IPM/P-2007/011
arXiv:hep-th/0703099v4 14 May 2007
On R2 Corrections for 5D Black Holes
Mohsen Alishahiha1
Institute for Studies in Theoretical Physics and Mathematics (IPM)
P.O. Box 19395-5531, Tehran, Iran
Abstract
We study higher order corrections to extremal black holes/black string in
five dimensions. These higher order corrections are due to supersymmetric
completion of R2 term in five dimensions. By making use of the results we
extend the notion of very special geometry when higher derivative terms are
also taken into account. This can be used to make a connection between total
bundle space of near horizon wrapped M2’s and wrapped M5’s in the presence
of higher order corrections. We also show how the corrected geometry removes
the singularity of a small black hole.
1
[email protected]
1
Introduction
In recent years our understanding of corrections to the black hole entropy has increased considerably. In a gravitational theory using the Wald entropy formula [1]
one can find the contribution of higher order corrections to the tree level BekensteinHawking area law formula. For extremal black holes in string theory taking into
account the higher derivative terms the corrected entropy has been evaluated in
several papers including [2–4] where it was shown that the results are in agreement
with microscopic description of entropy coming from microstate counting in string
theory.
Although at the beginning the corrections have mainly been studied for the supersymmetric black holes where the attractor mechanism [5] was observed, recently
it was generalized to non-supersymmetric cases as well (for example see [6–23]).
An extremal black hole in four dimensions has AdS2 × S 2 near horizon geometry,
while in five dimensions the near horizon geometry could be either AdS2 × S 3 or
AdS3 × S 2 . To compute the contributions of higher order corrections to a black hole
with near horizon geometry AdS2 × S 3 one may simplify the Wald formula leading
to the entropy function formalism [24]. While for those with AdS3 × S 2 near horizon
geometry it is useful to work within the c-extremization framework [25]. Using these
mechanisms one can find corrections to the Bekenstein-Hawking area law formula
coming from higher derivative terms. These corrections have to be compare with
the microstate counting in string theory/M-theory.
As far as the five dimensional black holes are concerned, We note that although
the microscopic origin of N = 8, 4 five dimensional rotating black holes has been understood for a decade [26, 27], the origin of the entropy for five dimensional N = 2
black hole has not been fully understood yet. Recently the microscopic accounting of the five dimensional rotating black hole arising from wrapped M2-branes in
Calabi-Yau compactification of M-theory has been studied in [28] where the authors
established a connection between this black hole and a well understood one by making use of an embedding of space-time in the total space of the U(1) gauge bundle
over near horizon geometry of the black holes. Of course it was done in an specific
case, namely near zero-entropy, zero-temperature and maximally rotating limits.
This is the aim of this article to further study higher derivative corrections to
N = 2 five dimensional black holes. To do this we will work with the full 5D
supersymmetry invariant four-derivative action, corresponding to the supersymmetric completion of the four-derivative Chern-Simons term which has recently been
obtained in [29].
The article is organized as follows. In section 2 we shall fix our notation where
we present the five dimensional action obtained in [29]. In section 3 we will consider
a five dimensional black string at R2 level. In section 4 we will generalize the notion
of very special geometry in the presence of higher derivative terms using the results
of section 3. This generalization can be used to extend the consideration of [28] to
the case where higher order corrections are also taken into account. In section 5 we
1
shall study higher order corrections to a five dimensional extremal black hole using
the fully supersymmetrized higher derivative terms. We then compare the result
with the corrections coming from the bosonic Gauss-Bonnet action. We shall also
see how the higher derivative terms remove the singularity of a small black hole.
The last section is devoted to discussions.
2
Basic setup
In this section we will fix our notation. We would like to study N = 2 supergravity in five dimensions. This model is usually studied in the context of very special
geometry. We note, however, that sometime it is useful to work with a more general context, namely the superconformal approach. This approach, in particular,
is useful when we want to write the explicit form of the action. In this approach
we start with a five dimensional theory which is invariant under a larger group
that is superconformal group and therefore we construct a conformal supergravity.
Then by imposing a gauge fixing condition, one breaks the conformal supergravity
to standard supergravity model.
The representation of superconformal group includes Weyl, vector and hyper
multiples. The bosonic part of the Weyl multiplet contains the vielbein eaµ , twoform auxiliary field vab , and a scalar auxiliary field D. The bosonic part of the
vector multiplet contains one-form gauge field AI and scalar fields X I , where I =
1, · · · , nv labels generators of a gauge group. The hypermultiplet contains scalar
fields Aiα where i = 1, 2 is SU(2) doublet index and α = 1, · · · , 2r refers to USp(2r)
group. Although we won’t couple the theory to matters, we shall consider the hyper
multiplet to gauge fix the dilataional symmetry reducing the action to the standard
N = 2 supergravity action.
In this notation at leading order the bosonic part of the action is [29]
Z
1
I=
d5 xL0 ,
(2.1)
16πG5
with
D
R
v2
I ab
+ (2ν − 3A2) + (6ν − A2 ) + 2νI Fab
v
4
8
2
g −1
1
I
K
νIJ (Fab
F J ab + 2∂a X I ∂ a X J ) +
CIJK ǫabcde AIa FbcJ Fde
,
(2.2)
+
4
24
where A2 = Aiα ab Aαi ab , v 2 = vab v ab and
L0 = ∂a Aiα ∂ a Aαi + (2ν + A2 )
1
1
ν = CIJK X I X J X K ,
νI = CIJK X J X K ,
νIJ = CIJK X K .
(2.3)
6
2
To fix the gauge it is convenient to set A2 = −2. Then integrating out the
auxiliary fields by making use of their equations of motion one finds
g −1 abcde
1
I
I
J K
F Jab − Gij ∂a φi ∂ a φj +
ǫ
CIJK Fab
Fcd
Ae .
L0 = R − GIJ Fab
2
24
2
(2.4)
The parameters in the action (2.4) are defined by
1
GIJ = − ∂I ∂J log ν|ν=1 ,
2
Gij = GIJ ∂i X I ∂j X J |ν=1 ,
(2.5)
where ∂i refers to a partial derivative with respect to the scalar fields φi . In fact
doing this, we recover the very special geometry underlines the theory in the leading
order.
We can also find higher derivative terms in the action using the superconformal
language. Actually, the supersymmetrized higher order action with four-derivative
has recently been obtained in [29]. The corresponding action is
c2I 1 −1
1
1
1
L1 =
g ǫabcde AIa C bcf g C def g + X I C abcd Cabcd + X I D 2 + F Iab vab D
24 16
8
12
6
1 I
1 Iab
8 I
ab cd
cd
b
ac
− X Cabcd v v − F Cabcd v + X vab D̂ D̂c v
(2.6)
3
2
3
4
2
4
+ X I D̂ a v bc D̂a vbc + X I D̂ a v bc D̂b vca − e−1 X I ǫabcde v ab v cd D̂f v ef
3
3
3
2 −1 Iab
cd
ef
−1 Iab
+ e F ǫabcde v D̂f v + e F ǫabcde v cf D̂ d v ef
3
1 Iab
4 Iab
cd
2
I
bc
da
I
ab 2
− F vac v vdb − F vab v + 4X vab v vcd v − X (vab v ) ,
3
3
where Cabcd is the Weyl tensor defined as
1 [a b]
4 [a b]
C abcd = Rabcd + Rδ [c δ d] − δ [c R d] .
6
3
(2.7)
The double covariant derivative of vab has curvature contributions given by
2
1
vab D̂ b D̂c v ac = vab D b Dc v ac + v ac vcb Rab + vab v ab R .
3
12
(2.8)
In general it is quite difficult to solve the equations of motion coming from the
action contains both L0 and L1 . Nevertheless as long as the supersymmetric solutions are concerned, it is useful to look at the supersymmetry transformations which
could give a simple way to find explicit solutions. The supersymmetry variations of
the fermions in Weyl, vector and hyper multiplets are
1
δψµi = Dµ εi + v ab γµab εi − γµ η i ,
2
δχi = Dεi − 2γ c γ ab εi Da vbc − 2γ a εiǫabcde v bc v de + 4γ · vη i,
1
1
δΩIi = − γ · F I εi − γ a ∂a X I − X I η i ,
4
2
δζ α = γ a ∂a Aαj − γ · vεAαj + 3Aαj η i ,
(2.9)
where garavitino ψµi and the auxiliary Majorana spinor χi come from the Weyl
multiple, while the gaugino ΩIi and ζ α come from vector and huper multiplets,
respectively.
3
3
Black string and c-extremization
In this section we will consider a five dimensional extremal black string solution
whose near horizon geometry is AdS3 × S 2 . Using the symmetry of the near horizon
geometry one may start with the following ansatz for near horizon solution
X I = cont.
2
ds = lA
ds2ADS3 + lS2 ds2S 2 ,
I
Fθφ
=
pI
sin θ.
2
(3.1)
By making use of the c-extremization [25] we can fix the parameters lA , lS and XI as
follows. In this method we will first define c-function whose critical points correspond
to the solutions of the equations of motion. Then evaluating the c-function at critical
points gives the average of the left and right moving central charges of the associated
CFT.
In our model at leading order the c-function is given by
3 2
c = −6 lA
lS L,
(3.2)
where L is the Lagrangian evaluated on the above ansatz. The parameters of the
ansatz are obtained by extremizing this function with respect to them
∂c
= 0,
∂lA
∂c
= 0,
∂lS
∂c
= 0.
∂X I
(3.3)
For the five dimensional action (2.4) the above equations can be solved leading to
1
lA = 2lS = ( CIJK pI pJ pK )1/3 ,
6
XI =
pI
.
( 61 CIJK pI pJ pK )1/3
(3.4)
Plugging these into (3.2), one finds the central charge for the black string at twoderivative level as c = CIJK pI pJ pK .
Let us now redo c-extremization for the black string solution (3.1) in the presence
of higher derivative terms. This correction has recently been studied in [30]. In
general one should evaluate the Lagrangian (2.2) and (2.6) for our ansatz (3.1) and
then extremize it with respect to the parameters. We note, however, that it is in
general difficult to solve the equations of motion explicitly. Nevertheless one may
use the supersymmetry transformations (2.9) to fix some of the parameters. The
remaining parameters can then be found by equation of motion of the auxiliary field
D. To be precise, from the supersymmetry transformations (2.9) for our ansatz (3.1)
one finds [30]
D=
12
,
lS2
3
V = − lA ,
8
8
pI = − V X I ,
3
lA = 2lS .
On the other hand the equation of motion of auxiliary field D is
V pI
c2I
1
I
I J K
DX + 4
CIJK X X X +
= 1,
6
72
lS
4
(3.5)
(3.6)
which, by making use of (3.5), can be recast to the following form
1
1
CIJK X I X J X K +
c X I = 1.
2 2I
6
12lA
(3.7)
Setting c2I = 0, it reduces to ν = 1 where one can define very special geometry
underlines the theory at leading order. Therefore we would like to interpret this
expression as a generalization of ν = 1 when the R2 correction is also taken into
account. This is analogous to the one loop correction to the prepotential of four
I
I J K
dimensional N = 2 supergravity which it is given by F = 16 CIJ K XX 0X X + Λ2 c2IXX0 .
It is then natural to define the dual coordinates XI as
1
1
XI = CIJK X J X K +
c ,
2 2I
6
12lA
(3.8)
I
such that XI X I = 1. From the supersymmetry conditions (3.5) one has X I = plA .
1
3
Plugging this into the above expressions we get lA
= 16 CIJK pI pJ pK + 12
c2I pI and
therefore
XI =
pI
( 16 CIJK pI pI pK +
1
c pI )
12 2I
,
1/3
XI =
1
1
C pI pK + 12
c2I
6 IJK
.
1
1
( 6 CIJK pI pI pK + 12 c2I pI )2/3
(3.9)
We will give more comments on this structure in the next section.
4
Very special geometry and black holes at R2
level
In the previous section we have considered the black string solution in the presence
of higher derivative terms. We have observed that adding higher derivative terms
will change the feature of the very special geometry in such a way that the leading
order constraint ν = 1 is not satisfied any more. Therefore the standard method of
very special geometry is not applicable beyond tree level and we will have to solve
the equations of motion coming from c-extremization directly.
In this section we would like to generalize the notion of very special geometry
when higher order corrections are also taken into account. It is possible to do
that due to new progresses which have recently been made in computing the higher
order correction using the fully supersymmetrized higher derivative terms. We note,
however, that the notion of generalized very special geometry depends on the explicit
solution we are considering. To be specified we will consider a black string solution
in five dimensions in the presence of supersymmetric higher derivative terms and
we define the generalized very special geometry for this case. Then we will be able
to reproduce all results of the previous section in this framework. We will give a
comment on how to generalize it for an arbitrary ansatz later in the discussions
section.
5
Since the leading order constraint ν = 1 defining the very special geometry
is, indeed, the equation of motion of the auxiliary field D, we will also define the
constraint for the generalized very special geometry by making use of the equation
of motion of field D in the presence of higher derivative terms. Using the black
string ansatz of the previous section and taking into account the supersymmetry
constraints (3.5) the corresponding equation can be recast to the following form
1
1
CIJK X I X J X K +
c X I = 1.
2 2I
6
12lA
(4.1)
Now the main point to define the generalized very special geometry is as follows.
When we are considering the supersymmetrized action, in the ansatz we are choosing
there is an auxiliary two-form field vµν which could be treated as an additional gauge
field in the theory with charge p0 . Accordingly, we could introduce new scalar field
0
X 0 such that in the near horizon geometry one may set X 0 = lpA . Using this notation
the above constraint may be written as
1
1
CIJK X I X J X K + c2I (X 0 )2 X I |p0 =1 = 1.
6
12
(4.2)
0
0
0
. Obviously working
One may also define Fθφ
= p2 sin θ such that vθφ = − 4l3A Fθφ
with this notation all expressions reduce to those in section 3 for p0 = 1, though
it is not a solution for p0 6= 1. Nevertheless we will work with p0 6= 1 with the
understanding that the solution is obtained by setting p0 = 1. It is worth noting
that in general the auxiliary field vµν cannot be treated as a gauge field, though it
can be seen that for the models we are going to study it may be considered as a
gauge field. Now we shall demonstrate how the solutions of previous section can be
reproduced in this framework.
To start, let us first introduce new indexes A, B, · · · such that they take their
values over 0 and I, J, · · · by which the equation (4.2) can be recast to the following
form
1
CABC X A X B X C = 1
(4.3)
6
where CABC = CIJK for A, B, C = I, J, K and C00I = c62I and the other components
are zero. Thus, it is natural to define XA and the metric CAB as follows
1
CAB = CABC X C .
6
1
XA = CABC X B X C ,
6
(4.4)
More explicitly one has
1
c2I 0 2
XI = CIJK X J X K +
(X ) ,
6
36
and
1
CIJ = CIJK X K ,
6
CI0 =
6
c2I 0
X ,
36
X0 =
c2I X I 0
X ,
18
(4.5)
c2I X I
.
36
(4.6)
C00 =
It is easy to verify that
XA X A = 1,
XA = CAB X B ,
CAB X A X B = 1.
(4.7)
Following the standard notion of very special geometry, the magnetic central charge
may be defined as Zm = XA pA and the near horizon parameters are fixed by extremizing it, i.e. ∂i Zm = ∂i XA pA = 0. Using (4.7) a solution would be X A = pA /Zm.
Plugging this into (4.3) we find
Zm =
1
CABC pA pB pC
6
1/3
=
(p0 )2
1
I J K
CIJK p p p +
c2I pI
6
12
1/3
.
(4.8)
To get our results in the previous section one needs to set p0 = 1 at the end of the
computations.
It is almost obvious, but worth to mention, that having had the analogous to very
special geometry does not mean that the action can also be given just by (2.2) with
the generalized constraint (4.3). Therefore we would like to treat this construction
as an auxiliary method to study N = 2 supergravity solutions in the presence of
higher order corrections.
For example in the rest of this section we would like to show how the generalized
very special geometry can be used to extend the consideration of [28] while higher
derivative terms are also taken into account. To do this, consider the black string
solution obtained in section 3. At R2 level the solution can be written as follows
2
lA
dr 2
pI
(dx2 − 2rdxdr + 2 + dθ2 + sin2 θ dφ2 ),
XI = ,
4
r
lA
I
p
1
3
0
AIθ = − cos θ dφ,
A0θ = − cos θ dφ,
with v = − lA Fθφ
, (4.9)
2
2
4
1/3
1
where lA = 16 CIJK pI pJ pK + 12
c2I pI
. In writing the metric we have used the
fact that AdS3 can be written as S 1 fibered over AdS2 . Now we would like to embed
this solution into the framework of the generalized very special geometry. In order
to do that, we start with the following solution with the understanding of p0 = 1,
ds2 =
2
Zm
dr 2
(dx2 − 2rdxdr + 2
4
r
pA
3
= − cos θ dφ,
Zm
=
2
ds2 =
+ dθ2 + sin2 θ dφ2),
AA
θ
1
CABC pA pB pC .
6
XA =
pA
,
Zm
(4.10)
Following [28] we will consider the total space of U(1) bundle over (4.10) to define
a six dimensional manifold with the metric
ds26 =
2
dr 2
Zm
(dx2 − 2rdxdt + 2 + σ12 + σ22 ) + (2XA XB − CAB )(dy A + AA )(dy B + AB ).
4
r
(4.11)
7
Here σi are right invariant one-forms such that σ12 + σ22 = dθ + sin2 θ dφ2 . Let us
define new coordinates z, ψ through the following expressions
1
y A = z A + (sin B − 1)X A XB z B − pA ψ,
2
x=
2 cos B
XA z A
Zm
(4.12)
where sin B is a constant which its physical meaning will become clear later. We
define the coordinates such that the new coordinates have the following identification
z A ∼ z A + 2πnA ,
ψ ∼ ψ + 4πm,
(4.13)
where m and nA are integers. Accordingly one can read the identifications of y and
x. Using the new coordinate ψ one can define the right invariant one-forms by
σ1 = − sin ψdθ + cos ψ sin θdφ,
σ2 = cos ψdθ + sin ψ sin θdφ,
σ3 = dψ + cos θdφ.
(4.14)
In terms of the new coordinates the six dimensional metric (4.11) reads
ds2 =
Z2
dr 2
[−(cos B rdt + sin Bσ3 )2 + 2 + σ12 + σ22 + σ32 ]
4
r
+ (2XA XB − CAB )(dz A + ÃA )(dz B + ÃB ),
(4.15)
A
where ÃA = − p2 (cos B rdt + sin Bσ3 ). The obtained six dimensional manifold can
be treated as the total space of a U(1) bundle over BMPV black hole at R2 level.
Therefore we can reduce to five dimensions to get BMPV black hole where higher
derivative corrections are also taken into account. The resulting five dimensional
black hole solution is
ds
2
ÃI
2
dr 2
pI
lA
2
2
2
2
I
[−(cos B rdt + sin Bσ3 ) + 2 + σ1 + σ2 + σ3 ], X = ,
=
4
r
lA
I
p
1
= − (cos B rdt + sin Bσ3 ),
Ã0 = − (cos B rdt + sin Bσ3 ), (4.16)
2
2
and the auxiliary field is given by v = − 34 lA F̃ 0 . From this solution we can identify
sin B as the angular momentum, J, of the BMPV solution through the relation
sin B = ZJ3 . As a result we have demonstrated that the total bundle space of near
horizon wrapped M2’s and wrapped M5’s are equivalent up to R2 level, generalizing
the tree level results of [28]. In fact one may go further to show that both of them can
be obtained from quotients of AdS3 × S 3 with a flat U(1)N −1 bundle, much similar
to tree level considered in [28]. It is then possible to use this connection to increase
our understanding of microstate counting of 5D supersymmetric rotating black hole
arising from wrapped M2-branes in Calabi-Yau compactification of M-theory. We
hope to come back to this point in our future publication.
8
5
Black hole solution
In this section we shall consider a five dimensional extremal black hole2 whose near
geometry is AdS2 ×S 3 . When we are dealing with an extremal black hole with AdS2
near horizon geometry, it is more appropriate to work with entropy function formalism [24]. In fact this approach has been used to study five dimensional extremal
black hole in Heterotic string theory in presence of higher derivative terms given by
Gauss-Bonnet action [6].3 It is shown that this bosonic term was enough to correctly
reproduce the microscopic entropy coming from microstate counting in string theory.
It is the aim of this section to study the five dimensional extremal black hole in the
presence of higher derivative terms which come from supersymmetrized action. We
will also compare the result with the case where only bosonic Gauss-Bonnet term is
present. In fact it is analogous to the consideration of [35,36] where four dimensional
extremal black hole has been studied in the presence of supersymmetrized action
using entropy function formalism.
Let us start with the following ansatz for near horizon geometry
2
ds = lA
ds2ADS2 + lS2 ds2S 3 ,
X I = cont.
FrtI = eI ,
vrt = V.
(5.1)
Then the entropy function is given by E = 2π(eI qI − f0 + f1 ) where f0 is the leading
order contribution coming from quadratic part of the action which is
1 2 3 ν−1
ν +3 3
1
2(3ν + 1) 2 4νI eI
νIJ eI eJ
f0 = lA lS
D+
− 2 −
V − 4 V−
, (5.2)
4
4
2
2
2
lS2
lA
lA
lA
2lA
and the higher order contribution, f1 , comes from the four-derivative terms that for
our ansatz, is
2
c2I 2 3 X I 1
1
4V 4 I 4V 3 I DV I D 2 I
f1 =
lA lS
X
−
X + 8 e − 4 e +
+
2
8
48
4 lS2
lA
lA
3lA
3lA
12
2V 2 X I
−
4
3lA
3
5
+ 2
2
lS lA
V eI
− 4
lA
1
1
− 2
2
lA lS
.
(5.3)
At leading order where only f0 contributes, one may integrate out the the auxiliary fields by extremizing the entropy function with respect to them, arriving at
1 2 3 6
1 2 3 6
2
1
2
GIJ eI eJ
I J
f0 = lA lS 2 − 2 + 4 (νI νJ − νIJ )e e = lA lS 2 − 2 +
.
4
2
lS
lA 2lA
2
lS
lA
lA
(5.4)
2
Explicit solutions of N = 2 5D supersymmetric black holes in the context of very special
geometry have been obtained for example in [31].
3
Higher order corrections to 5D BH have also been studied in [32–34].
9
It is easy to extremize the entropy function with respect to the parameters to find
lA , lS , X I and eI . In particular for STU model where C123 = 1 and the other components are zero one gets
(q1 q2 q3 )1/3
1 (q1 q2 q3 )1/2
, eI =
,
(5.5)
qI
2
qI
√
and the corresponding black hole entropy is S = 2π q1 q2 q3 .
In general it is difficult to do the same while the higher order corrections are also
taking into account. Therefore to proceed, the same as in the previous, we will use
the supersymmetry transformations to simplify as much as we can and then using
the equation of motion of field D, we find a supersymmetric solution. For our ansatz
(5.1) the supersymmetry conditions (2.9) lead to
lS = 2lA = (q1 q2 q3 )1/6 , X I =
D=−
3
,
2
lA
4
eI = − V X I ,
3
Using these relations we may set X I =
ν=
1
E,
3
lA
νI eI =
eI
lA
3
V = − lA ,
4
lS = 2lA .
(5.6)
and defining E = 16 CIJK eI eJ eK one has
3
E,
2
lA
νIJ eI eJ =
6
E.
lA
In this notation the equation of motion for auxiliary field D reads
3
DX I
V eI
lA
3
− 4 =0
E − lA + c2I
12
6
3lA
(5.7)
(5.8)
I
so that lA = 12 (8E − c2I6e )1/3 . By making use of the expressions for the parameters
D, V, X I and lS given above, the entropy function gets the following simple form
1
1
2
I
I
I
I J K
E = 2π(qI e − 4E + c2I e ) = 2π (qI + c2I )e − CIJK e e e
(5.9)
8
8
3
Extremizing the entropy function with respect to eI we get
1
2CIJK eJ eK = qI + c2I
8
(5.10)
which in principle can be solved to find eI in terms of the charges, c2I and parameters
CIJK ’s. The entropy is also given by S = 4π
q + eI 4 . In particular for STU model
3 I
we get eI = (q1+ q2+ q3+ )1/2 /2qI+ . Plugging this into the expressions of lA and X I , we
obtain
2lA =
4
(q1+ q2+ q3+ )1/6 (1
c2I 1 1/3
) ,
−
12 qI+
(q1+ q2+ q3+ )1/3
c2I 1 −1/3
X =
(1 −
)
. (5.11)
+
12 qI+
qI
I
We use a notion in which qI+ = qI + 81 c2I .
10
Finally the entropy is found to be
S = 2π
q
q1+ q2+ q3+ .
(5.12)
It is very interesting in the sense that the entropy of the black hole at R2 level has
the same form as the tree level except that the charges qI ’s are replaced by shifted
charges qI+ ’s.
This result can be used to see how the higher order corrections stretch horizon.
For example if we start by a classical solution with q1 = 0 we get vanishing horizon
(small black hole) and therefore the entropy is zero. But adding the R2 terms we
get a smooth solution with a non-zero entropy give by
r
c21 + +
S=π
q q .
(5.13)
2 2 3
This procedure could also be used to understand, upon dimensional reduction
to four dimensions, the single-charge small black hole studied in [37].
We note that in the above consideration we have used the supersymmetry transformations to simplify the computations and therefore the solution would be supersymmetric. Thus it does not exclude other solutions. In fact we would expect to
have another solution corresponding to non-BPS solution. Actually one could start
from a more general constraint than (5.6) as follows
D=
α
,
2
lA
XI = β
eI
,
lA
V = γlA ,
lS = 2lA
(5.14)
and solve the equations for parameters (α, β, γ). Doing so, for fixed lA given above,
we find two solutions: (−3, 1, −3/4) and (3, −1, −3/4). The first one is the solution
we have studied, but the second one which is not supersymmetric leads to the
following entropy function
3
2
I
I J K
E = 2π (qI − c2I )e − CIJK e e e .
(5.15)
8
3
It is straightforward to extremize the entropy function with respect to eI ’s to get
the parameters in terms of qI ’s, though we won’t do that here.
It is also instructive to compare the results with the case where the higher order
corrections are given in terms of the Gauss-Bonnet action. It is known that this
term cannot be supersymmetrized, but it is still interesting to see what would be
the corresponding corrections. In our notation the Gauss-Bonnet term is given by
c2I X I
Rabcd Rabcd − 4Rab Rab + R2 .
(5.16)
8
2
2 · 3π
It can be shown that with the specific coefficient we have chosen for the GaussBonnet action, the corresponding entropy function, fixing the auxiliary fields as
(5.6) for tree level action, is the same as (5.9) and therefore we get the same results
as those in supersymmetrized action.
LGB =
11
6
Discussions
In this paper we have studied black string and black hole solutions in the presence of
higher derivative terms. In the black string case, where the near horizon geometry
is AdS3 × S 2 , a proper method is c-extremization, whereas in the black hole solution
with AdS2 × S 3 near horizon geometry the better method is given by the entropy
function formalism. In both cases we have seen that adding higher derivative terms
will change the feature of the very special geometry in such a way that the leading
order constraint ν = 1 is not satisfied any more. Therefore the standard method of
very special geometry is not applicable at this level.
We have shown how to generalize the notion of very special geometry in the
presence of higher derivative terms. We have observed that the generalized very
special geometry depends on the solution we are considering. In particular we have
explicitly studied the generalization for the magnetic black string solution where we
have reproduced all results obtained from another method, e.g. c-extremization.
For an arbitrary solution, the generalized constraint which defines the very special geometry can be obtained from equation of motion of the auxiliary filed D. In
general we have
1
c2I I
CIJK X I X J X K +
(X D + F Iµν vµν ) = 1.
(6.1)
6
72
If we are interested in a supersymmetric solution, we can also use the supersymmetry
transformations to further simplify the constraint
1
c2I X I 3
I J K
a bc de
2
2
(6.2)
CIJK X X X +
ǫabcde γ v v − (γ · v) − v = 1.
6
54
2
An interesting application of this construction is to generalize the consideration
of [28]. In particular we have shown that the total bundle space of near horizon
wrapped M2’s and wrapped M5’s are equivalent up to R2 level, generalizing the tree
level results of [28]. Actually we could also show that both of them can be obtained
from quotients of AdS3 × S 3 with a flat U(1)N −1 bundle, much similar to tree level
considered in [28]. Although we have not pushed this observation any further, one
might suspect that using this connection could increase our knowledge of microstate
counting of 5D supersymmetric rotating black hole arising from wrapped M2-branes
in Calabi-Yau compactification of M-theory.
We have also studied higher order corrections to extremal black holes in five
dimensions in which the higher order corrections come from supersymmetric completion of R2 term. In this paper we have explicitly written the corrected solution
of 5D black holes for a specific model, namely, STU model. To extend the solution
for a general model we will have to solve a set of equations
CIJK xJ xK = aI ,
I = 1, · · · , N,
(6.3)
for given CIJK and aI . In general it is difficult to solve this equation and it may
not even have a unique solution. Nevertheless for a particular model this can be
12
solved explicitly (for example see [38]). An interesting observation we have made in
this paper is that the solution of the above equation can be used for both tree level
case and the case when the R2 corrections are added. The only difference is that in
the presence of R2 terms one just needs to replace aI by another constant which is
related to it by a constant shift.
In fact as far as the entropy is concerned we have seen that the corrected entropy
can simply be obtained by replacing the electric charges qI ’s with the shifted charges
qI+ ’s defined in footnote 2. In particular this result shows how one can get a smooth
solution with non-zero entropy out of a small black hole which in leading order
supergravity is singular with vanishing entropy.
J
Finally as an aside let us define C IJK in terms of CIJK by C IJK CKLM = δLI δM
+
I J
δM
δL . Although it is not clear whether this equation can be solved in general or
even if it has a unique solution, it can be used to write a solution for equation (6.3).
For example it terms of this parameter the corresponding black hole entropy could
be written as [39]
r
1 IJK
S = 2π
C
qI qJ qK .
(6.4)
6
In the presence of R2 correction the entropy has the same form, except that one
needs to replace qI by qI+ .
Note added: While we were in the final stage of the project we have received the
paper [40] where the five dimensional black hole in the presence of higher derivative
terms given by (2.6) has also been studied.
Acknowledgments
I would like to thank Hajar Ebrahim for discussions on the related topic and also
comments on the draft of the article.
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16