Nonabelian solutions in N=4, D=5 gauged supergravity

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by  Eugen Radu

Nonabelian solutions in = 4, D = 5 gauged supergravity
N
Eugen Radu
Department of Mathematical Physics, National University of Ireland Maynooth, Ireland
Abstract
We consider static, nonabelian solutions in N = 4, D = 5 Romans’ gauged supergravity model.
Numericalargumentsarepresentedfortheexistenceofasymptotically anti-deSitterconfigurations in
theN =4+ version of thetheory, with a dilaton potential presenting a stationary point. Considering
6 the version of the theory with a Liouville dilaton potential, we look for configurations with unusual
0 topology. A new exact solution is presented, and a counterterm method is proposed to compute the
0 mass and action.
2

n
a
J 1 Introduction

9
1 In the last few years, there has been increasing interest in solutions of various supergravity models with

nonabelian matter, following the discovery by Chamseddine and Volkov [1] of a nontrivial monopole-type
1
supersymmetricvacuainthecontextofthe =4D =4Freedman-Schwarzgaugedsupergravity[2]. This
v N
5 is one of the few analytically known configurations involving both non-abelian gauge fields and gravity
3 (for a generalreview of such solutions see [3, 4]). Its ten-dimensional lift was shown to represent5-branes
1 wrappedonashrinkingS2[5]. AsdiscoveredbyMaldacenaandNun˜ez,thissolutionprovidesaholographic
1
description for =1, D =4 super-Yang-Mills theory [6].
0 N
Chamseddine and Volkov looked also for five dimensional nonabelian configurations [7] in a version of
6
= 4 Romans’ gauged supergravity model [8] with a Liouville dilaton potential. The static spherically
0
N
/ symmetricsolutiontheyfound(althoughnotinaclosedform)possessestwounbrokensupersymmetriesand
h
hasbeenshownbyMaldacenaandNastasetodescribe,afterliftingtotendimensions,thesupergravitydual
t
- ofanNS5-branewrappedonS3 withatwistthatpreservesonly =1supersymmetryin2+1dimensions
p N
[9]. Both particle-like and black hole generalizations of the D = 5 Chamseddine-Volkov solution are
e
h discussed in [10] froma ten-dimensional perspective, where a backgroundsubtraction method to compute
: the mass and action of these configuration is also proposed. Although the dilaton potential V(φ) can be
v
i viewedasaneffectivenegative,position-dependentcosmologicalterm,thesesolutionsdonothaveasimple
X
asymptotic behaviour, the dilaton diverging at infinity.
r However, as discussed in this paper, the situation is different for the = 4+ version of the Romans
a N
model, with a dilaton potential consisting of the sum of two Liouville terms. In this case, the dilaton
field approaches asymptotically a constant value φ , which corresponds to an extremum of the potential
0
such that dV/dφ = 0 and V(φ ) < 0. This makes possible the existence of both regular and black
φ0 0
hole solutions approaching at infinity the anti-de Sitter (AdS) background. Topological black holes with
(cid:12)
nonabelian hair a(cid:12)re found as well, in which case the three-sphere is replaced by a three-dimensional space
of negative or vanishing curvature.
Considering next the case of the Romans’ model with a Liouville dilaton potential, it is natural to ask
if apartfromblackholes discussedin [10], whose horizonhas sphericaltopology,there arealso topological
blackholeswithnonabelianfields. SuchsolutionsareknowntoexistinanAbeliantruncationofthetheory
(seee.g. [11])andalsoinafourdimensionalEinstein-Yang-Mills(EYM)systemwithnegativecosmological
constant[12]. However,we find thatthe inclusionofnonabelianfields leads to apathologicalsupergravity
background: the factor multiplying the hyperbolic or flat surface gets negative for some finite values of
the radialcoordinate. The massand actionofthe sphericallysymmetric solutions with a Liouvilledilaton
potential is computed by using a boundary counterterm method, the standard background subtraction
results being recovered.
1
A general discussion of the D = 5 configurations admitting a translation along the fourth spatial
coordinateispresentedinSection5. Twonewexactsolutionsarefoundbyuplifitingtopologicallynontrivial
configurationsofthe =4 D =4 Freedman-Schwarzmodel. We give our conclusionsandremarksin the
N
final section.
2 General framework
2.1 Action principle and field equations
The bosonic matter content of the Romans’ gauged supergravity consists of gravity, a scalar φ, an SU(2)
Yang-Mills (YM) potential AI (with field strength FI = ∂ AI ∂ AI + g ǫIJKAJAK), an abelian
µ µν µ ν − ν µ 2 µ ν
potential W (f being the corresponding field strength), and a pair of two-form fields. These two form
µ µν
fields can consistently be set to zero, which yields the bosonic part of the action
1 1 1 1 1
I = d5x√ g ∂ φ∂µφ e2aφFI FIµν e 4aφf fµν (1)
5 4π − 4R− 2 µ − 4 µν − 4 − µν
ZM (cid:16)
1 1
ǫµνρστFI FI W V(φ) d4x√ hK,
−4√ g µν στ τ − − 8π −
− (cid:17) Z∂M
where a= 2/3. Here
p g2 g
V(φ)= 2 e 2aφ+2√2 1eaφ (2)
−
− 8 g
(cid:18) 2 (cid:19)
is the dilaton potential, g being the U(1) gauge coupling constant. The last term in (1) is the Hawking-
1
Gibbons surface term, necessary to ensure that the Euler-Lagrange variation is well defined, where K is
the trace of the extrinsic curvature for the boundary ∂ and h is the induced metric of the boundary.
M
As discussed in [8], the theory has three canonical forms, corresponding to different choices of the
gauge coupling constants in (2). The case g = 0 corresponds to = 40 theory, where the SU(2) U(1)
2
N ×
symmetry is replacedby the abeliangroupU(1)4; there is also a =4+ versionin whichg =g √2, and
2 1
N
=4 with g = g √2. Also, for this truncation of the theory with vanishing two forms, one can take
− 2 1
N −
g = 0 and find another distinct case. Note that for a nonvanishing g , one can set its value to one, by
1 2
using a suitable rescaling of the field.
Thefieldequationsareobtainedbyvaryingtheaction(1)withrespecttothefieldvariablesg ,AI, W
µν µ µ
and φ
1
R g R = 2 T ,
µν µν µν
− 2
a ∂V
2φ e2aφFI FIµν +ae 4aφf fµν = 0, (3)
∇ − 2 µν − µν − ∂φ
1
D (e 4aφfµν) ǫµνρστFI FI = 0,
ν − − 4√ g νρ στ
−
1
D (e2aφFIµν) ǫµνρστFI f = 0,
ν − 2√ g νρ στ
−
where the energy-momentum tensor is defined by
1
T = ∂ φ∂ φ g ∂ φ∂σφ g V(φ) (4)
µν µ ν µν σ µν
− 2 −
1 1
+e2aφ(FI FI gρσ g FI FIρσ)+e 4aφ(f f gρσ g f fρσ).
µρ νσ − 4 µν ρσ − µρ νσ − 4 µν ρσ
2
2.2 The ansatz
Restricting to static solutions, we consider a general metric ansatz on the form
ds2 =A2(r)dr2+B2(r)dΣ2 C2(r)dt2, (5)
3,k −
where dΣ2 = dψ2 + f2(ψ)dΩ2 denotes the line element of a three-dimensional space with constant
3,k k 2
curvature(dΩ2 =dθ2+sin2θdϕ2 beingtheroundmetricofS2). Thediscreteparameterk takesthevalues
1,0 and 1 and implies the form of the function f (ψ)
k
−
sinψ, for k =1
f (ψ)= ψ, for k =0 (6)
k 
sinhψ, for k = 1.

−
When k = 1, the hypersurface Σ represents a 3-sphere; for k = 1, it is a 3 dimensional negative
3,1 
− −
constantcurvaturespaceanditcouldbeaclosedhypersufacewitharbitrarlyhighgenusunderappropriate
identifications. For k =0, Σ is a three-dimensional Euclidean space.
3,0
For the matter fields ansatz, we start by choosing a purely-electric abelian field ansatz
f =f (r)dt dr, (7)
rt
∧
the dilaton field being also a function only on the coordinate r.
The computation of the most general expression for AI compatible with the symmetries of the line-
µ
element(5)isastraightforwardgeneralizationofthek =1casediscussedinAppendixAof[13]. Applying
the standardrulesforcalculatingthegaugepotentialsforanyspacetimegroup[14,15],andtakingAI =0
t
i.e. no dyons, one finds the ansatz (with τ the Pauli spin matrices)
a
1 df (ψ)
k
A= τ (ω(r)dψ+cosθdϕ) (τ dθ+τ sinθdϕ)+ω(r)f (ψ)(τ dθ τ sinθdϕ) , (8)
3 2 1 k 1 2
2 − dψ −
n o
the corresponding YM curvature being
1
F = ω τ dr dψ+ω f τ dr dθ f ω τ dr dϕ+(k w2)f τ dψ dθ (9)
′ 3 ′ k 1 k ′ 2 k 2
2 ∧ ∧ − ∧ − ∧
(cid:16)
+(k w2)f sinθτ dψ dϕ+(w2 k)f2sinθτ dθ dϕ ,
− k 1 ∧ − k 3 ∧
(cid:17)
where a prime denotes a derivative with respect to r.
For a purely magnetic YM ansatz, the equation for the Abelian field
1
e 4aφfνµ = ǫµνρστFI FI (10)
∇ν − 4√ g νρ στ
−
(cid:0) (cid:1)
have a total derivative structure, implying after the integration the simple expression for f
tr
e4aφ
f (r)= (2ω2 6kω+c)A2(r)C2(r), (11)
tr
√ g −
−
where c is a constant of integration.
3 = 4+ solutions
N
Thescalarpotentialforthe =4+ modelV(φ)= (e 2aφ+2eaφ)/8hasexactlyoneextremumatφ=0,
−
N −
corresponding to an the effective cosmologicalconstant
3
Λ =2V(0)= . (12)
eff
−4
AsdiscussedbyRomans[8],themaximallysymmetricAdS spacetimeisasolutionofthetheoryforφ 0
5
≡
and pure gauge fields, preserving the full =4 supersymmetry.
N
3
2
1.5 m(r)
σ(r)
1
0.5
φ(r)
0
ω(r)
-0.5
b=0.23, φ =0.1
0
-1
-1 -0.5 0 0.5 1 1.5 2 2.5 3
log (r)
10
Figure 1. The gauge function ω, the dilaton φ and the metric functions m(r), σ(r) are shown as function of
the coordinate r for a typical spherically symmetricregular solution.
Here we look for configurations with nonavanishing matter fields, approaching asymptotically the AdS
background.
In deriving numerical solutions, it turns out to be convinient to use the following parametrization of
the line element (5)
1
A2(r)= , B2(r)=r2, C2(r)=H(r)σ2(r) (13)
H(r)
The existence in the asymptotic region of an effective negative cosmological constant suggest to use the
following form of the metric function H(r)
4m(r) r2
H(r)=k + , (14)
− 3r2 ℓ2
(with Λ = 6/ℓ2 i.e. the characteristic length scale ℓ2 = 8), the function m(r) being related in the
eff
−
usual approachto the local mass-energy density, up to some numerical factor.
Inserting this ansatz into the action (1), the field equations reduce to
1 3 (ω2 k)2 1 (2ω3 6kω+c)2
m′ = φ′2Hr3+ e2aφr(Hω′2+ − )+ e4aφ − ,
2 2 r2 2 r3
σ 2 2
′ = rφ2+ e2aφω2, (15)
′ ′
σ 3 r
ω 1 6(2ω3 6kω+c)(ω2 k)
(e2aφσrHω′)′ = 2e2aφσ (ω2 k)+ e4aφ − − ,
r − 3 r3
ω2 k)2 σ(2ω3 6kω+c)2 a
(σr3Hφ′)′ = 3ae2aφσr(Hω′2+ − )+2ae2aφ − + (e−2aφ eaφ)r3σ,
r2 r r3 4 −
while the U(1) gauge field is given by (11).
These equations present both globally regular and black hole solutions.
4
2
k=0 r =2
h
ω =0.5, φ =1.
h h
1.5 m(r)
σ(r)
1
0.5
φ(r)
0
ω(r)
-0.5
0.5 1 1.5 2 2.5
log (r)
10
Figure 2. The gauge function ω, the dilaton φ and the metric functions m(r), σ(r) are shown as function of
the coordinate r for a typical k=0 topological black hole solution.
Regular configurations exist for k=1, c=4 only, and have the following expansion near the origin
1
m(r) = 3b2e2aφ0 (e 2aφ0 +2e 2aφ0 3) r4+O(r5), σ(r)=σ (1+4b2e2aφ0)r2+O(r2),
− − 0
− 32 −
a
φ(r) = φ(cid:0) + 3ab2e2aφ0 + (e 2aφ0 e 2aφ0(cid:1)) r2+O(r4), ω(r)=1 br2+O(r4), (16)
0 − −
16 − −
(cid:16) (cid:17)
with b, φ , σ real constants.
0 0
We are also interested in black hole solutions having a regular event horizon at r = r > 0. The field
h
equations implies the following behaviour as r r in terms of three parameters (φ ,σ ,ω )
h h h h
→
3 r2
m(r) = r2(k+ h)+m (r r )+O(r r )2, σ(r)=σ +σ (r r )+O(r r )2, (17)
4 h ℓ2 1 − h − h h 1 − h − h
φ(r) = φ +φ (r r )+O(r r )2, ω(r)=ω +ω (r r )+O(r r )2,
h 1 h h h 1 h h
− − − −
where we defined
1 1
m = e4aφh(2ω3 6kω +c)2+r2(3e2aφh(ω2 k)2 r4(e 2aφh +2e 2aφh 3)) ,
1 2r3 h− h h h− − 4 h − − −
h
(cid:16) (cid:17)
6r2ω (ω2 k)+6e2aφh(2ω3 6kω +c)(ω2 k)
ω = h h h− h− h h− , (18)
1 2r2( 2m +3kr +6r3/ℓ2)
h − 1 h h
3(2ae4aφh(2ω3 6kω +c)2+r2(3ae2aφh(ω2 k)2+r4V (φ )
φ = h− h h h− h ′ h .
1 2r4( 2m +3kr +6r3/ℓ2)
h − 1 h h
Note that since the equations (15) are invariant under the transformation ω ω one can set ω(0) >
→ −
0, ω(r )>0 without loss of generality.
h
Using the above initial conditions, the equations (15) were integrated for a large set of b, φ (ω , φ
0 h h
respectively) and several values of r . For all solutions presented here we set c = 4, although we studied
h
black holes with other values of c also. The overall picture we find combines features of the pure five
dimensional EYM-Λ system discussed in [13] and the four dimensional EYM-dilaton solutions with a
potential approaching a constant negative value at infinity, considered in [16].
5
1.5
σ(0)
0 ω
0
α
-1.5
β
-3
φ =-0.5
0
-4.5
0 0.1 0.2 0.3 0.4 0.5
b
Figure 3. The asymptotic parameters ω0, α, β, and the value of the metric function function σ(r) at the
origin areshown asafunction ofbfor spherically symmetricregularsolutions with φ0=−0.5. Forb=0onefinds
α≃0.984, β≃−4.166 and σ(0)≃0.951.
A continuum of monopole solutions is obtained for compact intervals of the initial parameters. The
gaugefield interpolates monotonicallybetweenthe initial value at the r =r (with r =0 orr ) andsome
i i h
asymptotic value ω . The value at the origin/event horizon of the metric function σ(r) (which is a result
0
of the numerical integration) decreases along these intervals and, at some stage a singularity appears,
corresponding to σ(r ) 0. Typical configurations are presented in Figure 1 (a spherically symmetric
i
→
regular solution) and Figure 2 (a k =0 black hole solution).
For all configurations we have studied, the function m(r) diverges as r . There are two distinct
→ ∞
sources of this behavior. Considering first the dilaton sector, we note that the scalar field mass µ is given
by
1
µ2 =V (0)= ,
′′
−2
which saturates the Breitenlohner-Freedman bound [17]. Thus, the scalar field behaves asymptotically as
α βlogr
φ(r)= + +..., (19)
r2 r2
where α, β are real constants. For such solutions, due to the back reaction of the scalar field, the usual
ADM mass diverges logarithmically with r as r [18, 19].
→∞
Thereisalsoasecondlogarithmicdivergenceofthefunctionm(r)associatedwiththenonabeliangauge
field ω(r). For generic solutions, ω2 ω2 = k as r , which from (15) implies that asymptotically
→ 0 6 → ∞
m(r) (ω2 k)2logr. Afinite contributiontothe totalmassisfoundforω2 =k only. However,wefailed
∼ 0− 0
to find such solutions and it seems that, similar to the EYM-Λ case [13], only a pure gauge configuration
has this asymptotics.
We remark that for any value of φ(r ) it is possible to find a initial value of the gauge field such that
i
β = 0, which removes the divergencies associated with the scalar field. Also, as seen in Figure 3 for a
set of regular configurations, the solution with b = 0 (i.e. ω(r) 1) and a nonzero φ is not the vacuum
0
≡
AdSspacetime. Thus,fortheseasymptoticstherearegloballyregularsolutionsevenwithoutanonabelian
field. One finds also black holes with scalar hair without a gauge field. Similar solutions have been found
6
inothermodelswithadilatonfieldpossessinganontrivialpotentialapproachingaconstantnegativevalue
at infinity (see e.g. [16, 18]).
A systematic analysis reveals the following expansion of the solutions at large r:
1 β2 3
m(r) = M + β(β 4α)logr log2r+ (ω2 k)2logr+..., (20)
16 − − 8 2 0−
1 logr 1
logσ(r) = β(4α β+2βlogr) (β2 4αβ+8α2) +...,
−3 − r4 − − 2r4
logr c α logr
ω(r) = ω ℓ2ω (ω2 k) + 1 +..., φ(r)= +β +...,
0− 0 0− r2 r2 r2 r2
where c is a constant. Note that the asymptotic metric still preserves the usual SO(4,2) symmetry, and
1
the spacetime is still asymptotically AdS, despite the diverging m(r).
We close this section remarking that in the absence of a gauge field, it is possible to define a total
mass by using the approach in [18, 19]. By employing an Hamiltonian method, the divergencies from the
gravityandscalarpartscancelout, yielding afinite totalcharge. Itwouldbe interesting togeneralizethis
approachin the presence of a nonabeliangauge field in the bulk and to define a finite mass and action for
these configurations, too.
4 g = 0 solutions
1
The properties of the solutions are very different if we set g = 0 in (2), which leads to a Liouville-type
1
dilaton potential
1
V(φ)= e 2aφ. (21)
−
−8
In this case we found convinient to use the following parametrization of the metric ansatz (5)
A(r)=eaφ(r)+Y(r) X(r), B(r)=eaφ(r)R(r), C(r)=eaφ(r)+X(r), (22)
−
which yields the reduced action
3
L= e3aφ+2X Y(R2+3aRφR +aR2φX +RRX +R2φ2+ke 2X+2Y) (23)
− ′ ′ ′ ′ ′ ′ ′ ′ −
2
3 ω2 k)2 1 (2ω3 6kω+c)2 1
e3aφ+Y(e2X−2YRω′2+ − ) e3aφ+Y − + e3aφ+YR3.
−2 R − 2 R3 8
We remark that (23) allows for the reparametrizationr r˜(r) which is unbroken by our ansatz.
→
In this way, we find that the field equations are (here we fix the metric gauge by setting Y = 0 and
define e2X =ν)
2(2ω3 6kω+c)2 2k 4 ν R 2R2 2ω2
R′′ = − + (ω2 k)2 ′ ′ ′ ′ √6R′φ′,
− νR5 νR − νR3 − − ν − R − R −
2ω 3(2ω3 6kω+c)
(e3aφRνω′)′ =e3aφ(ω2 k) + − ,
− R 2 R3
(k ω2)2 k (cid:16)ν R2 ω2 (cid:17)
′ ′ ′
R′′+ − +R′( +2φ′)+ + =0, (24)
νR3 − νR ν R R
(k ω2)2 k ν R R2 2Rφ
′ ′ ′ ′ ′
φ′′ − + =0,
− νR4 νR2 − νR − R2 − R
X αeY X 2s =0,
′ − −
−
where α is an integration constant. In the ”extremal” case α = 0 we can set X = 0, without loss of
generality. For α=0, we find black hole solutions with a nontrivial ν =e2X metric function.
6
7
4.1 BPS configurations
As discussedin [7], [10], for α=0, k =1 the system (24) presents configurationspreservingsome amount
of supersymmetry and solving first order Bogomol’nyi equations.
Tofindsuchsolutionsforanyvalueofk,itisconvinienttointroducethenewvariables(also,toconform
with other results in literature, we note here c=4κ)
2 3 1
φ= s g X, R=eg . (25)
3 − 2 − 6
r r r
With this choice, the lagrangian(23) becomes
2 1 X 2
L=eX+2s s2 g2 ′ eX+2s 2gω2 e X+2s 4g(ω2 k)2 (26)
′ ′ − ′ − −
3 − 2 − 3 − − −
(cid:18) (cid:19)
1 1
(2κ 3kω+ω3)2e X+2s 6g +ke X+Y+2s 2g e X+2s,
− − − − −
−3 − − 12
which can be written in the form
dyidyk
L=G (y) U(y), (27)
ik
dr dr −
where yi =(s,g,w) and G =e2sdiag(2, 1, e 2g). The potential U can be represented as
ik 3 −2 − −
∂W ∂W
U = Gik , (28)
− ∂yi ∂yk
where the superpotential W has the expression
W = 1e g+2s 1e2g 6(ω2 k)+18e 2g(ω2 k)2 e 2g(2ω3 6kω+4κ) 3ω 2. (29)
6 − 2 − − − − − − − −
r
(cid:0) (cid:1)
As a result, we find the first order Bogomol’nyi equations dyi/dr =Gik∂W/∂yk
1
s = e g+Y (e2g+6(2k+ω2)+8e 4g(2κ 3kw+w3)2+12e2g(w4 4ωκ+3k2),
′ 2√2 − − − −
e g+pY( e4gω+4e2g(κ ω3)+4(k ω2)(2κ 3kω+ω3))
−
ω′ = − − − − , (30)
√2 e6g+6e4g(2k+ω2)+8(2κ 3kω+ω3)2+12e2g(3k2 4κω+ω4))
− −
e2g( e4gω+4e2g(κ ω3)+4(k ω2)(2κ 3kω+ω3))
p
g′ = − − − − ,
√2 (e4g(2k+ω2)+4(2κ 3kω+ω3)2+4e2g(3k2 4κω+ω4)
− −
whichsolvealsothpe second-ordersystem. Itcanbe proventhat, afterasuitable redefinition,thesearethe
equations derived in [3] for k =1, by using a Killing spinor approach.
Unfortunately, no exact solution of these equations can be found in the general case, except for the
specialvalues(k =0, κ=0). ForagaugechoiceY =g,wefindthethe newexactsolutionofthe Romans
model
dt2
ds2 =e2βr/3+4s0/3(2(β2 e 2γ0(r r0)))1/3( +dr2+dΩ2 ), (31)
− − − −2(β2 e 2γ0(r r0)) 0,3
− − −
e4βr/3+4s0/3
w(r)=e β(r r0), e2aφ = ,
− −
(2(β2 e 2γ0(r r0)))2/3
− − −
where s , β, γ , r are arbitrary real constants.
0 0 0
A direct computation reveals that the above line element presents a curvature singularity for a finite
value of the radial coordinate, r = r (with e 2γ0(r∗ r0) = β2). This singularity appear to be repulsive:
∗ − −
8
no timelike geodesic hits it, though a radial null geodesic can. Thus our solution violate the criterion of
[20] because g in the Einstein frame is unbounded at the singularity and thus they cannot accurately
tt
describe the IR dynamics of a dual gauge theory.
Ofcourse,inother cases,the equations(30) canbe solvednumerically. However,adirectinspectionof
theaboverelationsrevealthattheabsenceofsolutionswitharegularoriginfork =1isagenericproperty
6
ofk =0, 1BPSsolutions(we callregularoriginthe pointr =r , where the function R(r) vanishes but
0
−
all curvature invariants are bounded (without loss of generality we can set r =0)).
0
It can be proven that this is a generic feature of all k = 1 configurations. Considering solutions of
6
the second order equations (24), one finds that it is also not possible to take a consistent set of boundary
conditions at the origin without introducing a curvature singularity. This fact has to be attributed to the
particular formofthe potential termV =e2aφ(ω2 k)2/(2R)in the reducedlagrangeanofthe system.
YM
−
Solutions with regularoriginmay exist for k =1, κ=1 only and are parametrizedby the value of the
parameter b appearing in the expansionof ω(r) at the originω(r)=1 br2+O(r4). As discussed in [10],
−
globallyregularsolutionsextendingtoinfinity (i.e.anunboundedR(r)) existsfor0<b<1only,the BPS
solution found in [7] corresponding to b = 1/3. For b 1, the function R(r) goes to zero again at some
≥
finite value of r.
4.2 Black hole solutions
A natural way to deal with the type of singularities we have found for k =0, 1 is to hide them inside an
−
event horizon. To implement the black hole interpretation we restrict the parameters so that the metric
describes the exteriorof a blackhole with a non-degeneratehorizon. That implies the existence ofa point
r = r where e2X = ν vanishes, while all other functions are finite and differentiable. Without loss of
h
generality we can set r =0.
h
The field equations (24) give the following expansion near the event horizon
e 3aφh
R(r) = R +R r+O(r2), φ(r)=φ +φ r+O(r2), ν(r)=α − r+O(r2),
h 1 h 1 R3
h
2e3aφh
ω(r) = w + (ω2 k)(6R2ω +2ω3 6kω +c)r+O(r2), (32)
h 3αR − h h h− h
h
where
ae3aφh
φ = R6 12kR4 +36R2(ω2 k)2+20(2ω3 6kω +c) ,
1 8αR3 h− h h − h− h
h
(cid:16) (cid:17)
e3aφh
R1 = 2αR2 Rh2(kRh2 +(ωh2 −k)2+(2ωh3−6kωh+c)2−12e−3aφhαaφ1Rh .
h
(cid:16) (cid:17)
The solutions present three free parameters: the value of the dilaton at the horizon φ , the event horizon
h
radius R and the value of the gauge potential at the horizon ω . For any k, one can set α = 1 without
h h
loss of generality, since this value can be obtained by a global rescaling of the line element (5).
Usingthe initialconditionsonthe eventhorizon(32),the equations(24)wereintegratedfor arangeof
values of φ , R and varying ω . The numerical analysis shows the existence of a continuum of solutions
h h h
for everyvalue of(k,R ,ω ,φ ). Also,foreverychoiceofφ andagiven(k,R ,ω ),we findqualitatively
h h h h h h
similar solutions (different values of φ lead to global rescalings of the solutions).
h
The results we found for k = 1 are similar to those derived in [10] from a ten-dimensional point of
view. One can show that in this case ω is restricted to ω 1, while the value of c is not restricted.
h h
| | ≤
Spherically symmetric black hole solutions are found for any set of values (φ ,R ,ω ). For R2 +ω2 >1,
h h h h h
R(r) as r , while for R2 +ω2 < 1, as numerically found in [10], the asymptotic is different:
→ ∞ → ∞ h h
R(r) vanishes at some finite value of r, where there is a curvature singularity.
Unfortunately, the situationfor a nonsphericallysymmetric eventhorizonresemblesthis lastcase. For
each set (R , w , φ ) we find a solution living in the interval r [0,r [, where r has a finite value. For
h h h ∗ ∗
∈
fixed R , φ , the value of r decreases when increasing ω . A typical k = 1 solution is presented in
h h ∗ h
−
Figure 4 for c=1, α=1.
9
3
k=-1
2.5 Rh=2.1, wh=0.52, φh=0.01
φ(r)
2
R(r)
1.5
1
ω(r)
0.5
ν(r) x 100
0
-3 -2.5 -2 -1.5 -1
log (r)
10
Figure 4. The gauge function ω, the dilaton φ and the metric functions R, ν = e2X are shown as function
of thecoordinate r for a typicalk=−1 solution with an event horizon at Rh=2.1.
This fact can be understood by noticing that, for k =1, the boundary conditions (32) imply that R(r) is
6
a strictly decreasing function near the event horizon (i.e. R(r )<0). In all cases we studied, R(r) keeps
′ h
decreasing for increasing r, vanishing at r where there is a curvature singularity. For r > r , a wrong
∗ ∗
signature spacetime is found.
Therefore,itseemsthat,foraLiouvilledilatonpotential,allnonabelianconfigurationswithnonspheri-
caltopologypresentsomepathologicalfeatures. Giventhepresenceofthenakedsingularities,thephysical
significance of these solutions is not obvious. A similar property has been noticed in [21] for k = 0, 1
−
solutions of =4 D =4 Freedman-Schwarz gauged supergravity model which possesses also a Liouville
N
dilaton potential.
4.3 A counterterm proposal
In this section we’ll concentrate on the computation of mass and action of the generic k = 1 spherically
symmetric solutions for which R(r) as r . We start by presenting the asymptotic expression of
→∞ →∞
these solution [10], which is shared by both globally regular and black hole configurations
γ2 1
R = √2r +... +√2 r3/4e−2r(1+ +...)+ (e−3r), (33)
− 4√2r3/2 P r O
(cid:18) (cid:19)
φ 2 1 3 5√3γ2 3 1
φ = ∞ + r logr+ +... + r1/4e−2r(1+ +...)+ (e−3r),
√6 r3 − 4r2 64√2r2 ! r2P 2r O
γ K
ω = (1+...)+ r1/2e 2r(1+...)+ (e 3r), ν =1 e 2r+aφ∞(1+....),
− − −
√r C O − 25/2r3/4
where γ, K, , φ and are free parameters.
C ∞ P
Theconstructionoftheconservedquantitiesforthistypeofasymptoticsisaninterestingproblem. We
start by evaluating the on-shell action. By using the scalar and abelian field equations, one can show the
volume term in the action (1) reduces to the integral of a total derivative
1 1
I = d5x√ g 2φ ∂ (√ ge 4aφfµνW ) , (34)
B µ − ν
4π − −3a∇ − −
ZM (cid:18) (cid:19)
10