1; s2; s3g,
1
V ! V;
M ! 6 CIJK sI sJ sK M;
1
L
I
KI
! 2 CIJK sJ sK LI ;
! sI KI ;
fs1; s2; s3g 2 R3:
(3.2)
HJEP10(27)
They preserve the regularity of the solution. Indeed, all the horizonless
conditions (2.22) are still satis ed and the bubble equations and the quartic invariant
are multiplied by an overall factor s1s2s3 and (s1s2s3)2 respectively while H does not
change. Thus, one chooses the three sI to be the smallest integers needed to obtain
integer charges from the rational charges.
6. The factors sI are usually large numbers, so multiplying the harmonic functions LI
and M by them makes their constant terms very large. Asymptotic atness of the
ve-dimensional metric (2.3) demands having the constant terms of all LI equal to
one.12 To obtain such solutions one again has to change by hand the constant terms
of all the LI . As explained in [
54
], such a change can always be done for scaling
solutions, and results in a global dilatation of the multicenter con guration. To
make the inter-center distances small again, we simply ne-tune the value of some of
the dipole charges (keeping them integer) to make the solution scale [9].
This method produces asymptotically- at, scaling solutions with four
GibbonsHawking centers that have integer charges. Using this systematic procedure we can build a
huge number of four-GH-center solutions and obtain the variation of the entropy parameter
H as one moves in the parameter space spanned by k1; k2; k3; Q(3) ;
1
Q(1)
2
Q(2)
3
Q(2) ;
1
Main results of the analysis
We divided our analysis in three parts, considering the e ect of modifying three sets
of parameters: the Gibbons-Hawking charges (q0; q1; q2), the supertube dipole charges
clusions:
(k1; k2; k3) and the supertube charge ratios ( Q(23) ;
1
Q(1)
Q(2)
3
Q(2) ;
1
Q(3)
Q(21) ). We reach the following
con3
The entropy parameter approaches zero drastically when the absolute value of the
Gibbons-Hawking charges is large. The optimal value we observed for the
GibbonsHawking charges is 1,1,1 and -2.
For the initial supertube dipole charges, we observed that con gurations with k2
negative and k1 and k3 positive are the optimal ones.
With the two other sign
con gurations, we did not nd domains of charge ratios with an entropy parameter
12Actually only their product has to be equal to one, but this subtlety is not particularly relevant.
bigger than 0.1. We also noticed that the entropy parameter does not depend signi
cantly on k2 and it depends essentially on kk13 . Furthermore, we observed that for any
charge ratios one can
nd a particular dipole ratio kk13 where the entropy parameter
is maximal and the upper bound seems to be H
0:3.
With the optimal con guration of dipole charge signs and Gibbons-Hawking charges,
we have found several domains of charge ratios where the entropy parameter is
above 0.2.
Moreover, we performed an analysis to study the impact of the hierarchy of scales. In
gure 1, we show one of the main results of the analysis. It illustrates how the entropy
parameter can signi cantly increase with the aspect ratios. The entropy parameter is
represented with respect to two variables, one of the charge ratios and the order of magnitude
of the hierarchy m, which is de ned as
z2
z3
The rest of parameters are chosen to optimize the entropy parameter, according to the
numeric results just presented (see appendix B for more details). The graph shows that when
m is around 0 the solutions are near-maximally spinning, with H very close to 0, recovering
the results of [2]. Furthermore, in all the solutions we examined the entropy parameter
increases as the hierarchy between the distances gets more pronounced, converging toward
a value below one. We have con rmed that this is a general behavior for several other
domains of the parameter space.
The analysis performed supports the conclusion that microstate geometries with an
angular momentum that is at a
nite fraction of the cc bound must have a di erence in
scale between their inter-center distances.
3.2
Here we give the explicit form of the harmonic functions characterizing a BPS scaling
microstate geometry with four Gibbons-Hawking centers. The solution has been found
following the recipe detailed in section 3.1.1, taking the initial parameters from the region
that optimizes the value of the entropy parameter according to the results of the numerical
analysis.
The solution is determined by the following harmonic functions,
V =
K1 =
K2 =
K3 =
L1 = 1
The bubble equations can be solved numerically for the location of the centers,
Performing an asymptotic expansion of ZI and
we can obtain the three electric charges
and the angular momentum of the solution, which can be read from the O(r 1) coe
cients [9, 24]
For these values of asymptotic charges the entropy parameter is
Q1 = 1993340
Q2 = 29014
Q3 = 229906
J =
87655680:
H = 0:42 : : :
While this value is not close to 1, we can de nitely a rm that it is far from 0. Thus,
this microstate geometry corresponds to a rotating black hole whose angular momentum
is signi cantly below the cc bound.
(3.4)
(3.6)
(3.7)
100.00046
100.0004639
5:0152
4:6445
1:9403
1:3199
3:5190
524:33
524:33
524:33
524:33
524:33
150:38
150:38
150:38
150:38
150:38
In general, one might need to break the axisymmetry of the con guration to scale the
solutions. However, as it was proposed in [9], axisymmetry can be preserved in the scaling
process by slightly modifying the values of one of the parameters in the KI functions. Here,
we choose to dial the value of k21 but any other dipole charges could have worked. At each
step one can check the bubble equations and the absence of ctc's. The scaling process is
summed up in table 1.
As explained in [9], in the scaling process the microstate geometry develops a \throat"
that resembles the near-horizon geometry of an extremal black hole to increasing accuracy.
The depth of this throat gets larger and larger as the cluster of centers shrinks. Since
a BPS black hole has an in nite throat, during the scaling process the bubbling solution
becomes more and more similar to the exterior of the black hole solution. Therefore, we
have found a speci c example of an asymptotically
at, scaling microstate geometry with
four Gibbons-Hawking centers that corresponds to a microstate of a BMPV black hole with
H = 0:42.
4
A supertube with three Gibbons-Hawking centers
As we already mentioned in section 2.4, BPS scaling solutions with one supertube and
three Gibbons-Hawking centers can be generated from three-supertube solutions in
TaubNUT. These con gurations are interesting because they correspond to smooth horizonless
microstate geometries in the D1-D5-P frame. We follow the same approach as in previous
section. First, we explain how such solutions can be systematically generated and we
perform a numerical analysis of the dependence of the entropy parameter on the initial
parameters. Second, we present explicit examples of solutions with and without scale
di erences between the four centers.
To obtain our solutions one only needs to apply two generalized spectral ows to the
original system of three supertubes. As we have seen, spectral ow transformations are
responsible for decreasing the entropy parameter, so one may hope to
nd solutions with
low angular momentum even without imposing a hierarchy of scales in the inter-center
distances.
4.1
Systematic generation of solutions
We start from solutions that do not have a hierarchy of scales:
z1
z2
z3
z3
z2
z3
1;
1:
(4.1)
The technique to generate these con gurations is very similar to the method of [2],
reviewed in detail in section 3.1.1. The only di erence is that one of the generalized spectral
ows is not applied. In a nutshell, the starting point is a three-supertube con guration
with a Taub-NUT base space satisfying (4.1), which is characterized by seven parameters
Q(1) and q0. Then, we apply any two generalized spectral ows. The
corresponding parameters, say J and
K , are xed by imposing a particular value for the
Gibbons-Hawking integer charges generated in the process qJ and qK which are free as long
as P qa = 1. Then, we apply three gauge transformations to cancel the constant terms
of the KI . Finally, we truncate the constant term of the harmonic function V to obtain
a base space asymptotic to R4 and we round to integers all the charges in the harmonic
functions. By systematizing this procedure it is possible to scan vast classes of solutions,
parameterized by k1; k2; k3; Q(3) ;
2
1
Q(1)
Q(2)
Q(2) ;
3
1
The same procedure can also be applied to solutions with a hierarchy of scales. As
we saw in previous section, increasing the scale di erence has a signi cant impact on the
entropy parameter of four-GH-center solutions, which can reach values of the order of
0:5. It is natural to ask how large this parameter can be for solutions with three
H
Gibbons-Hawking centers and one supertube.
4.1.2
Main results of the analysis
The details of the numerical analysis are contained in appendix C. After scanning relevant
domains of the space of parameters, we have reached the following conclusions when looking
for the best value of H:
The optimal location of the supertube is the outermost one: (0; 0; z1).
The Gibbons-Hawking charges qa should have the smallest possible absolute value,
jq2j = jq3j = jq0j = 1, in agreement with what we found in section 3.1.2.
All sign con gurations for the initial dipole charges ka appear to be equally favored.
We nd that, when k2 is taken negative, the entropy parameter reaches a maximum
for a particular value of kk13 , regardless of the values of the other parameters.
For aspect ratios satisfying (4.1), the maximal value of H is around 0:25.
The analysis con rms what we anticipated: when only two generalized spectral ows
are performed, the resulting solutions have lower angular momentum. Thus, one can reach
a nite value of H even without a hierarchy of scales.
k2 = k3 = 1, QQ((2113)) = 0:85 and QQ((2331)) = 0:009.
the order of magnitude of the inter-center distance ratio. The other parameters are q0 = q2 = 1,
Of course, we just found in the previous section that hierarchic con gurations can
improve the value of the entropy parameter, at least for four GH centers. So we would like
to investigate how adding a hierarchy of scales a ects the angular momentum of solutions
with one supertube. For that purpose, let us de ne the variable m as we did in the previous
section,
z1
z2
z2
z3
10m
10m:
(4.2)
We can then evaluate the value of the entropy parameter for a large set of solutions with
di erent values of m and the charge ratio Q(2)
values according to the analysis performed for m
0. The result is very surprising. As
the value of m increases the value of the entropy parameter improves signi cantly and can
stay arbitrarily close to H = 1 in a large region of the moduli space. This maximal value
is obtained for m
1:5, so the hierarchy of scales is not too pronounced. Unexpectedly,
the value of the entropy parameter decreases if we go beyond that optimal hierarchy, see
Q(32) . The other parameters are xed to optimal
1
gure 2.
Solutions with m
1:5 are non-spinning. Indeed, one can nd ctc-free scaling
solutions with one supertube and three Gibbons-Hawking centers for which the spectral ow
transformations completely annihilate the original angular momentum. However, those
solutions typically have irrational charges.
To obtain solutions with integer charges and uxes, one has to rst round these charges
to nearby rational ones, and this typically brings back some angular momentum. However,
the value of this angular momentum is proportional to the rounding, and hence can be
made arbitrarily small by tightening the rounding. Hence, one can
nd regular scaling
solutions with an entropy parameter in nitesimally close to one. In section 4.3 we give an
explicit example of such solutions.
V =
K1 =
K2 =
K3 =
+
r1
+
+
1836
r2
r3
+
r1
2194116
24187
r3
294543
r2
+
1916188
r3
23769590
10014462
64192298
r2
+
r3
:
where ra are Euclidean three-dimensional distances measured from the centers at (0; 0; za).
These locations are obtained solving numerically the bubble equations, which yield
z1 = 1:0635 : : : 10 2 ;
z2 = 7:1863 : : : 10 3 ;
z3 = 3:5109 : : : 10 3:
(4.4)
The three global electric charges and the angular momentum are
Following the procedure outlined in section 4.1.1 we can easily construct solutions with one
supertube and three GH centers. For example:
(4.3)
0:24;
The entropy parameter of this solution is
which means that the angular momentum, J , is at 87% of its maximal value for those
electric charges.
4.2.1
Scaling solutions
Following the procedure outlined in section 3.2.1, we scale the solution by ne-tuning the
value of k21. At each step in the scaling process, we solve the bubble equations and check
for the absence of ctc's. The results are summed up in table 2.
k
-184.000034
-184.00003411
-184.000034117
-184.000034117128
1:2834
3:6513
2:2225
4:0366
4:6524
0:98237 1:0469
0:98236 1:0469
with aspect ratios of order one.
A solution with very small angular momentum
Here we build a solution with one supertube and three Gibbons-Hawking centers which has
an entropy parameter H
1. For this purpose, we choose appropriately the scale di erence
between the inter-center distances and the values of the initial charges and dipole charges
of the three supertubes to maximize the entropy parameter. Our procedure allows us to
ne-tune the parameters to have H in nitesimally close to 1, and we present an example
with H = 0:999997:
2
3
4
5
6
(4.7)
(4.9)
V =
K1 =
K2 =
K3 =
The bubble equations give the positions of the centers:
The three charges and the angular momentum are:
z1 = 7:3189 : : : 10 2 ;
z2 = 3:6046 : : : 10 3 ;
z3 = 9:7241 : : : 10 5:
(4.8)
+
73830
r3
18981
r0
381142
r1
+
22479930
r2
+
3100860
r3
:
J =
16021;
2
3
4
5
k
-113.99999583
-113.999995825
-113.9999958247
3:0729
9:2980
5:3346
7:5857
-113.9999958246957
4:8195
20:304
20:304
20:304
20:304
20:304
37:068
37:068
37:068
37:068
37:068
with parametrically-large aspect ratios.
giving, as advertised, an entropy parameter
H = 0:999997 : : : :
(4.10)
Thus, the angular momentum is at 0:17% of the cc bound.
We scale the solution by ne-tuning the value of k21. At each step, we solve the bubble
equations and check the absence of closed timelike curves. The scaling process is summed
up in table 3.
Acknowledgments
We are indebted to Tomas Ort n, David Turton and Nick Warner for interesting discussions.
The work of IB and PH is supported by the ANR grant Black-dS-String. The work of PH
is supported by an ENS Lyon grant. The work of PFR was supported by the Severo
Ochoa pre-doctoral grant SVP-2013-067903. This work has been supported by the Spanish
Goverment grant FPA2015-66793-P (MINECO/FEDER, UE) and the Centro de Excelencia
Severo Ochoa Program grant SEV-2016-0597.
A
Solving the BPS equations
The action of the STU model of N = 1, d = 5 supergravity is completely determined
by the constant symmetric tensor CIJK = j"IJK j. All the timelike-supersymmetric- eld
con gurations of this theory have a conformastationary metric [
55
]
ds2 =
(Z1Z2Z3) 2=3 (dt + k)2 + (Z1Z2Z3)1=3 hmndxmdxn ;
where hmndxmdxn is the metric of a hyper-Kahler manifold, while ZI and k are respectively
three functions and a 1-form taking values in this four-dimensional space. The remaining
bosonic content consists of three vector elds satisfying
(A.1)
(A.2)
AI =
(dt + k) + BI ;
1
ZI
where BI is a 1-form in the hyper-Kahler space. These eld con gurations become solutions
when the following set of BPS equations, de ned on the four-dimensional manifold, is
satis ed
dBI = ?(4)dBI ;
r(24)ZI = CIJK ?(4) dBJ
^ dBK ;
dk + ?(4)dk = ZI dBI ;
(A.3)
(A.4)
(A.5)
(A.6)
(A.8)
(A.9)
(A.10)
(A.11)
(A.12)
(A.13)
and two scalars that can be conveniently parametrized as
e
Therefore, the requirement that the solution is supersymmetric drastically simpli es
the equations of motion of the theory to a linear system of PDE's on a manifold with
Euclidean signature. Still, for general hyper-Kahler spaces this problem is a hard nut to
crack. This is why, in order to make further progress, one usually chooses a speci c, yet
very general family of hyper-Kahler manifolds admitting a triholomorphic isometry. These
are Gibbons-Hawking spaces [56], whose metric is given by
hmndxmdxn = V 1 (d
The integrability condition of the equation above implies that V is harmonic in R3. We
can make further progress if we assume that all matter elds are also independent of the
isometric coordinate
. Then the functions and forms that characterize the solution can
be further decomposed,
B
I =
V 1KI (d
I
+ ) + A ;
k =
+ ) + ! :
Upon substitution in the system of BPS equations we nd a set of di erential equations
for the three-dimensional seeds
?(3)dKI = dAI ;
?(3)d! = V dM
M dV +
KI dLI
LI dKI =< ; d
> ;
and the following algebraic expressions for the building blocks that make up the solution,
1
2
1
2
= M +
V 1LI KI +
V 2CIJK KI KJ KK ;
where LI and M are harmonic functions in R3, r(23)LI = r(23)M = 0. Therefore,
supersymmetric solutions admitting a spacelike isometry are completely speci ed in terms
of 8 harmonic functions,
= (V; KI ; LI ; M ). Notice that the integrability condition of
equation (A.11) yields the bubble equations
b
rab
It is convenient to de ne the quartic invariant I4 which must satisfy the following
inequality to avoid the presence of closed timelike curves [23, 24]
This condition can be understood from the fact that the metric can be written as
where we write f 3
Z1Z2Z3.
V 2
I4
2
+ f 1
V
d~x d~x
; (A.16)
!2
I4
B
Numerical analysis of the entropy parameter of four-GH-center
solutions
The aspect ratios of the solutions are xed to:
z1
z2
z2
z3
102
102:
(A.14)
(A.15)
(B.1)
(B.2)
By generating such solutions using numerics, we want to describe the evolution of the
analyze the entropy parameter by varying the initial supertube charges Q(1)
k3, Q(1)
Q(23) ,
1
Q(32) ,
1
entropy parameQte(3r) H as a function of the nine degrees of freedom of the solutions k1, k2,
Q(2)
Q(21) , q0, q1 and q2. We decompose our analysis in three parts. We rst
3
with all the other parameters xed. Then, we analyze the entropy parameter when varying
q0, q1 and q2. Finally, we analyze the entropy parameter as we vary the three initial dipole
Q2(3) , Q(2)
3
1 1
Q(2) and QQ((231)) ,
3
charges k1, k2 and k3.
Each of the graphs is made by generating 2500 solutions following the procedure
detailed in section 3.1.1. Because a con guration of parameters k1, k2, k3, QQ((213)) , Q(2)
1 1
Q(32) , Q(3)
3
Q(21) , q0,
q1 and q2 can give di erent four-GH-center solutions, we take the
nal solution with the
highest entropy parameter. Moreover, for readability reason, we smooth all the discrete
graphs we initially obtained to have at the end a continuous curve.
The graphs in gure 3 show the variations of the entropy parameter with the three
ratios of supertube charges. The other parameters have been xed to
The entropy parameters can be greater than 15% in many domains of charge ratios
and more than 25% in some small others.
k1 =
The graphs in gure 4 illustrate the variation of the entropy parameter as a function
of q0, q1 and q2. We suppressed the values zero in the graphs. They correspond to
three-GH-center and one-supertube solutions. The six other parameters have been
xed to
k1 =
9 2
Q(3) =
However, we observed the same features for di erent values of charge ratios and
dipole charges. The graphs show that for any value of q0 the entropy is maximum
when the absolute values of the charges are close to one. Furthermore the minimal
Gibbons-Hawking charges (1,1,1 and -2) are the best choice to obtain four-GH-center
solutions with low angular momentum. This is an unexpected feature. Indeed, in
the ve-center solution of [9], the GH charges are close to each other and large. Our
solutions do not share this feature.
(b) q0 = 9
(c) q0 = 13
(d) q0 = 20
kF3igaurereeq4u.aTlthoe 1enatnrodpQQy((31p22))ar=am2,etQQe((23r31))H=a1s aanfudnQQct((2113io))n=o0f:t9h.e charges of V, q0, q1 and q2 with k1, k2,
k3. We vary also one charge ratio, Q(1)
For the initial supertube dipole charges, we observed that the sign con guration given
by (B.2) (k2 negative, k1 and k3 positive) is the optimal one. With the two other sign
con gurations, we did not nd domains of charges where the entropy parameter is
above 0.1. For the rest of the analysis we focus on con gurations with k2 negative and
k1 and k3 positive. By doing a quick analysis, we observed that the entropy parameter
does not depend on the absolute value of k2. The graphs in
gure 5 illustrate how
the entropy parameter depends on the absolute value of the dipole charges k1 and
Q(23) , keeping the other parameters xed:
1
(B.4)
We remark that the entropy parameter depends essentially on the ratio kk13 and the
entropy is maximum and far from 0 for one particular value of kk13 . We observed the
same kind of graph for di erent values of charge ratios. If one varies the value of QQ((213)) ,
1
q0 = q1 = q2 = 1;
1 Q(32)
2 Q(2) =
1
(c) QQ((13)) = 1:8
2
1
(b) QQ((13)) = 0:9
2
1
(d) QQ((13)) = 3:6
2
1
Q(32) , Q(3)
1 3
was detailed in section 3.1.2.
ratio Q(21)
Q(13) with q0, q1, q2 equal to 1 and QQ((3122)) = 2, QQ((2331)) = 1.
the particular value of kk13 changes but the maximum value of the entropy parameter
remains the same whereas if one varies the two other charge ratios both change. The
maximum value of entropy parameter we observed is 0.3.
To conclude, the numerical analysis shows that there exist large domains of
supertubecharge ratios and supertube dipole charges where the entropy parameter of solutions
satisfying (C.1) is maximal and around 0.3. The only necessary conditions to have an angular
momentum signi cantly below the cc bound is that the Gibbons-Hawking charges must
be minimal and the dipole charge con guration of the generating three-supertube solution
must be k1 and k3 positive and k2 negative. Moreover, increasing the di erence in scale
between the inter-center distances does not a ect how the entropy parameter varies with
Q(21) , q0, q1 and q2. It a ects only the maximal value reachable as it
Numerical analysis of the entropy parameter of solutions with one
supertube and three Gibbons-Hawking centers
We proceed the same way to analyse the entropy parameter of solutions with three
GibbonsHawking centers and one supertube. We focus on solutions without scale di erences
between the inter-center distances:
z1
z2
z3
z3
1:
(C.1)
According to the method used to generate them (see section 4.1.1), the solutions depends
on eight free parameters and the aspect ratios (C.1). We will also decompose our analysis
in three parts. We rst vary the initial supertube charges Q(1)
Q(23) , Q(2)
1
Q(2) and Q(3)
3
1 3
Q(21) , with all the
other parameters xed. Then, we analyze the entropy parameter as a function of q0 and
qJ , where J is 1, 2 or 3 depending on which center is the supertube. Finally, we vary the
three initial dipole charges k1, k2 and k3. All the graphs have been generated as explained
in the previous section.
First of all, we noticed that the localization of the supertube center compared to the
three Gibbons-Hawking centers has a signi cant impact on the entropy parameter.
The best con guration is when the supertube is not located between the
GibbonsHawking centers.
With our conventions, this means that the supertube center is
the rst center given by (0; 0; z1). Indeed, we have found several domains of charges
and dipole charges where the entropy parameter is above 0.15 for the three possible
supertube locations. However, we have found that H has much higher values when
the supertube is located at the rst center.
The graphs in gure 6 give the variations of the entropy parameter with the three
initial charge ratios when the supertube is located at the rst center. We have xed
the other parameters to be
k1 =
q0 =
is small, the entropy parameter can reach 0.25. This is the upper bound we found
for a con guration which satis es (C.2) and (C.1).
Regarding the variation of the entropy parameter as a function of q0 and q3 (q2 is
xed to satisfy
qa = 1), we have observed the same features as in solutions with
four Gibbons-Hawking centers: the higher the absolute value of the Gibbons-Hawking
charges is, the lower is the entropy parameter. The graph in gure 7 shows the
variation of the entropy parameter as a function of q0 and q3 for solutions satisfying (C.1)
k1 =
We have observed similar variations for di erent initial charge ratios and dipole
charges. Thus, q0 = 1, q2 = 1 and q3 =
1 is the best con guration to optimize
the entropy parameter.
Varying the initial supertube dipole charges, we have again observed exactly the
same features as in solutions with four Gibbons-Hawking centers. The best sign
con guration is when k2 is negative and when k1 and k3 are positive. Moreover, the
entropy parameter does not depend signi cantly on the absolute value of k2 and it
only depends on kk13 . It also reaches a maximum for a particular value of the ratio kk13 .
The value and the location of the maximum depends on the values of the supertube
charge ratios. The graphs in gure 8 illustrate these conclusions. We built solutions
and computed their entropy as a function of the absolute value of the dipole charges
are equal to 1 and
Q(32)
Q(12) = 4,
Q(23)
Q(31) = 0:06 and
Q(21)
Q(13) = 0:5.
HJEP10(27)
Q
Q
(1)
1
ratio
Q(21)
Q(13) with q0, q1, q2 equal to 1 and
Q(32)
Q(12) = 4,
Q(23)
Q(31) = 0:06.
Q(23) . The other parameters have been xed to
1
We have analyzed the entropy parameter for charge ratios di erent from the one
above. The upper bound of all the maxima we observed is 0.25.
The numerical analysis shows that solutions with one supertube and three
GibbonsHawking centers do not need to have a scale di erence between the inter-center distances
to have an entropy parameter above 0.1. If one chooses minimal Gibbons-Hawking charges
and k2 negative, k1 and k3 positive, one can nd domains of parameters where the entropy
is around 0.2.
Open Access.
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any medium, provided the original author(s) and source are credited.
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