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Published for SISSA by Springer

Received: February 19, 2016

Revised: July 7, 2016

Accepted: August 28, 2016

Published: September 1, 2016

Dyonic AdS4 black hole entropy and attractors via

entropy function

Prieslei Goulart

Instituto de F́ısica Teórica, UNESP-Universidade Estadual Paulista,

R. Dr. Bento T. Ferraz 271, Bl. II, Sao Paulo 01140-070, SP, Brazil

Max-Planck-Institut für Physik (Werner Heisenberg Institut),

Föhringer Ring 6, D-80805 Munich, Germany

E-mail: prieslei@ift.unesp.br

Abstract: Using the Sen’s entropy function formalism, we compute the entropy for the

extremal dyonic black hole solutions of theories in the presence of dilaton field coupled to

the field strength and a dilaton potential. We solve the attractor equations analytically and

determine the near horizon metric, the value of the scalar fields and the electric field on the

horizon, and consequently the entropy of these black holes. The attractor mechanism plays

a very important role for these systems, and after studying the simplest systems involving

dilaton fields, we propose a general solution for the value of the scalar field on the horizon,

which allows us to solve the attractor equations for gauged supergravity theories in AdS4

spaces. In particular, we derive an expression for the dyonic black hole entropy for the

N = 8 gauged supergravity in 4 dimensions which does not contain explicitly the gauge

parameter of the potential.

Keywords: Black Holes, Black Holes in String Theory

ArXiv ePrint: 1512.05399

Open Access, c© The Authors.

Article funded by SCOAP3.
doi:10.1007/JHEP09(2016)003

mailto:prieslei@ift.unesp.br
http://arxiv.org/abs/1512.05399
http://dx.doi.org/10.1007/JHEP09(2016)003


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Contents

1 Introduction 1

2 The dilaton action 3

3 Entropy function and attractor equations 3

4 Entropy function for the dilaton action with a potential 4

5 AdS4 static dyonic black holes 7

6 Summary and conclusions 12

A U(1)4 gauged supergravity in 4 dimensions 13

1 Introduction

The structure of charged black holes solutions in string theory differs a lot from the usual

Reissner-Nordstrom solution. Such a difference is due to a nontrivial coupling between

the dilaton field and the Maxwell field Fµν required to describe a low-energy effective

Lagrangian in string theory. The Reissner-Nordstrom solution describes black holes con-

taining both electric and magnetic charges at the same time (dyonic black hole), or just one

of them separately. For low-energy Lagrangians coming from string theory the nontrivial

coupling between the dilaton and the Maxwell field makes difficult to obtain analytical so-

lutions for the resulting Einstein’s equations when the black hole is dyonic. The situation

gets even worse when the chosen compactification scheme introduces a dilaton potential

with a complicated form, as is the case of gauged supergravities in AdS spacetimes.

The first achievements in obtaining such solutions in a closed form in the presence

of the dilaton field was due to Gibbons and Maeda in [1], and later, the same solution

was independently found by Garfinkle, Horowitz and Strominger in [2]. In both cases, the

solution corresponds to a magnetically charged black hole that is related to the electrically

charged black hole via a duality transformation. Later, the solution in the presence of both

electric and magnetic charges was found in [3], and they were only possible to be obtained by

requiring the existence of another scalar field, the axion field, that also couples non-trivially

to the Maxwell field. In the presence of both the dilaton and the axion, the equations of

motion are invariant under SL(2, R) transformations, which are a generalization of the

previous duality transformation. In the absence of an axion, the dyonic black hole solution

for theN = 4 ungauged supergravity was found in [4]. In [1–3] the black hole has the metric

and thermodynamic properties of a Schwarzschild solution, whereas the solution found in [4]

has the metric and thermodynamic properties of a Reissner-Nordstrom black hole.

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The attractor mechanism for black holes states that the value of the scalar fields on

the horizon is independent of their values at infinity, and it depends only on the charges of

the black hole. It was first discovered in the context of supergravity [5–7], but later it was

shown that supersymmetry does not play an important role in the attractor phenomenon,

and even after the inclusion of higher derivatives terms in the action such mechanism still

occurs (see for instance [8] and references therein), and it is believed to be a universal

feature of any gravity theory containing scalars. The attractor mechanism became a so

powerful tool that it allowed Sen [8] to develop a whole formalism to calculate not only

the value of the scalar fields on the horizon but also the entropy of extremal black holes,

for theories containing also higher derivative terms. The importance of this result resides

in the classical and quantum aspects of the black holes, since entropy is related to the

counting of their microstates.

It is undeniable that the gauge-gravity duality [9, 10], not only reinforced the interest

in black hole solutions in supergravities in AdS spacetimes, but also has given new in-

sights to the study of strongly coupled systems. Applications of holographic methods

to study superconductivity has recently been a subject of intense investigation. The

first model for a holographic superconductor was created by Gubser in [11] and reviewed

in [12] for instance. This theory is just gravity with a negative cosmological constant

coupled to a Maxwell system and a charged massive scalar. Efforts were made to embed

holographic superconductor models in string theory [13–16], but the situation gets more

complicated due to the nonminimal coupling of the scalar fields and also the more com-

plicated forms of the scalar potentials coming from supergravity. Although the scalar

fields considered in this paper are not charged and not minimally coupled, we believe

that understanding the near horizon properties of these complicated systems might be a

crucial step towards a better description of not only the counting of microstates of the

black hole but also of the holographic superconductors, as the infrared properties are

important to figure out phenomenologically viable models for superconductivity. Also,

from the gauge-gravity duality point of view, the moduli flow is interpreted as an RG

flow [17, 18].

In the context of N = 2 gauged supergravity, black holes in AdS4 spaces have been

extensively studied, and supersymmetric black hole solutions with running scalar fields

were found in [19], and explored with more details in [20–22]. Extensions for the non-

supersymmetric case can be found for instance in [23–26], and references therein.

In this work we compute the entropy for extremal black holes with spherical symmetry

using the entropy function formalism, by assuming that the near horizon geometry of these

theories is AdS2 × SD−2. The paper is organized as follows. In section 2 we derive the

equations of motion for a general dilaton theory in which the dilaton couples nontrivially

to the Maxwell field and has a dilaton potential. In section 3 we review the Sen’s entropy

function formalism and write the attractor equations. In section 4 we apply this formalism

to compute the entropy for the extremal black hole for different potentials. We derive

the near horizon metric, the value of the scalars on the horizon, the electric field and the

entropy follows directly. In section 5 we apply this for AdS4 black holes, and we speculate

about the generality of the ansatz for the dilaton on the horizon, which allowed us to

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determine explicitly the solution for the attractor equations and the entropy of this dyonic

black hole. In section 6 we present a summary and conclusions.

2 The dilaton action

The general dilaton action in the presence of a scalar potential is

S =

∫
d4x
√
−g (R− 2∂µφ∂

µφ−W (φ)FµνF
µν − V (φ)) . (2.1)

Here, we define the field strength as

Fµν = ∂µAν − ∂νAµ, (2.2)

and we take units in which 16πG = 1. The action is written in terms of a function of the

dilaton field, W (φ), coupled to the field strength, and a dilaton potential V (φ). This is a

general form of the bosonic part of some gauged supergravity theories, where W (φ) is in

general an exponential function of the dilaton, and V (φ) is a generic potential whose form

may vary due to the choice of the gauge for scalar fields in truncated supergravity theories.

One could also have more than one scalar or gauge field, as the case of U(1)4 gauged

supergravity described in the text and in the appendix, but all the formulas obtained are

easily generalized for these cases.

The equations of motion are:

• for the metric:

Rµν = 2∂µφ∂νφ−
1

2
gµνW (φ)FρσF

ρσ + 2W (φ)FµρFν
ρ +

1

2
gµνV (φ); (2.3)

• for the dilaton:

∇µ(∂µφ)− 1

4

∂W (φ)

∂φ
FµνF

µν − 1

4

∂V

∂φ
= 0; (2.4)

• for the gauge field:

∇µ (W (φ)Fµν) = 0. (2.5)

We also have the Bianchi identities:

∇[µFρσ] = 0. (2.6)

3 Entropy function and attractor equations

The basic assumption we have to make to compute the Sen’s entropy function is that the

spherically symmetric extremal black hole solution has the near horizon metric given by

ds2 = v1

(
−r2dt2 +

dr2

r2

)
+ v2(dθ2 + sin2 θdφ2), (3.1)

where the constants v1 and v2 are the AdS2 radius and the S2 radius respectively. This

means that we are assuming that the near horizon geometry is AdS2 × S2. It is believed

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that an AdS2×SD−2 is a general feature of extremal black holes with spherical symmetry

in D dimensions, and this was proven in 4 and 5 spacetime dimensions [27]. The scalar

and vector fields are constants for this geometry and are written as [8]

φs = us, F
(A)
rt = eA, F

(A)
θφ =

pA
4π

sin θ, (3.2)

where eA and pA are related to the integrals of the magnetic and electric fluxes, which are

in turn related to the electric and magnetic charges respectively. The metric (3.1) has the

SO(2, 1) × SO(3) symmetry of AdS2 × S2. The function f(us, vi, eA, pA) is defined as the

Lagrangian density
√
− det gL evaluated for the near horizon geometry (3.1) and integrated

over the angular variables,

f(us, vi, eA, pA) =

∫
dθdφ

√
− det gL. (3.3)

We extremize this function with respect to us, vi and eA by

∂f

∂us
= 0,

∂f

∂vi
= 0,

∂f

∂eA
= qA, (3.4)

where the first two equations are the equations of motion for the scalar and the metric

respectively.1 Next, one defines the entropy function,

E(~u,~v,~e, ~q, ~p) ≡ 2π[eAq
A − f(~u,~v,~e, ~p)]. (3.5)

The equations that extremize the entropy function are

∂E
∂us

= 0,
∂E
∂v1

= 0,
∂E
∂v2

= 0,
∂E
∂eA

= 0 , (3.6)

and are called the attractor equations. At the extremum (3.6), this new function equals

the entropy of the black hole

SBH = E(~u,~v,~e, ~q, ~p). (3.7)

4 Entropy function for the dilaton action with a potential

In this section we compute the entropy of the extremal black holes considered in the text.

The theories we consider do not make use of any explicit functional form for the coupling of

the dilaton field with the field strength, i.e. W (φ). Also, with the exception of the constant

potential, the potentials V (φ) we consider here are written in terms of these couplings

in a specific way, which leads us to make generic assumptions about the solutions to the

attractor equations for the case of U(1)4 theories considered in the next section.

The quantities of interest for us are the Riemann tensor

Rαβγδ = −v−1
1 (gαγgβδ − gαδgβγ), α, β, γ, δ = r, t

Rmnpq = v−1
2 (gmpgnq − gmqgnp), m, n, p, q = θ, φ, (4.1)

1Notice that in the absence of axionic couplings in the action, i.e. aF F̃ , where F̃ is the dual of F , qA is

simply interpreted as the electric charge. In the symplectic covariant formalism of N = 2, qA has the same

meaning as qΛ defined in equation 2.5 of reference [21] for instance. For more details, see Sen’s review [28].

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and, from it we can obtain easily the Ricci scalar

R = − 2

v1
+

2

v2
. (4.2)

We define the components of the field strength as Frt = e and Fθφ = P sin θ. Also, as

mentioned before, v1 is the AdS2 radius, v2 is the S2 radius, and here we define the value

of the scalar on the horizon as uD. Following the procedure for obtaining the entropy

function we plug our ansatze for the metric and fields in the Lagrangian, and then we

integrate it over the angular variables to obtain

f = 4πv1v2

[
− 2

v1
+

2

v2
+W (uD)

(
−e

2

v2
1

+
P 2

v2
2

)
− V (uD)

]
, (4.3)

and the entropy function is just E = 2π[Qe− f ], which gives

E = 2π

[
Qe− 8π(v1 − v2)− 4πv1v2W (uD)

(
e2

v2
1

− P 2

v2
2

)
+ 4πv1v2V (uD)

]
. (4.4)

Taking the proper derivatives, we obtain the attractor equations

Q− 8πv1v2W (uD)
e

v2
1

= 0, (4.5)

−2 + v2W (uD)

(
e2

v2
1

+
P 2

v2
2

)
+ v2V (uD) = 0, (4.6)

2− v1W (uD)

(
e2

v2
1

+
P 2

v2
2

)
+ v1V (uD) = 0, (4.7)

∂W (uD)

∂uD

(
e2

v2
1

− P 2

v2
2

)
− ∂V (uD)

∂uD
= 0. (4.8)

One can check that these equations are the equations of motion derived in the previous

section for the near horizon geometry. Using (4.5) we can eliminate the term containing Q

in (4.4), and the result is

E = 2π

[
−8π(v1 − v2) + 4πv1v2W (uD)

(
e2

v2
1

+
P 2

v2
2

)
+ 4πv1v2V (uD)

]
. (4.9)

Multiplying (4.6) by v1 and adding to (4.7) multiplied by v2 we have a formula for the po-

tential

v1v2V (uD) = (v2 − v1). (4.10)

This formula holds for general scalar fields φI and gauge fields AIµ, and we will also use it

in the next section. We can eliminate the potential from (4.9) to obtain

E = 2π

[
−4π(v1 − v2) + 4πv1v2W (uD)

(
e2

v2
1

+
1

v2
2

p2

(4π)2

)]
. (4.11)

Now, multiplying (4.6) by v1 and subtracting (4.7) multiplied by v2 we have

v1v2W (uD)

(
e2

v2
1

+
P 2

v2
2

)
= (v1 + v2), (4.12)

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and eliminating this term from (4.11) we have the entropy written only in terms of the

S2 radius,

E = 16π2v2. (4.13)

One can recover the correct constant by dimensional analysis, meaning 16πG ≡ 1, and

show that this corresponds to the usual A/4 term of the black hole entropy. Notice that

we can solve (4.5) directly, giving

e

v1
=

Q

8πv2W (uD)
. (4.14)

We will analyze the solutions of the attractor equations for some specific potentials, by

making some considerations about its functional dependence on the dilaton field. We list

the solutions case by case.

• V (φ) = 0. For this case (4.8) gives directly the value of the function W (uD) on the

horizon, and the constants v1 and v2 and the entropy are

W (uD) =
Q

8πP
, v1 = v2 =

QP

8π
, e = P, (4.15)

E = 2πQP. (4.16)

This means that the value of the electric field is equal to the value of the magnetic

charge on the horizon. Again, one should notice that this result is independent of

the functional form of W (uD). This should be contrasted to the GHS solution [2], in

which the dilaton coupling to the field strength is W (φ) = e−2φ. The same result for

the entropy was derived before by other method in [4].

• V (φ) = 2Λ. The solution for the attractor equations for this case is

W (uD) =
Q

8πP
, e = P

v1

v2
, (4.17)

v1 =
1

Λ

1
2Λ

(
1±

√
1 + ΛQP

2π

)
− QP

8π

QP
4π −

1
2Λ

(
1±

√
1 + ΛQP

2π

) , v2 =
1

2Λ

(
1±

√
1− ΛQP

2π

)
, (4.18)

E =
8π2

Λ

(
1−

√
1− ΛQP

2π

)
. (4.19)

We left the value of the electric field in an implicit form, and took the minus sign

in v2 in order to write the entropy. When we Taylor expand for small values of the

cosmological constant, we recover the previous result for zero potential when Λ → 0

only with the negative sign.

• V (φ) = βW (φ). For this case, (4.8) gives directly

W 2(uD) =
Q2

(8π)2

1

(P 2 + βv2
2)
. (4.20)

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Replacing this in (4.7) we find v2 and all the rest can be found after some algebraic

manipulation

W (uD) =
Q

8πP

(
1− βQ2

(8π)2

)1/2

e =
P(

1− βQ2

(8π)2

)3/2
(4.21)

v1 =
QP

8π

1(
1− βQ2

(8π)2

)3/2
, v2 =

QP

8π

1(
1− βQ2

(8π)2

)1/2
, (4.22)

E =
2πQP(

1− βQ2

(8π)2

)1/2
. (4.23)

• V (φ) = β
W (φ) . This case is a bit simpler to be obtained. The solutions are

W (uD) =
Q

8πP

1

(1− βP 2)1/2
e =

P

(1− βP 2)1/2
(4.24)

v1 =
QP

8π

1

(1− βP 2)3/2
, v2 =

QP

8π

1

(1− βP 2)1/2
, (4.25)

E =
2πQP

(1− βP 2)1/2
. (4.26)

Notice that we can achieve the zero potential case by setting the constant β to zero.

Notice also that the AdS2 and S2 radii, v1 and v2, are related in equations (4.22) and (4.25),

by the exchange Q/(8π)↔ P . Of course, one can try other kinds of potentials containing

different combinations of these two examples.

It is good to emphasize again that none of the results depend on the functional form

of the coupling W (φ), and, in order to obtain the black hole entropy, we have assumed

that the scalar potential depends on this function in a specific form. These cases illustrate

some possible examples that are relevant for gauged supergravities for instance, since the

coupling to the field strength is a function of the dilaton field, and the potential may

be written as being a function of these couplings. In the next section we show how this

works through an explicit computation for the black hole entropy for a specific theory in

AdS4 space.

5 AdS4 static dyonic black holes

We apply these ideas to the case of AdS4 static black holes. Such system was studied in [29]

by Morales and Samtleben via entropy function, when the gauge field has only electric

charge. The solution found by the authors is written in an implicit form, depending on the

values of a set of parameters rather than the electric charges. Also, the entropy function

was used as a criteria to prove the existence of regular extremal black holes with finite

horizon area for a purely electric system [30]. Here, we consider both electric and magnetic

charges, meaning that we have a dyonic black hole, and also in the presence of a potential.

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The notation is explained in the appendix, as the definition of the field XI may change a

lot in the literature, and also the definition of the scalar potential. The theory is the U(1)4

gauged supergravity in four dimensions, which follows from a truncation of the maximal

N = 8, SO(8) supergravity down to the Cartan subgroup of SO(8). The bosonic action is

given by [31]

S =

∫
d4x
√
−g

[
R− 1

32

(
3

4∑
I=1

(∂µλI)
2 − 2

∑
I<J

∂µλI∂
µλJ

)
− 1

4

4∑
I=1

X2
I (F Iµν)2 − V (X)

]
,

(5.1)

where I = 1, . . . , 4, 16πG = 1, and

F Iµν = ∂µA
I
ν − ∂νAIµ, V (X) = −g

2

4

∑
I<J

1

XIXJ
, X1X2X3X4 = 1. (5.2)

Notice that we can set the potential term to zero by choosing g = 0. The near horizon

fields will be

XI = uI , F
I
rt = eI , F Iθφ = pI sin θ. (5.3)

As the kinetic terms give zero contribution to the entropy function, one can easily do some

identifications and add a summation in the entropy function (4.11) in order to obtain

E = 2π

[
eIq

I − 4πv1v2

(
− 2

v1
+

2

v2
+

1

2

4∑
I=1

u2
I

(
e2
I

v2
1

−
p2
I

v2
2

)
+ 4g2

4∑
I<J

uIuJ

)]
. (5.4)

The attractor equations (3.6) are

eI
v1

=
qI

(4π)u2
Iv2

, (5.5)

2− v2

2

4∑
I=1

u2
I

(
e2
I

v2
1

+
p2
I

v2
2

)
+ v2V (u) = 0, (5.6)

−2 +
v1

2

4∑
I=1

u2
I

(
e2
I

v2
1

+
p2
I

v2
2

)
+ v1V (u) = 0, (5.7)

uI

(
e2
I

v2
1

−
p2
I

v2
2

)
− ∂V (u)

∂uI
= 0. (5.8)

• V(X)=0. We first discuss the case in which the potential is null. Notice that, if we

also set the magnetic charges equal to zero, equation (5.8) will give zero dilaton or

zero gauge field on the horizon, and the attractor equations give a trivial solution.

This means that for zero potential and zero magnetic charges the attractor equations

do not make sense. For nontrivial magnetic charges, one can easily find the solutions:

eI = pI , u2
I =

qI

4πpI
, v1 = v2 =

1

2(4π)

4∑
I=1

qIpI , (5.9)

E = 2π

4∑
I=1

qIpI . (5.10)

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From the results of the previous section we see that this is expected, as we have done

nothing but add indices to our fields.

• V (X) = −g2

4

∑4
I<J

1
XIXJ

. For this case, equation (5.8) gives

4∑
I=1

u2
I

e2
I

v2
1

=

4∑
I=1

u2
I

p2
I

v2
2

− 2V (u), (5.11)

and replacing this directly in (5.6) we obtain a quadratic equation

v2
2V (u) + v2 −

1

2

4∑
I=1

u2
Ip

2
I = 0, (5.12)

whose solution is

v2 =
−1±

√
1 + 2V (u)

∑4
I=1 u

2
Ip

2
I

V (u)
. (5.13)

In order to check if this result has the correct limit we expand it for V (u)→ 0, giving

v2 =
1

V (u)

(
−1±

(
1 + V (u)

4∑
I=1

u2
Ip

2
I +O(V (u)2)

))
. (5.14)

Taking the plus sign we can recover the previous case setting the potential to zero.

This is again an indication that the case for zero potential should be recovered from

this case by setting the potential to zero, i.e. g2 → 0.

Inserting also (5.11) in (5.7) we find directly

v1 =
2v2

2∑4
I=1 u

2
Ip

2
I

. (5.15)

Also, from (5.5) we can obtain

4∑
I=1

u2
I

e2
I

v2
1

=
4∑
I=1

(
qI

4π

)2
1

u2
Iv

2
2

. (5.16)

Inserting this in (5.11) we have the following equation

4∑
I=1

(
qI

4π

)2
1

u2
I

=

4∑
I=1

u2
Ip

2
I − 2v2

2V (u). (5.17)

We can eliminate the last term containing v2
2 using (5.12), and obtain

v2 =
1

2

4∑
I=1

(
qI

4π

)2
1

u2
I

. (5.18)

In order to have a solution for the attractor equations one should use (5.18) and replace

v2 in (5.12) or in (5.17), and then solve it for uI , i.e. write uI in terms of the electric

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and magnetic charges. It turns out that finding solutions for the resulting equations is

a non-trivial task due to the amount of scalar fields and all summations involving them.

Based in the analysis made in the previous section, we know that the black hole entropy

obtained in (5.10) should be recovered when the potential is set to zero for this case, and

so, all other parameters obtained here should also match the ones obtained before in (5.9)

in the same limit. By direct observation of the value of the general coupling W (uD) on the

horizon of the black hole obtained in all the previous cases, we are then led to consider a

solution for the dilaton, given by

u2
I =

qI

4πpI
F (q, p)1/2, (5.19)

where F (q, p) is a function of the charges that will be fixed by the attractor equations.

With this solution we obtain for the other constant fields

v2 =
1

2(4π)

1

F (q, p)1/2

4∑
I=1

qIpI , (5.20)

v1 =
1

2(4π)

1

F (q, p)3/2

4∑
I=1

qIpI , (5.21)

eI =
pI

F (q, p)3/2
. (5.22)

They will have the correct limit if, at zero potential, the function F (q, p) → 1. In order

to obtain this function, we insert these possible solutions into the attractor equations. It

is easy to check that (5.5) gives an identity. Equation (5.6), (5.7), (5.8), lead to the same

equation. This shows the correctness of our solution (5.19). The resulting equation is

F (q, p)2 − F (q, p) +
g2

8

(
4∑
I=1

qIpI

)(∑
J<K

√
pJpK
qJqK

)
= 0, (5.23)

whose solution is

F (q, p) =
1

2

1±

√√√√1− g2

2

(
4∑
I=1

qIpI

)(∑
J<K

√
pJpK
qJqK

) . (5.24)

In the limit of zero potential we must have g → 0, and, as alluded before, by consistency

F (q, p) → 1, so the positive sign is the correct one. So, the entropy of the extremal black

hole in the presence of electric and magnetic charge for this theory is

E = 2π

(
4∑
I=1

qIpI

)1

2

1 +

√√√√1− g2

2

(
4∑
I=1

qIpI

)(∑
J<K

√
pJpK
qJqK

)−1/2

. (5.25)

As the dilaton fields are not all independent we can also use the constraint for the scalar

fields, X1X2X3X4 = 1, and compute the function F (q, p) in a simpler but not explicit way.

This is written as

F (q, p) = (4π)2

√
p1p2p3p4

q1q2q3q4
, (5.26)

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and the entropy will be

E =
1

2

(
4∑
I=1

qIpI

)(
q1q2q3q4

p1p2p3p4

)1/4

. (5.27)

Written in this form the entropy does not show explicitly the coupling constant g2. One

can obtain it as a function of the charges of the black hole by using (5.24) and (5.26), i.e.

g2 =
8(4π)4

(√
p1p2p3p4

q1q2q3q4 − p1p2p3p4

q1q2q3q4

)
(∑4

I=1 q
IpI

)(∑
J<K

√
pJpK
qJqK

) . (5.28)

This means that, for this geometry, the charges must be chosen in such a way to satisfy

this specific condition on the horizon of the extremal black hole. This is due to the fact

that, for gauged supergravities, in order to obtain a consistent truncation, the scalar fields

should be constrained (like X1X2X3X4 = 1 in the U(1)4 case), and this constraints the

values of the electric and magnetic charges on the horizon of the extremal black hole, since

g2 is a parameter defining the theory that one can freely vary. In other words, (5.28) is

a condition on the charges of the black hole, and not on the coupling constant g2. In the

absence of such constraint the electric and magnetic charges can also be chosen freely.

If the dyonic black hole solution is invariant under duality transformation, uI → u−1
I

and (qI/4π)↔ pI , then the dilaton field is invariant, since F → F−1.2 But notice that the

entropy changes to

E = 8π2

(
4∑
I=1

qIpI

)(
p1p2p3p4

q1q2q3q4

)1/4

. (5.29)

If one makes the rescaling qI ≡ ε, then the electric charge disappears from this formula,

and the result is just

E = 8π2

(
4∑
I=1

pI

)(
p1p2p3p4

)1/4
. (5.30)

In order to obtain the same result from equation (5.25), one should take the limit of

magnetic black holes, i.e. ε → 0. A formula for purely magnetic black hole was found for

an N = 2 supersymmetric theory in reference [19], which is given by

S = 2π2

(
4∏
I=1

pI

gI

)1/4

, (5.31)

where gI = gξI , for some constant ξI . As pointed out in the same reference, such a

model can be embedded in N = 8 gauged supergravity. The gravitino is charged in

2As was pointed out in [31], for the case of purely electric or purely magnetic black holes, there is a direct

correspondence between the supersymmetry properties of the electric and magnetic solutions in the absence

of gauging (g = 0), which apparently does not happen for the gauged theory. However it is not clear yet if

the dyonic solution of this theory preserves the supersymmetry properties under electric-magnetic duality.

Here we just take the dual in order to show that it is possible to recover the entropy computed for purely

magnetic black holes.

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gauged supergravity, and so, for topological consistency in a magnetic backgroung field,

the charges should satisfy the Dirac quantization condition: for BPS configurations this

is just
∑
gIp

I ∈ κ, where κ is the curvature of the horizon geometry. In the case of an

S2 horizon we have κ = 1. For the U(1)4 theory the Dirac quantization is written as

−8π
∑
gIp

I = Z. Using this condition, it is easy to see that equations (5.30) and (5.31)

agree for some choice of constant ξI , which shows that one can recover the purely electric

case from the dyonic one, if the dyonic solution is invariant under electric-magnetic duality

in the presence of gaugings. It should be emphasized that the Sen’s formalism allows

us to obtain information about the fields in the near horizon for general extremal black

holes: they do not necessarily need to be BPS, and this is the reason why we should use

−8π
∑
gIp

I = Z, instead of the Dirac quantization for BPS black holes.

The question that naturally arises in this work is whether solutions of the kind (5.19)

represent a general feature of the scalar fields for any theory of the kind (2.1). All the results

obtained here depended strongly on the functional form of the dilaton potential. Some

complications in the equations might arise when the potential is a complicated function

(like logarithmic or exponential)of the couplings to the field strength (W (φ) in (2.1) and

u2
I in (5.1)). Potentials having polynomial functional forms of these couplings are the

most common ones in gauged supergravities, and, at least in 4 spacetime dimensions, it

seems that (5.19) will allow one to solve the attractor equations for any of these theories.

Of course, one should check it case by case, since the results presented here are only an

illustration of why this should work, but not a complete proof. Also, for more general

theories in D-dimensions containing Chern-Simons terms and higher order derivatives, like

RF 2 and R2 for instance, the attractor equations might present higher powers of v1 and

v2, and such solutions are much more difficult to be obtained, but of course, in the limit

when the coupling constants of such terms go to zero, we must recover the case for zero

potential. How our results change in such cases may be a subject of future work.

6 Summary and conclusions

We have shown that Sen’s entropy function allows us to obtain the entropy for the extremal

dyonic black hole with near horizon geometry given by AdS2×S2, whenever the action is of

the type (2.1) with the potential having a specific (polynomial) form. After analysing some

general examples, we have shown explicitly how to solve the attractors the equations for

the case of U(1)4 gauged supergravity in 4 dimensions. One should write all the attractor

equations in terms of the dilaton fields, and then assume that the dilaton fields on the

horizon are given by

u2
I =

qI

4πpI
F (q, p)1/2. (6.1)

Then, after inserting this solution in the attractor equations, one solves the attractor

equations for F (q, p) and obtain the solution for all the fields, uI , v1, v2 and eI . The

function F (q, p) may not be unique in the sense that it is a solution of a polynomial

equation (quadratic equation for U(1)4 case). The solution that must be chosen is the one

that satisfies

lim
V→0

F (q, p) = 1. (6.2)

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One can see that this is achieved in (5.24) by choosing the + sign. This shows that the

scalar attractors change in a specific way that depends crucially on the functional form of

the potential, and we speculated about (6.1) being a universal feature of the sort of theories

treated here. We emphasize that this should be understood as a parametrization, since

we have just singled out a factor (or factors) of q
4πp from the function F (q, p), and not an

ansatz. No restriction on the function F (q, p) was made. As the attractor equations are a

system with the same number of variables and linearly independent equations, the system

is completely determined.

We have obtained the values of the dilaton and the electric field on the horizon of the

extremal dyonic black hole for the U(1)4 gauged supergravity theory in 4 dimensions. We

have also derived explicitly the near horizon metric and the entropy of these black holes,

and showed that the constraint X1X2X3X4 = 1 fixes the electric and magnetic charges

on the horizon to satisfy (5.28) (this may be associated to the mass m2 of the dilaton

field for some potentials). Through the U(1)4 example, we also showed the power of the

entropy function formalism in determining the near horizon metric and the entropy for

dyonic black holes. This approach might give some indications about how the full metric

for the non-extremal dyonic dilatonic black hole should be, since the near horizon metric

computed here should be obtained by taking the extremal limit in the full non-extremal

solution. None of these results could have been obtained without assuming that the scalars

on the horizon are written as (6.1).

Acknowledgments

The author would like to thank Thiago R. Araujo, Johanna Erdmenger, Daniel Fernandez

and Horatiu Nastase for reading the final version of this paper and for useful discussions.

The author is grateful for the hospitality of the Max-Planck-Institut für Physik (Werner

Heisenberg Institut), where part of this work was developed. The work of PG is supported

by FAPESP grant 2013/00140-7 and 2015/17441-5.

A U(1)4 gauged supergravity in 4 dimensions

Due to some different notation in the definition of the dilaton field, we present a brief

review to explain our definitions for the case of U(1)4 supergravity in 4 dimensions. The

bosonic action with the same field definition and coefficients given in [31] is

S =

∫
d4x
√
−g
[
R− 1

2

(
(∂φ(12))2 + (∂φ(13))2 + (∂φ(14))2

)
− V

−2
(
e−λ1(F (1)

µν )2 + e−λ2(F (2)
µν )2 + e−λ3(F (3)

µν )2 + e−λ4(F (4)
µν )2

)]
, (A.1)

where the scalar combinations λ are given by

λ1 = −φ(12) − φ(13) − φ(14),

λ2 = −φ(12) + φ(13) + φ(14),

λ3 = φ(12) − φ(13) + φ(14),

λ4 = φ(12) + φ(13) − φ(14),

(A.2)

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and the scalar potential is

V = −4g2
(

coshφ(12) + coshφ(13) + coshφ(14)
)
. (A.3)

Notice that the scalars are not all independent,

λ1 + λ2 + λ3 + λ4 = 0. (A.4)

We can write the scalar φ(ij) in terms of the fields λ as

φ(12) =
1

4
(−λ1 − λ2 + λ3 + λ4), (A.5)

φ(13) =
1

4
(−λ1 + λ2 − λ3 + λ4), (A.6)

φ(14) =
1

4
(−λ1 + λ2 + λ3 − λ4). (A.7)

We can rewrite parts of the kinetic terms as

∂µφ
12∂µφ12 =

1

16

[
(∂µλ1)2 + (∂µλ2)2 + (∂µλ3)2 + (∂µλ4)2

+ 2∂µλ1∂
µλ2 − 2∂µλ1∂

µλ3 − 2∂µλ1∂
µλ4

−2∂µλ2∂
µλ3 − 2∂µλ2∂

µλ4 + 2∂µλ3∂
µλ4] , (A.8)

∂µφ
13∂µφ13 =

1

16

[
(∂µλ1)2 + (∂µλ2)2 + (∂µλ3)2 + (∂µλ4)2

− 2∂µλ1∂
µλ2 + 2∂µλ1∂

µλ3 − 2∂µλ1∂
µλ4

−2∂µλ2∂
µλ3 + 2∂µλ2∂

µλ4 − 2∂µλ3∂
µλ4] , (A.9)

∂µφ
14∂µφ14 =

1

16

[
(∂µλ1)2 + (∂µλ2)2 + (∂µλ3)2 + (∂µλ4)2

− 2∂µλ1∂
µλ2 − 2∂µλ1∂

µλ3 + 2∂µλ1∂
µλ4

+2∂µλ2∂
µλ3 − 2∂µλ2∂

µλ4 − 2∂µλ3∂
µλ4] . (A.10)

The full kinetic term is

(∂µφ
12)2 + (∂µφ

13)2 + (∂µφ
14)2 =

1

16

[
3(∂µλ1)2 + 3(∂µλ2)2 + 3(∂µλ3)2 + 3(∂µλ4)2

− 2∂µλ1∂
µλ2 − 2∂µλ1∂

µλ3 − 2∂µλ1∂
µλ4 − 2∂µλ2∂

µλ3

−2∂µλ2∂
µλ4 − 2∂µλ3∂

µλ4] . (A.11)

The potential term is written as

V = −2g2

(
eφ

12
+ eφ

13
+ eφ

14
+

1

eφ12 +
1

eφ13 +
1

eφ14

)
, (A.12)

and in terms of the fields λ we have

V = −2g2
[
e

1
2

(λ1+λ2) + e
1
2

(λ1+λ3) + e
1
2

(λ1+λ4) + e
1
2

(λ2+λ3) + e
1
2

(λ2+λ4) + e
1
2

(λ3+λ4)
]
.

(A.13)

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In order to have the Maxwell term written with a factor 1/4, we redefine the exponential

of the fields as
XI√

8
≡ e−

λI
2 ⇒ ∂µλI = −2

∂µXI

XI
. (A.14)

The potential is then

V = −g
2

4

∑
I<J

1

XIXJ
. (A.15)

The action is finally rewritten as

S =

∫
d4x
√
−g

[
R− 1

32

(
3

4∑
I=1

(∂µλI)
2−2

∑
I<J

∂µλI∂
µλJ

)
− 1

4

4∑
I=1

X2
I (F Iµν)2−V

]
. (A.16)

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License (CC-BY 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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	Introduction
	The dilaton action
	Entropy function and attractor equations
	Entropy function for the dilaton action with a potential
	AdS(4) static dyonic black holes
	Summary and conclusions
	U(1)**(4) gauged supergravity in 4 dimensions