On timelike supersymmetric solutions of Abelian gauged 5dimensional supergravity
HJE
On timelike supersymmetric solutions of Abelian
Samuele Chimento 0
Instituto de F´ısica Teo´rica UAM/CSIC 0
0 C/ Nicol ́as Cabrera , 1315, C.U. Cantoblanco, E28049 Madrid , Spain
We consider 5dimensional gauged N = 1 supergravity coupled to Abelian vector multiplets, and we look for supersymmetric solutions for which the 4dimensional K¨ahler base space admits a holomorphic isometry. Taking advantage of this isometry, we are able to find several supersymmetric solutions for the ST[2, nv + 1] special geometric model with arbitrarily many vector multiplets. Among these there are three families of solutions with nv + 2 independent parameters, which for one of the families can be seen to correspond to nv + 1 electric charges and one angular momentum. These solutions generalize the ones recently found for minimal gauged supergravity in JHEP 1704 (2017) 017 and include in particular the general supersymmetric asymptoticallyAdS5 black holes of Gutowski and Reall, analogous black hole solutions with noncompact horizon, the three near horizon geometries themselves, and the singular static solutions of Behrndt, Chamseddine and Sabra.
Black Holes; Black Holes in String Theory; Supergravity Models

2
3
4
5
1
1 Introduction
3.1
Summary
Solutions
4.1
4.2
4.3
Ansatz
Introduction
Abelian gauged N = 1, d = 5 supergravity
2.1
Timelike supersymmetric solutions
Timelike supersymmetric solutions of Abelian gauged N
supergravity with one additional isometry
Solutions for the ST[2, nv + 1] model
Supersymmetric black holes
Conserved charges
Static solutions
still an open problem for rotating supersymmetric black holes in AdS5, see e.g. [2–4]).
However, while assuming unbroken supersymmetry makes the problem more tractable,
it is usually not enough to find explicit solutions, and one has to make some additional
assumptions or to impose a specific ansatz in order to solve the equations.1
An approach that has proven to be very successful in ungauged 5dimensional
supergravity, with or without vector multiplets, is to assume that the 4dimensional base space,
which for that theory has to be hyperKa¨hler, admits one triholomorphic isometry. In this
case the base space has a GibbonsHawking metric [6, 7], and it turns out that the
solutions can be completely characterized in terms of a small number of building blocks, namely
harmonic functions on 3dimensional flat space [8, 9]. The same ansatz has also been
effective for N = 1, d = 5 supergravity with vector multiplets and nonAbelian gaugings [10],
but without FayetIliopoulos terms, in which case the base space is again a 4dimensional
hyperKa¨hler space.
Recently [11] a similar ansatz was applied to the case of minimal d = 5 gauged
supergravity, where a U(1) subgroup of the SU(2) Rsymmetry group is gauged by adding
a FayetIliopoulos term to the bosonic action. In this case the base space is just K¨ahler,
instead of hyperKa¨hler, and the ansatz consists in assuming that it admits a holomorphic
isometry. The metric of the base space can then be written in terms of two functions [12]
in a form that generalizes the GibbonsHawking metrics, and the problem of finding
supersymmetric solutions is reduced to that of solving a system of fourth order differential
equations for these two functions plus a third one.
The aim of this paper is to apply the same ansatz in the case of N
gauged supergravity [11]. They are studied in some detail in subsection 4.3, where the
conserved charges are computed for one of the families, and it is shown that they include as
particular cases black holes with compact or noncompact horizon, as well as static singular
solutions. In subsection 4.4 we give the explicit expression of the fields for supersymmetric
black holes not included in the solutions of subsection 4.3, despite being very similar to a
subcase of them. We conclude in section 5 with some final remarks.
1For a comprehensive review of supersymmetric solutions of supergravity theories with many references
see, e.g. ref. [5].
2By superficially asymptoticallyAdS we mean that the metric components approach those of AdS in an
appropriate limit, which however does not guarantee that the solutions are globally asymptoticallyAdS.
– 2 –
HJEP07(21)59
Abelian gauged N
= 1, d = 5 supergravity
In this section we give a brief description of the bosonic sector of a general theory of
N = 1, d = 5 supergravity coupled to nv vector multiplets in which a U(1) subgroup of the
SU(2) Rsymmetry group has been gauged by the addition of FayetIliopoulos (FI) terms.
The U(1) subgroup to be gauged and the gauge vector used in the gauging are determined
by the tensor PI r, as we are going to explain.3 Our conventions are those in refs. [
13, 14
]
which are those of ref. [
15
] with minor modifications.
The supergravity multiplet is constituted by the graviton eµa , the gravitino ψµi and the
graviphoton Aµ . All the spinors are symplectic Majorana spinors and carry a fundamental
SU(2) Rsymmetry index. The nv vector multiplets, labeled by x = 1, . . . , nv consist of a
real vector field Aµx, a real scalar φx and a gaugino λi x.
It is convenient to combine the matter vector fields Aµx with the graviphoton Aµ ≡ Aµ0
into a vector (AµI ) = (Aµ0 , Aµi ). It is also convenient to define a vector of functions of the
scalars hI (φ). N = 1, d = 5 supersymmetry requires that these nv + 1 functions of the nv
scalars satisfy a constraint of the form
CIJK hI (φ)hJ (φ)hK (φ) = 1 ,
where the constant symmetric tensor CIJK completely characterizes the ungauged theory
and the Special Real geometry of the scalar manifold. In particular, the kinetic matrix of
the vector fields aIJ (φ) and the metric of the scalar manifold gxy(φ) can be derived from
it as follows: first, we define
hI ≡ CIJK hJ hK ,
hI hI = 1 ,
The metric of the scalar manifold gxy(φ), which we will use to raise and lower x, y
3Although its origin is different, it can be understood as a particular example of embedding tensor.
– 3 –
and
√
h
I
x ≡ −
3hI ,x ≡ −
√ ∂hI
3
∂φx
,
Then, aIJ is defined implicitly by the relations
It can be checked that
indices is (proportional to) the pullback of aIJ
We will use the completeness relation
hIx ≡ + 3hI,x ,
⇒
hI hIx = hI hIx = 0 .
⇒
√
hI = aIJ hI ,
hIx = aIJ hJ x .
aIJ = −2CIJK hK + 3hI hJ .
gxy ≡ aIJ hI xhJ y = −2CIJK hIxhyJ hK .
hI hJ + gxyhx I hyJ = δI J .
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
The FI gauging of any model of N = 1, d = 5 supergravity coupled to vector multiplets
is completely determined by the choice of PI r, where r = 1, 2, 3 is a su(2) index. In the
Abelian case, this tensor can be factorized as follows:
PI r = gcI dr ≡ gI dr ,
where g is the gauge coupling constant, dr (which we can normalize drdr = 1) chooses
a direction in S3 or, equivalently, a u(1) ⊂ su(2) to be gauged and cI (also normalized
cI cI = 1) dictates which linear combination of the vector fields, cI AI µ , acts as gauge field.
gI = gcI is a convenient combination of constants that we will use. We will not make any
specific choices for the time being.
The bosonic action is given in terms of aIJ , gxy and CIJK and PI r
S =
Z d5x√g
R + 12 gxy∂µ φx∂µ φy − V (φ) − 14 aIJ F Iµν F J µν
+
CIJK εµνρσα
12√3√g
F I µν F J ρσAK α ,
where the Abelian vector field strengths are F I µν = 2∂[µ AI ν] and the scalar potential V (φ)
is given by
V (φ) = − 4hI hJ − 2gxyhIxhyJ PI rPJ r = −4CIJK hI PJ rP r
K .
The equations of motion for the bosonic fields are
∇ν aIJ F J νµ
+ √
found in refs. [16–18]. In what follows we are going to review it using the notation and
results of ref. [
14
] in which general nonAbelian gaugings were considered,5 but restricting
to Abelian FI gaugings.
The building blocks of the timelike supersymmetric solutions are the scalar function
fˆ, the 4dimensional spatial metric hmn,6 an antiselfdual almost hypercomplex structure
Φˆ (r)mn,7 a 1form ωˆm, the 1form potentials AˆI m and the scalars of the theory combined
4A timelike (commuting) spinor ǫi is, by definition, such that the real vector bilinear constructed from
it iVμ ∼ ǫ¯iγμǫi is timelike.
5Even more general gaugings were considered in [19] with the inclusion of tensor multiplets.
6m, n, p = 1, · · · , 4 will be tangent space indices and m, n, p = 1, · · · , 4 will be curved indices. We are
going to denote with hats all objects that naturally live in this 4dimensional space.
7That is: the 2forms Φˆ(r)mn r, s, t = 1, 2, 3 satisfy
Φˆ(r) mn = − 1 εmnpq Φˆ(r)pq ,
2
Φˆ(r) mn Φˆ(s) np = −δrsδmp + εrst Φˆ(t) mp .
or
Φˆ(r) = − ⋆4 Φˆ(r) ,
– 4 –
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
(2.13)
(2.14)
(2.15)
HJEP07(21)59
into the functions hI (φ). All these fields are defined on the 4dimensional spatial manifold
usually called “base space”. They are timeindependent and must satisfy a number of
conditions:
1. The antiselfdual almost hypercomplex structure Φˆ(r)mn, the 1form potentials AˆI m
and the base space metric hmn (through its LeviCivita connection) satisfy the
equation
∇ˆ m Φˆ(r)np + εrstAˆI mPI sΦˆ (t)np = 0 .
hI FˆI+ = √2 (fˆdωˆ)+ ,
3
FˆI− = −2fˆ−1CIJK hJ PK rΦˆ (r) .
2. The selfdual part of the spatial vector field strengths FˆI
the function fˆ, the 1form ωˆ and the scalars of the theory by
≡ dAˆI must be related to
3. while the antiselfdual part is related to the almost hypercomplex structure by8
4. Finally, all the building blocks are related by the equation
1
2 3 aIK − 2CIJK h
J FˆK · (fˆdωˆ)− = 0 ,
(2.20)
where the dots indicate standard contraction of all the indices of the tensors.
Once the building blocks that satisfy the above conditions have been found, the physical
5dimensional fields can be built out of them9 as follows:
1. The 5dimensional (conformastationary) metric is given by
2. The complete 5dimensional vector fields are given by
ds2 = fˆ2(dt + ωˆ)2 − fˆ−1hmndxmdxn .
√
AI = −
3hI e0 + AˆI , where e0 ≡ fˆ(dt + ωˆ) ,
so that the spatial components are
and the 5dimensional field strength is
A m = AˆI m −
I
√
3hI fˆωˆm ,
F I = −
√3d(hI e0) + FˆI .
– 5 –
(2.16)
(2.17)
(2.19)
(2.21)
(2.22)
(2.23)
(2.24)
(2.18)
useful relations
by using the condition eq. (2.1).
8In this equation the indices of CIJK have been raised using the inverse metric aIJ and one has the
CIJKhK = hI hJ − 1 gxyhIxhyJ =
3 hI hJ − 1 aIJ .
2
2
2
9In the ungauged case the above conditions determine the quotients hI /fˆ from which fˆ can be found
where we have defined
3. The scalar fields φx can be obtained by inverting the functions hI (φ) or hI (φ). A
parametrization which is always available is
As it has already been observed in refs. [16, 18] choosing dr = δr1 we see that eq. (2.16)
gives us additional information: it splits into
other two equations is11
The first equation means that the metric hmn is K¨ahler with respect to the complex
structure Jˆmn ≡ Φˆ (1)mn. Taking this fact into account,10 the integrability condition of the
ˆ ˆ
Rmn = −2∇ˆ [mPn] = −gI FˆI mn .
This equation must be read as a constraint on the 1form potentials AˆIm posed by the
choice of base space metric.
Eq. (2.19) takes a simpler form as well:
FˆI− = −2fˆ−1CIJK hJ gK Jˆ,
⇒
( gI FˆI− = 12 fˆ−1V (φ)Jˆ,
hI FˆI− = −2fˆ−1gI hI Jˆ.
(2.25)
(2.26)
(2.27)
(2.28)
(2.29)
(2.34)
(2.35)
(2.36)
(2.30)
(2.31)
(2.32)
(2.33)
which leads to the relation between the Ricci and Riemann tensors
The Ricci 2form, defined as
is, therefore, related to the Riemann tensor by
Rˆ = −2V /fˆ.
Rˆmnpq = RˆmnrsJˆrpJˆsq ,
Rˆmn = − 21 RˆmprqJˆrqJˆpn .
Rˆ mn ≡ RˆmpJˆpn ,
Rˆ mn = 12 RˆmnpqJˆpq .
Tracing the first of these equations and eq. (2.34) with Jˆmn one finds a relation between
the Ricci scalar of the base space metric, the scalar potential and the function fˆ:
10We use the integrability condition of eq. (2.26)
11If Pm vanishes (for instance, in the ungauged case), then we have a covariantly constant hyperK¨ahler
structure and, then, the base space is hyperKa¨hler.
– 6 –
The last equation to be simplified by our choice is eq. (2.20). Substituting in it
eq. (2.35) and using eqs. (2.18) and the completeness relation eq. (2.7) one finds
1
1
In order to make progress one has to start making specific assumptions about the base
space metric. In the ungauged [8, 13] and the nonAbelian gauged cases [10] it has proven
very useful to assume that the base space metric has an additional isometry because, then,
it depends on a very small number of independent functions. Recently the same assumption
was made for pure gauged supergravity [11], where the base space can be a general K¨ahler
metric, allowing to reduce the problem of finding supersymmetric solutions to a system
of fourth order differential equations for three functions. In what follows we are going to
make the same assumption for the case at hand, in which vector multiplets are present, in
the attempt to simplify the task of finding supersymmetric solutions.
3
Timelike supersymmetric solutions of Abelian gauged N
supergravity with one additional isometry
Any fourdimensional K¨ahler metric with one holomorphic isometry can be written locally
as [12]:
ds2 = H−1 (dz + χ)2 + H (dx2)2 + W 2[(dx1)2 + (dx3)2] ,
with the functions H and W , and the 1form χ, depending only on the three coordinates
xi and satisfying the constraints:
whose integrability condition is
In a frame defined by the Vierbein
(dχ)12 = ∂3H ,
(dχ)23 = ∂1H ,
(dχ)31 = ∂2 W 2H ,
D2H ≡ ∂12H + ∂22 W 2H
+ ∂32H = 0 .
e♯ = H−1/2 (dz + χ) ,
e2 = H1/2dx2 ,
e1,3 = H1/2W dx1,3 ,
(Jˆmn) =
02×2
1
− 2×2
1
2×2
02×2
!
the conserved complex structure is given by
The Ricci tensor and Ricci scalar of the 4dimensional metric can be expressed in terms of
the functions H and W 2 in a compact form,
Rˆmn = ∇ˆ m∇ˆ n log W + JˆmpJˆnq∇ˆ p∇ˆ q log W ,
Rˆ = ∇ˆ 2 log W 2 ,
where the 4dimensional Laplacian acts on zindependent functions as
and ∇2 is the Laplacian operator associated with the 3dimensional metric
The expression for the Ricci scalar should be compared with eq. (2.36).
We will take the base space metric hmndxmdxn to be of the form (3.1), and we will
make the identification Φˆ(1) = Jˆ. We can solve for Pˆm in eqs. (2.27) and (2.28) if we choose
a particular form for the complex structures Φˆ (2,3). Without loss of generality they can be
,
where σ2 is the second Pauli matrix
Then we find that the flat components of P can be written in the compact form
On the other hand, recalling the definition of Pˆm eq. (2.29) we find for the gauge vector
and its field strength
gI AˆI m = Jˆmn ∂n log W ,
gI FˆI mn = −Rmn = −2∇ˆ [m∇ˆ p log W Jˆpn] .
written in terms of a 1form living on the 3dimensional space ϑ = ϑidxi as
Every (anti)selfdual 2form F ± on the four dimensional K¨ahler base space can be
F ± = (dz + χ) ∧ ϑ ± H ⋆3 ϑ .
The 2forms we consider here are also zindependent and so will the components of the
corresponding 1forms be. Thus, we introduce the zindependent 3dimensional 1forms
ΛI , ΣI , Ω
± defined by
1
1
FˆI+ = − 2 (dz + χ) ∧ Λ − 2
I
FˆI− = − 2 (dz + χ) ∧ ΣI +
H ⋆3 ΛI ,
I
H ⋆3 Σ ,
1
1
2
(dωˆ)± = (dz + χ) ∧ Ω± ± H ⋆3 Ω± ,
– 8 –
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
(3.16)
(3.17)
(3.18)
(3.19)
Comparing the expression of FˆI− with eq. (2.35) and those of hI FˆI+ and (dω)+ with
eq. (2.17) we conclude that
Σ
which means that, locally,
for some functions KI .
equation, one also gets
From the same condition, using eq. (3.3) and the definition of the operator D2 in that
D2KI = 2 ∂2 HW 2ΣI2 .
Using eq. (3.15) and its full contraction with Jˆ one finds
where an integration constant reflecting the possibility of adding to the solutions KI of
eq. (3.24) solutions of the homogeneous equation has been set to zero without loss of
generality, since from (3.23) it is clear that the KI ’s are defined up to a constant times H.
Using these relations, eq. (3.24) contracted with gI is automatically satisfied, leaving nV
independent equations.
It is convenient to rewrite ωˆ as
in terms of which
From eqs. (3.21) and (3.23) we find that
and, then, from eq. (3.27) we find that
ωˆ = ωz (dz + χ) + ω ,
ω = ωidxi ,
1
Ω± = ± 2
H−1 (ωz ⋆3 dχ + ⋆3dω) − 2
1
dωz .
Ω+ = − 4 f
ˆ
√
3 hI d KI /H
− Σ
I ,
Using either of the last two equations in eq. (3.27) one gets an equation for ω:
Ω− = −Ω+
− dωz =
√
4 f
ˆ
3 hI d KI /H
− Σ
I
− dωz .
dω = H ⋆3 dωz − ωzdχ −
3 hI H ⋆3 d KI /H
− Σ
– 9 –
and its integrability equation is just13
Before calculating its integrability condition it is convenient to make a change of
variables (identical to the one made in the ungauged case) to (partially)
“symplecticdiagonalize” the righthand side. Thus, we define LI and M through
hI /fˆ ≡ LI + 112 CIJK KJ KK /H ,
√
ωz ≡ M + 43 LI KI /H +
241√3 CIJK KI KJ KK /H2 .
Substituting these two expressions into eq. (3.30) and using the relation between the
1form χ and the functions H and W , eqs. (3.2), the equation for ω takes the form12
dω = ⋆3
HdM − M dH +
KI dLI − LI dKI
− H
ωz∂2 log W 2
− 2 3hI gI fˆ−2 dx2 ,
3
√
4
√
√
4
3
2
1
H∇ M − M ∇ H +
2
KI ∇2LI − LI ∇2KI
− W 2 ∂2 nHW 2 ωz∂2 log W 2
− 2√3hI gI fˆ−2 o = 0 .
This equation can be simplified by using the equations satisfied by the functions H and
KI (3.3) and (3.24), respectively. We postpone doing this until we derive the equation for
the functions LI , which follows from eq. (2.37). First of all, observe that, with our choice
of complex structure eq. (3.7)
Jˆ · (dωˆ) = 4(dωˆ)0−2 = 4Ω2− = √3 hˆI ∂2 KI /H
f
I
− Σ2 − ∂2ωz .
On the other hand, we have
ˆ 2 hI /fˆ =
∇
1
H ∇
2
hI /fˆ ,
FˆJ · ˆ⋆FˆK = ΛJmΛKm − ΣJmΣKm = ∂m H
CIJK H∂m H
KJ
KK
∂m H
= CIJK
" 2
∇
KJ KK
2H
KJ
+
KK
∂m H
− 2∂m
K(J
H
ΣKm) ,
KJ KK
2
2H2 ∇ H −
KJ 2
∇ KK #
H
and, using all these partial results into eq. (2.37), and (not everywhere, for the sake of
simplicity) the new variables eqs. (3.31), we arrive at
2
∇ LI − CIJK
" 1 KJ KK
12
H2
2
∇ H +
12We have left one ωz in order to get a more compact expression.
(3.31)
(3.32)
(3.33)
(3.34)
(3.35)
(3.36)
We can now use the relation between the 3dimensional Laplacian and the D2 operator
and the equations for the functions H and KI (3.3) and (3.24)
2
∇ H =
∇2KI =
D2H
and the equation for LI becomes
2
∇ LI +
C3WIJK2 ∂2 W 2KJ Σ2K − 4
1
H−1KJ KK ∂2W 2
+gI H
= 0 .
This equation, once substituted in eq. (3.33), gives
CIJK
2
∇ M = − 48√3W 2 ∂2 H−2KI KJ KK ∂2W 2 +
In this subsection we summarize for convenience the recipe to find a solution. One has to
solve the system of equations given by (3.3), (3.24), (3.38) and (3.39),
D2H ≡ ∂12H + ∂22 W 2H + ∂32H = 0 ,
D2KI = 2 ∂2 HW 2ΣI2 ,
2
∇ LI +
C3WIJK2 ∂2 W 2KJ Σ2K − 4
1
H−1KJ KK ∂2W 2
+ gI H
with hfˆI and ωz given by (3.31),
1
√
3
4
hI /fˆ ≡ LI +
12 CIJK KJ KK /H ,
ωz ≡ M +
LI KI /H +
1
24√3 CIJK KI KJ KK /H2 ,
(3.37)
(3.38)
(3.39)
(3.40)
(3.41)
(3.42)
(3.43)
(3.44)
HJEP07(21)59
for the functions H, W 2, KI , ΣI , LI and M while imposing the constraints (3.20)
2
and (3.25),
Σ
This is still a very difficult problem, in particular because the constraint (3.45) involves
the symmetric tensor CIJK with raised indices, which in general is not constant and cannot
be written in a simple way in terms of, for instance, the functions hfˆI .
To simplify the task one could assume that CIJK is constant, as is the case for several
interesting models, in which case (3.45) and (3.44) allow to write ΣI2 in terms of H, KI
and LI . One could then proceed as follows: first choose two functions H and W 2 solving
equation (3.40), which amounts to choosing a base space, and subsequently solve the system
of second order equations given by (3.41), (3.42) and (3.43) for KI , LI and M , subject to
the algebraic constraints (3.46).
Once all these functions are known, eq. (3.44) gives hˆI and ωz, equations (3.2)
f
and (3.30) can be integrated to give respectively χ and ω, ωˆ is given by (3.26) and fˆ can be
obtained from the functions hfˆI using the special geometric constraint CIJK hI hJ hK = 1.
At this point one has all the ingredients to write explicitly the metric (2.21), the scalar
fields (2.25) and the gauge field strengths (2.24), using equations (3.17), (3.18) and (3.23).
4
4.1
Solutions
Ansatz
Assume14 for simplicity that H only depends on the ̺ coordinate, H = H(̺), and that W 2
factorizes as W 2 = Ψ(̺)Φ(x1, x3). Then from (3.40)
H =
a̺ + b
Ψ
We will also assume a 6= 0,15 in which case one can set a = 1 and b = 0 by shifting and
rescaling the coordinate ̺, so that
Inspired by the pure supergravity case [11] we will take Ψ to be a third order polynomial
in ̺. In particular eq. (3.46), which implies
14In what follows we will rename the coordinate x2 to ̺, both for improved readability and for the natural
interpretation as “radial” coordinate.
15For pure supergravity taking a = 0 leads to Go¨dellike solutions [11].
We expect that for the
ST[2, nv + 1] model we will be considering in this paper, which admits a truncation to pure
supergravity, this choice would give a generalization of those solutions. We will leave the study of this possibility for
future work.
(4.1)
(4.2)
(4.3)
suggests to introduce nv + 1 polynomials
such that Ψ = gI ΨI and
Eq. (3.41) can be integrated to give
where αI are integration constants, which we will take to be independent of x1 and x3.
Eq. (3.46) implies then that Φ must be a solution of Liouville’s equation
with k given by
It is possible to choose without loss of generality k = 0, ±1 and
Φ = Φ(k) ≡
4
{1 + k [(x1)2 + (x3)2]}2
Equation (3.2) then determines χ up to a closed 1form,
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
(4.10)
(4.11)
(4.12)
HJEP07(21)59
One can then, after computing the functions LI from the definition (3.44), use
equation (3.42) to obtain an expression for ∂̺M . Since the expression must be the same for
16This is the case for instance when the scalar manifold is a symmetric space.
dχ = Φ dx3 ∧ dx1
=⇒
χ = χ(k) ≡ 1 + k [(x1)2 + (x3)2] .
2 x3dx1 − x1dx3
We now focus our attention on special geometric models for which the totally symmetric
tensor with raised indices CIJK is constant.16 Comparing the expression for ΣI in (4.6)
with the one in (3.45) it seems a natural choice to introduce nv + 1 first order polynomials
in ̺, QI , such that
with eq. (4.6) implying the constraints
hˆI =
f
QI
8̺
,
c3I =
c2I =
6
1
2
QI ≡ q0I + q1I ̺ ,
1 CIJK gJ q1K
αI + CIJK gJ q0K .
each of the nv + 1 equations (one for each value of I), the following proportionality
conditions must be met:
CIJK c3J c3K
∝ gI
4CIJK c3J (αK
− 2c2K ) + gJ c3J q0I ∝ gI
4CIJK (αJ
− 2c2J )c1K + 3gJ c1J q0I ∝ gI
2CIJK c1J c1K
− 3gJ c0J q0I ∝ gI .
After this, all that remains to do is to substitute ∂̺M in eq. (3.43) (we also assume for
simplicity M = M (̺)) and solve the resulting algebraic equation.
HJEP07(21)59
Solutions for the ST[2, nv + 1] model
In order to find explicit solutions we will consider a specific model, namely the ST[2, nv + 1]
model defined by
C0xy = C0xy =
ηxy ,
√
3
2
where x, y = 1, . . . , nv, ηxy is the Minkowski nvdimensional metric, and the other
components of CIJK vanish. This model reduces to pure supergravity for nv = 1 and h
and includes as a special case the STU model for nv = 2. In what follows xtype indices
will be raised and lowered with ηxy and their contraction will be denoted by a dot (e.g.
g ·c1 ≡ gxc1x). The constraints (4.12) become
c30 = 4√1 3 g ·q1
c20 = α20 + 43 g ·q0
√
c3x = 4√1 3 (gxq10 + g0q1x)
√
c2x = α2x + 43 (gxq00 + g0q0x) .
(4.13)
(4.14)
1 = h0,
(4.15)
The conditions (4.13) and equation (3.43) are satisfied for an arbitrary choice of gauging
constants gI only if one of the following sets of conditions is met:
1. q00 = 4 (c10)2 q0 ·q0 , c1x = √23 gIc1c00I q0x , q1x = qg100 gx
√3 gI c0I
2. q1x = qg100 gx , c10 = g ·q0 = g ·c1 = q0 ·c1 = c1 ·c1 = gI c0I = 0
3. q1x = − qg100 gx , q0I = 0 ∀I , c1x = 0 ∀x
4. q1x = − qg100 gx , q0I = 0 ∀I , c10 = c1 ·c1 = 0
5. q1 ·q1 = −( qg100 )2 g ·g , g ·q1 = 0 , q0I = 0 ∀I , c1x = 0 ∀x
6. q1 ·q1 = −( qg100 )2 g ·g , g ·q1 = 0 , q0I = 0 ∀I , c10 = c1 ·c1 = 0
7. q00 = 4 (c10)2 q0 ·q0 , c1x = √23 gIc1c00I q0x , q10 = g ·q0 = g ·q1 = q0 ·q1 = q1 ·q1 = 0
√3 gI c0I
8. q10 = c10 = g ·q0 = g ·q1 = q0 ·q1 = q1 ·q1 = g ·c1 = q0 ·c1 = c1 ·c1 = gI c0I = 0
For special choices of the gauging there are some other possibilities.
1. q00 = 43 (cg1·c00)2 q0 ·q0 , c1x = √23 gc·1c00 q0x , q10 = 0
√
2. q10 = c10 = g ·q0 = g ·c1 = q0 ·c1 = c1 ·c1 = g ·c0 = 0
The function fˆ can be computed from (4.11) using the special geometric
constraint (2.1), giving
We are interested in particular in asymptotically antide Sitter solutions. Given that the
line element of AdS5 (with radius ℓ) can be written in standard supersymmetric form as [11]
2
ds2 =
dt +
̺ dz + χ(k)
− ̺ k +
4
ℓ2
̺
one expects that for such solutions as ̺ → ∞ fˆ tends to a constant and Ψ diverges like ̺3.
These conditions translate to
q10 q1 ·q1 6= 0
and
gI c3I =
√ (2g0g ·q1 + q10g ·g) 6= 0 ,
1
excluding all the solutions above except the first six for arbitrary gauging. Out of these,
however, only the first two are actually asymptotically AdS, at least locally, since in the
other cases ωz does not present the correct behavior, being proportional to ̺−1 (one can
also check that their scalar curvature does not tend to a constant as ̺ → ∞). In the
following we will analyze some properties of these two cases.
(4.17)
(4.18)
We will now analyze in detail the solutions with parameters satisfying the conditions
The functions fˆ and Ψ become
while ωz can be obtained from eq. (3.44) after integrating ∂̺M ,
where d is an arbitrary constant, and ω from eq. (3.30)
ωz =
3
64
gI c0I 1
q0 ·q0 c10 ̺2 +
g0
(q10)2 g·g ̺ + d
+ q0 ·q0 + √
√
3 gI c0I
!
#
− d χ(k) .
2
g·g ̺2
!#1/3
, (4.20)
HJEP07(21)59
(4.19)
(4.21)
(4.22)
(4.23)
(4.24)
(4.25)
(4.26)
Since ω is of the form ω˜χ with ω˜ constant, it is always possible to reabsorb ω in ωz with a
shift in the t coordinate, t → t + ω˜z, leading to ω = 0 and
ωz =
64̺2
g0
3 " (q10)2 g·g ̺3 + q10
g0
√
!
2
+ q0 ·q0 + √
4 q10 c10
.
c1I → αc1I , c0I → α2c0I . Since we are assuming q10 6= 0 we can use this freedom to set
The full solution is invariant under the rescaling t → t/α, ̺ → α̺, q10 → q10/α,
where we introduced for convenience the constant ℓ defined by17
so that fˆ → 1 for ̺ → ∞.
17The solutions presented here are superficially asymptotically AdS5, with AdS radius ℓ.
8
q10 = √3 g0ℓ ,
ℓ3g0 g·g = 2 ,
with
ds2 = fˆ2 dt + ωz dz + χ(k)
ωz = ̺ +
2
ℓ
!
2
+ q0 ·q0 +
̺
4
Using the parametrization (2.25) the physical scalars are given by
The full gauge potentials are given, according to eq. (2.22), by
φx =
hx =
h0
hx/fˆ
h0/fˆ
=
8gxℓ̺ + √3q0x
8g0ℓ̺ + 34 (gcI1c00)I2 q0 ·q0
.
AI = −√3hI fˆ dt + ωz dz + χ(k)
+ AˆI ,
where the 4dimensional part AˆI can be obtained from (3.17), (3.18), (3.23),
Aˆ0 =
Aˆx =
g·gℓ̺ +
2g0gxℓ̺ +
while since hI = CIJK hJ hK ,18
√
3
4 g·q0 +
1 c10 !
2 ̺
√
3
4 g0q0x +
√3 gI c0I q0x !
4 c10 ̺
.
Pure supergravity is recovered by choosing gx = g0δx1, q0x = q00δx1 and q1x = q10δx1.
With this choice one recovers the class of asymptotically AdS solutions of minimal gauged
N = 1, d = 5 supergravity found in [11].
For each value of k the solutions are determined by nv + 2 parameters, q0x, c10 and
gI c0I . The metric however only depends on the q0x’s through the combinations g ·q0 and
q0·q0, so it is always determined by four parameters, independently of the number of vector
multiplets nv.
18Note that here hx 6= ηxyhy.
If an event horizon exists, it must be situated in ̺ = 0, where fˆ = 0 and the supersymmetric
Killing vector ∂t becomes null. Since fˆ, H and ωz only depend on ̺, it is possible to perform
a coordinate change such that
after which the metric takes the form
The combination (fˆ−1H−1 − fˆ2ωz2) tends to a constant in the limit ̺ → 0, so the
hypersurface ̺ = 0 is null, and is thus a Killing horizon, if fˆ2ωz goes to zero. The only possibility
to satisfy this condition without giving rise to singularities is to take the scaling limit
in which case the functions that determine the metric become
√
√
3
4
(4.36)
(4.37)
(4.38)
(4.39)
(4.40)
(4.41)
(4.42)
(4.43)
(4.44)
(4.45)
gI c0I = √3 q0 ·q0
√
(2g0g·q0 + q00g·g) ,
#
3
1
2
,
√ (2g0g·q0 + q00g·g) ̺ +
(g0q0 ·q0 + 2q00g·q0) .
For k = 1 these are the supersymmetric black holes of [17] with the choice (4.14), while for
k = 0 and k = −1 one gets a generalization of the black holes with noncompact horizon
found in [11] for pure gauged supergravity.
For them to be regular, any curvature singularity should lie behind the horizon ̺ = 0.
Since the curvature scalars diverge when fˆ−3 vanishes, then the zeroes of (4.40) must be
negative, which translates to the conditions
and either
in which case there is only one real root, or
q00 g·g > 0 ,
q0 ·q0 g·g > 0 ,
(g·q0)2 < q0 ·q0 g·g ,
(g·q0)2 ≥ q0 ·q0 g·g and g·q0 g0 > 0 ,
in which case all roots are negative. Further constraints on the parameters come from the
requirement
that also implies H > 0.
The near horizon geometries of these black holes are themselves supersymmetric
solutions and are included in the class of solutions we presented. They can be obtained from
equations (4.20), (4.21) and (4.24) by taking the limit (4.39) and choosing q10 = 0. They
are analogous to the three near horizon geometries obtained in [20] for pure supergravity,
in particular one can easily see from (4.38) that dimensional reduction along v gives the
geometries AdS2 × S2, AdS2 × H2 or AdS2 ×
E2, and that the horizon geometry is given
HJEP07(21)59
by a homogeneous Riemannian metric on the group manifolds SU(2) (in which case the
metric is that of a squashed S3), SL(2, R) or N il respectively for k = 1, −1 or 0. The
entropy density is
A(k)
3πV(k)
3
− 16
1 √3 (√
where A(k) is the area of the horizon and
V(k) ≡
Z
Φ(k)dv ∧ dx1 ∧ dx3 ,
so that the entropy for the k = 1 case is S(1) = 16π2s(1), in agreement with the horizon
area computation in [17].
4.3.2
Conserved charges
For k = 1 the class of solutions we presented is asymptotically globally AdS5 according to
the definition given by Ashtekar and Das in [21].19 It is then possible to use the prescription
in the same paper20 to compute the AD mass and angular momenta.
The mass is the conserved charge associated with the timelike Killing vector field
V =
∂
∂t
+
2 ∂
ℓ ∂z
.
This is the correct vector rather than the one associated with supersymmetry, since in
coordinates adapted to V the metric of AdS5, and in particular the metric on the conformal
boundary, is written in static form. The value of the mass is
M =
g0ℓ2
2√3 g ·q0 +
1 gI c0I
8g0ℓ (c10)2 q0 ·q0
+
3
32ℓ
q0 ·q0 −
19See [11] for a discussion of the asymptotics of a similar class of solutions in pure gauged supergravity.
20The AshtekarDas paper extends to higher dimensions the original fourdimensional results obtained
in an earlier paper by Ashtekar and Magnon [22]. The formalism is thus sometimes referred to as AMD
(AshtekarMagnonDas) formalism.
2
z = ψ + ϕ + t ,
x1 = tan cos ϕ ,
x3 = tan sin ϕ ,
θ
2
2
ℓ
Before computing the angular momenta, we perform the coordinate change
The angular momenta are the conserved charges associated with the Killing vectors ∂ϕ and
1
64
aIJ ∗ F J
2g0 −
2g0gx −
√3 gI c0I
√3 gI c0I
(c10)2 g·q0 − 4 (gcI1c00)I2 q0 ·q0 ,
(c10)2 (gxg·q0 − g·gq0x) + √3 q0x . (4.57)
16
(4.51)
(4.52)
(4.53)
.
(4.54)
(4.55)
(4.56)
(4.58)
(4.59)
(4.60)
(4.61)
(4.62)
HJEP07(21)59
2
⇒
M − ℓ J  = 4ℓg˜I QI 
1
g0ℓ2
g˜0 =
, g˜x =
2 gx
ℓ2 g·g
.
where we have defined
g˜I ≡ ̺l→im∞ aIJ gJ
4.3.3
Static solutions
in terms of fˆ,
With the choice c10 = 332ℓ q0 ·q0 the functions Ψ and ωz can be expressed in a simple way
It is straightforward to verify that the following BPS condition, obtained in [17] for
the black hole limit of the solutions, is satisfied for all values of the parameters:
with fˆ given by
Ψ = ℓ2
ωz = 2 ̺fˆ−3 ,
ℓ
4 ̺3fˆ−3 + k̺2 ,
ϕ = α − β ,
θ = 2ϑ ,
ds2 = − fˆ2
−1 + 2 ̺fˆ−3 dt2 −
4
l
d̺2
− fˆ
− cosh2 ϑdβ2 + dϑ2 + sinh2 ϑdα2 .
dψ + χ(1)
2 + dΩ(22,1) = 4dΩ2S3 = dψ˜ + cos θdϕ
2
+ dθ2 + sin2 θdϕ2 .
change
one has
For k = 1 one can see that substituting the chosen value of c10 in (4.54) the angular
momentum vanishes as expected. In this case the threedimensional part of the metric
contained in the square brackets is just the metric of a 3sphere, with the coordinate
ψ = ψ˜ + ϕ ,
x1 = tan
cos ϕ ,
x3 = tan
sin ϕ ,
(4.63)
(4.64)
(4.65)
(4.67)
(4.68)
(4.69)
(4.70)
(4.71)
(4.72)
where
HI ≡ 3 gI − ̺
QI
and the QI ’s, that for k = 1 are the electric charges (4.56) and (4.57), are
Q0 = −
32ℓ2 gI c0I ,
The gauge potentials and scalar fields can also be written in a simple way in terms of the
A0 = − 3H0
dt
Ax = − 3 H·H
φx = Hx .
H0
For k = ±1 it is possible to remove from the metric the cross term proportional to
dt(dz + χ) by performing a simple shift in the z coordinate, z = ψ + ℓ2k t, and rewrite the
4
l
k + 2 ̺fˆ−3 dt2 −
d̺2
̺fˆ k + l42 ̺fˆ−3
̺ h
− fˆ k dψ + χ(k)
2 + dΩ(22,k)i . (4.66)
Note that these coordinates are static for k = 1 but not for k = −1, since in that case the
time coordinate is actually ψ, while t is spatial. However the metric can still be rewritten
in static form making first the coordinate change
ψ = ψ˜ − ϕ ,
x1 = tanh
cos ϕ ,
x3 = tanh
sin ϕ ,
dψ + χ(−1) = dψ˜ − cosh θdϕ ,
dΩ(22,−1) = dθ2 + sinh2 θdϕ2 ,
functions HI ,
solutions as
ds2 =
so that
followed by a second change,
after which it takes the form
θ
2
θ
2
2
θ
2
This solution was first found in [
23
], and can be seen as a generalization in the presence of
vector multiplets of the BPS limit of the ReissnerN¨ordstromAdS5 black hole, to which it
reduces in the pure supergravity case.
For k = 0 it is not possible to eliminate the cross term in a simple way, and the metric is
ds2 = fˆ2dt2 +
4 ̺
ℓ fˆ
dt dz + χ(0) − 4
ℓ2 fˆ2d̺2
̺2
̺
In the pure supergravity case this reduces to a metric without free parameters and having
constant curvature scalars [11]. Here this is not true in general, and only happens if
HJEP07(21)59
H·H =
2
in which case the metric is the same as in the pure supergravity case, but it is still possible
to have independent vector fields and nontrivial scalar fields.
are almost identical to the black hole limit of the ones in subsection 4.3, given in
equations (4.40), (4.41) and (4.42), with the additional constraint g·q0 = 0. However there is an
additional term in the 4dimensional gauge potentials Aˆx proportional to the constants c1x,
which were zero in the aforementioned limit. These constants are not completely arbitrary,
being constrained by the relations g ·c1 = q0 ·c1 = c1 ·c1 = 0.
After the rescaling (4.25) the functions determining the metric are
while the scalars are
√
√
3
4
q00g ·g
,
3
!
64g ·gℓ2 ̺2
8gxℓ̺ + √3q0x ,
8g0ℓ̺ + √3q00
and the gauge potentials are of the form (4.32), with
Aˆ0 =
g ·gℓ̺ +
"
1 c10
2 ̺
√
4
3
Aˆx = 2g0gxℓ̺ +
(q00gx + g0q0x) +
(4.73)
(4.74)
(4.76)
(4.77)
(4.78)
(4.79)
(4.80)
(4.81)
and
h0fˆ = √3(8g0 ℓ̺ + √3q00)
,
8̺
.
For k = 1, the mass, angular momenta and electric charges are
M =
g0 +
√
3
Keeping into account the constraints to which the constants q0x and c1x are subject, it is
easy to check that the relation (4.58) is satisfied.
5
In this paper we have adapted the equations that determine the timelike supersymmetric
solutions of N = 1, d = 5 Abelian gauged supergravity coupled to vector multiplets to the
assumption that the K¨ahler base space admits a holomorphic isometry. While the resulting
system of equations is much more involved than in the pure supergravity case, we were
able, thanks in part to the experience gained in this latter case, to obtain several
supersymmetric solutions. Of these, the more interesting ones are three classes (for k = 0, ±1) of
superficially asymptoticallyAdS (globally asymptoticallyAdS for k = 1) solutions, which
are a direct generalization of the similar solutions found for pure supergravity in [11], and
which include various already known solutions.
It is worth noting that the special geometric model ST[2, nv + 1] considered here admits
as a special case the socalled U(1)3 model, which is just the STU model with equal gauging
parameters gI . This means that in this particular subcase our solutions can be oxidized to
typeIIB supergravity as described in [
24
].
The solutions constructed here only have one independent angular momentum, however
there are in the literature examples of supersymmetric black holes with two independent
angular momenta in N = 1, d = 5 Abelian gauged supergravity, both without and with
vector multiplets [
25, 26
]. It would be interesting to study whether less restrictive
assumptions than those made in this paper could lead to solutions generalizing these black
holes.21 Another possible extension of our work would be to consider more general
gaugings, for instance a combination of the Abelian FayetIliopoulos gauging considered here
21We have checked for instance [27] that the supersymmetric black holes of [
25
] can be written in the
same form as the black hole solutions in the present paper or in [11], but with the function Φ not satisfying
Liouville’s equation and fˆ consequently acquiring a dependence on one of the coordinates x1, x3.
and nonAbelian gaugings of the scalar manifold isometries. Work along these lines is in
progress [27].
Acknowledgments
20120249.
The author would like to thank Tom´as Ort´ın for his initial collaboration in this work,
useful comments and discussions. This work has been supported in part by the Spanish
Ministry of Science and Education grants FPA201235043C0201 and FPA201566793P
(MINECO/FEDER, UE) and the Centro de Excelencia Severo Ochoa Program grant
SEV
Open Access.
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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