Euclidean analysis of the entropy functional formalism
Euclidean analysis of the entropy functional formalism
Óscar J. C. Dias,1 Pedro J. Silva,2
1 Departament de F́ısica Fonamental, Universitat de Barcelona,
Av. Diagonal 647, E-08028 Barcelona, Spain,
2 Institut de Ciències de l’Espai (IEEC-CSIC) and
Institut de F́ısica d’Altes Energies (IFAE),
E-08193 Bellaterra (Barcelona), Spain
odias@ub.edu, psilva@ifae.es
ABSTRACT
The attractor mechanism implies that the supersymmetric black hole near horizon solution is
defined only in terms of the conserved charges and is therefore independent of asymptotic moduli.
Starting only with the near horizon geometry, Sen’s entropy functional formalism computes the
entropy of an extreme black hole by means of a Legendre transformation where the electric fields
are defined as conjugated variables to the electric charges. However, traditional Euclidean methods
require the knowledge of the full geometry to compute the black hole thermodynamic quantities.
We establish the connection between the entropy functional formalism and the standard Euclidean
formalism taken at zero temperature. We find that Sen’s entropy function f (on-shell) matches the
zero temperature limit of the Euclidean action. Moreover, Sen’s near horizon angular and electric
fields agree with the chemical potentials that are defined from the zero-temperature limit of the
Euclidean formalism.
http://arxiv.org/abs/0704.1405v3
Contents
1 Introduction 1
1.1 Attractor mechanism and entropy functional formalism . . . . . . . . . . . . . . . . . 2
1.2 Zero temperature limit and chemical potentials . . . . . . . . . . . . . . . . . . . . . 2
1.3 Entropy functional formalism from an Euclidean perspective . . . . . . . . . . . . . . 3
1.4 Main results and structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Entropy functional formalism revisited 5
3 Euclidean zero-temperature formalism: BPS black holes 9
4 Euclidean zero-temperature and entropy functional formalisms 14
4.1 Near-horizon and asymptotic contributions to the Euclidean action . . . . . . . . . . 14
4.2 Relation between chemical potentials in the two formalisms . . . . . . . . . . . . . . 17
5 Extremal (non-BPS) black holes 19
5.1 Extreme three-charged black hole with ergoregion . . . . . . . . . . . . . . . . . . . . 20
5.2 Extreme three-charged black hole without ergoregion . . . . . . . . . . . . . . . . . 21
6 Discussion 23
A Three-charged black hole: solution and thermodynamics 24
A.1 The D1-D5-P black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
A.2 The near-BPS limit of the D1-D5-P black hole . . . . . . . . . . . . . . . . . . . . . 28
B Explicit agreement for other black hole systems 29
B.1 Four-charged black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
B.1.1 BPS black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
B.1.2 Extreme (non-BPS) black hole: ergo-free solution . . . . . . . . . . . . . . . . 32
B.1.3 Extreme (non-BPS) black hole: ergo-branch solution . . . . . . . . . . . . . . 33
B.2 Extreme Kerr-Newman black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1 Introduction
Black holes (BH) are one of most interesting laboratories we have to investigate quantum gravity
effects. Due to their thermodynamic behavior these objects have been associated to ensembles of
microstates in the fundamental quantum gravity theory where ideally, quantum statistical analy-
sis should account for all the BH coarse-grained thermodynamical behavior. In particular, many
important insights in the classical and quantum structure of BH have been obtained studying
supersymmetric configurations in string theory. Supersymmetric BH have many important prop-
erties that turn out to be crucial to obtain all the new results. Basically, supersymmetry triggers a
number of non-renormalization mechanisms that protect tree level calculations from higher order
loop corrections. Moreover, this kind of behavior has also been found in some non-supersymmetric
extreme solutions.
1.1 Attractor mechanism and entropy functional formalism
In this context we have the so called attractor mechanism [1]. It was originally thought in the
context of four dimensional N = 2 supergravity, where we have that the values of the scalar fields
at the horizon are given by the values of the BH conserved charges and are independent of the
asymptotic values of the scalars at infinity. For these BH (and others) it has been checked that
the Bekenstein-Hawking entropy agrees with the microscopic counting of the associated D-brane
system. Not only in the supergravity approximation, but also after higher derivative corrections
are added to the generalized prepotential [2]. These results motivated a conjecture where the BH
partition function equals the squared of the associated topological string partition function i.e.,
ZBH = |ZTop|2 [3]. Lately, the attractor mechanism has been extended to other directions, and
applied to several gauged and ungauged supergravities (see, e.g., [4, 5, 6]).
Importantly, the attractor mechanism has provided a new way to calculate the BH entropy. In
a series of articles [7, 8, 9], Sen recovered the entropy of D-dimensional BPS BH using only the near
horizon part of the geometry. Basically, in this regime the solution adopts the form AdS2 ⊗ SD−2
1 plus some electric and magnetics fields. The entropy S is obtained by introducing a function f
as the integral of the corresponding supergravity Lagrangian over the SD−2. More concretely, an
entropy function is defined as 2π times the Legendre transform of f with respect to the electric
fields ei. Then, an extremization procedure fixes the on-shell BPS values of the different fields of
the solution and in particular determines the BPS value of the entropy,
Sbps = 2π
. (1.1)
Note that in the above definition the different near horizon electric fields take the role of “conjugated
chemical potentials” to the BH charges. This formalism has also been extended to extreme non-BPS
The attractor mechanism, both for asymptotically AdS or flat BH, implies that in the near
horizon geometry we have a dual CFT theory where the microscopic structure can be studied. We
expect that not only the entropy but all the statistic properties of such supergravity systems should
be described in terms of their dual CFT states.
1.2 Zero temperature limit and chemical potentials
Supersymmetric BH in asymptotically AdS spaces have also been studied using the AdS/CFT cor-
respondence [10, 11, 12, 13]. For the AdS5 case we still do not have a CFT microscopic derivation
of its entropy that reproduces the supergravity result. Nevertheless, in [12, 13] it was showed that
the phase space of this supersymmetric sector can be scanned in both sides of the correspondence
showing a rich structure with phase transitions and Hagedorn alike behavior2. In fact, observables
in both dual pictures agree up to numerical factors, a very non-trivial result since the CFT calcu-
1The analysis of the near horizon geometry has been applied to more general BH that define squashed AdS2⊗SD−2
geometries like in [9, 4].
2These T = 0 phase transitions were analyzed both in the strong and weak coupling regimes. Remarkably, it was
found that their properties resemble the well-known finite temperature phase transitions, where the Hawking-Page
phase transition in the strong coupling corresponds to the deconfinement/confinement transition at weak coupling
[10, 13].
lations are performed at zero coupling only3. In order to study the full statistical properties (so
that we could in principle do more than just account for the entropy), in [12, 13] it was found
how to define the different chemical potentials µi that control the supersymmetric BH partition
function in the grand canonical ensemble. The basic input comes from the thermodynamics of the
dual CFT theory, where the BPS partition function is obtained from the finite temperature one,
by sending the temperature to zero. This also sends the several chemical potentials to their BPS
values. The associated dual limiting procedure in the supergravity regime corresponds also to send
the temperature to zero. Done carefully, this defines the supergravity chemical potentials that are
dual to the the CFT ones and, more generally, the statistical mechanics of supersymmetric BH
that is free of divergencies. These chemical potentials are the next to leading order terms of the
zero temperature expansion of the horizon angular velocities and electric potentials. The resulting
supergravity partition function is given, as expected, by the exponentiation of the regularized Eu-
clidean action I evaluated at the BH solution. In this paper we call “Euclidean zero-temperature
formalism” to the zero-temperature limit in the supergravity system that determines the Euclidean
action, entropy and the chemical potentials. After some algebra we arrive to the supersymmetric
quantum statistical relation (SQSR) [14] where the Euclidean action I can be rewritten as the Leg-
endre transform of the entropy S with respect to the different supersymmetric chemical potentials
Ibps = µi q
bps − Sbps , (1.2)
where qibps’s represent the conserved BH charges conjugated to the µi’s (later, we will use the
notation qi ≡ {Qi, J i} and µi ≡ {φi, ωi}). As said above, these supergravity chemical potentials
are closely related to the dual CFT chemical potentials. Therefore, they provide a very clear
picture of the BPS BH as dual to a supersymmetric CFT in the grand canonical ensemble. This
approach also defines the finite supersymmetric Euclidean action (1.2), and in fact allows to study
the statistical mechanics of BPS black holes. A similar analysis can be done for extreme non-BPS
systems.
1.3 Entropy functional formalism from an Euclidean perspective
Sen’s entropy functional formalism is formulated only with the knowledge of the near horizon
geometry. But, since it computes the BH entropy, which is a thermodynamic quantity, it should
be possible to understand it starting from a traditional thermodynamical Euclidean analysis of the
black hole system.
In fact, the strong resemblance between equations (1.1) and (1.2) is evident. In other words, it
would be strange if string theory produces two unrelated functions in the same supergravity regime
that calculate the BH entropy. Looking into both definitions with more care, we find that the
entropy is defined as the Legendre transform of the BH charges in the saddle point approximation
of the supergravity theory. Nevertheless, in (1.1) the vacuum solution is just the near horizon
geometry with conjugated potentials related to the electric fields, and f is the on-shell Lagrangian
over only SD−2. Instead, in (1.2), the vacuum is the entire BH solution; the conjugated potentials
are associated to gauge potentials rather than field strengths; and I is the on-shell full Euclidean
action. The main goal of this paper is to understand the connection between these two approaches.
3In [10] the CFT partition function was calculated at zero coupling. Also, an index was considered to count
supersymmetric states but unfortunately it turns out to be blind to the BH sector.
One of the key points of our analysis relies in the natural splitting of the Euclidean action into
two parts corresponding basically to: i) the near horizon part of space, and ii) the asymptotic region.
Then we find that in the extremal cases (without ergoregion), the asymptotic part vanishes, and the
near horizon part reduces to Sen’s function 2πf . Also, the conjugated chemical potentials found in
both methods agree, due to an argument that relates differences of gauge potentials produced by
variations of near-BPS parameters with variations of the potential on the radial coordinate.
1.4 Main results and structure of the paper
As stated above, the main goal of this article is to provide a bridge between Sen’s entropy functional
formalism and standard Euclidean analysis of the thermodynamics of a black hole system. While
doing so, we also find that the supergravity conjugated potentials defined in Sen’s formalism map
into chemical potentials of the dual CFT.
We obtain a unifying picture where:
1)We are able to recover the entropy function of Sen from the zero temperature limit of the
usual BH thermodynamics and the statistical mechanics definitions of the dual CFT theory. The
supergravity and their dual CFT chemical potentials are identified with the surviving Sen’s near
horizon electric and angular fields. The Euclidean action is identified with Sen’s function 2πf .
2)As a byproduct of the above analysis we have understood how to calculate the BPS chemical
potentials that control the statistical properties of the BH using only the BPS regime, i.e., without
needing the knowledge of the non-BPS geometry. The CFT chemical potentials are dual to the
supergravity ones. Traditionally, to compute the latter we have to start with the non-BPS solution
and send the temperature to zero to find the next to leading order terms in the horizon angular
velocities and electric potentials expansions that give the chemical potentials. This requires the
knowledge of the non-BPS geometry. Unfortunately, sometimes this is not available and we only
know the BPS solution. But, from item 1) we know that the near horizon fields, that Sen computes
with the single knowledge of the BPS near horizon solution, give us the supergravity chemical
potentials. So now we can compute the supergravity chemical potentials of any BPS BH solution,
regardless of its embedding into a family of non-BPS solutions, while still keeping the relation with
the dual CFT.
3)It is known that the attractor mechanism seems to work also for non-supersymmetric but
extremal BH 4. We have tested the Euclidean zero temperature formalism for many of these BH,
always finding a well defined limit and agreement with Sen’s results for extremal non-BPS BH5.
This is a non-trivial fact since there is no supersymmetry protecting the limit. Therefore, in general,
the supergravity regime should not give the correct statistical relations. We interpret this result as
another confirmation that there is a protecting mechanism for extremal non-supersymmetric BH.
The plan of the paper is the following. In section 2 we review Sen’s entropy functional approach
using the D1-D5-P system as an illuminating example. In the beginning of section 3 we review
the main ideas and results of the Euclidean zero temperature formalism for BH in the AdS/CFT
4See [9, 25] and references there in.
5Actually at the level of two derivative theory, Euclidean T = 0 formalism is well defined only for BH with no
ergoregion. For BH with ergoregion we have an ill-defined limit, that nevertheless allows to define the entropy and
all chemical potentials. This is telling us that these geometries are not fully protected from string corrections. The
same caveats and conclusions are also obtained using Sen’s approach, and this is related to the fact that for these
BH the attractor mechanism is only partial since there is dependance on the asymptotic data [9].
framework. Then, we apply this formalism to the most general rotating D1-D5-P system. We
analyze the connection between the entropy functional and Euclidean formalisms in section 4,
identifying how and why both prescriptions are equivalent. In section 5 we discuss the application
of the Euclidean T = 0 formalism to extreme non-BPS BH and again find agreement with Sen’s
results. Section 6 is devoted to a short discussion on the results and possible future avenues to
follow. In Appendix A we review the D1-D5-P BH solution in detail, including its thermodynamics.
In Appendix B, we write the chemical potentials and Euclidean action for some other BH systems
not considered in the main body of the text. We consider the four charged system of type IIA
supergravity, and the Kerr-Newman BH. We confirm that for these BH the relation established
in section 4 between the entropy functional and Euclidean formalisms holds. This agreement also
extends to AdS black holes as is explicitly confirmed in the context of 5D gauged supergravity in
[15].
Note: While we where proof-reading this article, the paper [16] appeared in the arXives. It
contains relevant discussions and results connected to our work, regarding Sen’s approach and
Wald’s method for AdS BH.
2 Entropy functional formalism revisited
As we pointed out in the introduction, Sen developed a simple method – the entropy functional
formalism – to compute the entropy of supersymmetric BH in supergravity [7]. Lately, this approach
has been applied to rotating BH in gauged and ungauged supergravity (see, e.g., [9, 6]). Here, we
will review some of the key aspects of this formalism that we will use latter. We just need to
address non-rotating cases, but we will comeback to rotating attractors at the end of this section,
for completeness.
Sen’s entropy functional formalism assumes that: (i) we start with a Lagrangian L with gravity
plus some field strengths and uncharge massless scalar fields; and (ii) due to the attractor mechanism
the near horizon geometry of a D-dimensional BH is set to be of the form AdS2 ⊗SD−2. From the
above input data, the general form of the near horizon BH solution is
ds2 = v1
−ρ2dτ +
+ v2dΩ
D−2 ,
F (i)ρτ = ei , H
(a) = paǫD−2 ,
φs = us , (2.1)
where ǫD−2 is the unit-volume form of S
D−2, and (ei, pa) are respectively the electric fields and the
magnetic charges of the BH. Note that (~u,~v,~e, ~p) are arbitrary constants up to now and therefore
the solution is off-shell. Next, it is defined the following function
f(~u,~v,~e, ~p) =
−gL , (2.2)
where L is the string frame Lagrangian of the theory (see, e.g., (A.19)). After minimizing f(~u,~v,~e, ~p)
with respect to (~u,~v) we obtain the exact supersymmetric near horizon BH solution in terms of
(~e, ~p). In fact, the field equations are reproduced by this minimization procedure. Furhermore,
minimization with respect to ~e gives the electric charges ~q. Explicitly, the on-shell values of ~u,~v,~e
that specify (2.1) for a given theory described by (2.2) are found through the relations,
= 0 ,
= 0 ,
= qi . (2.3)
Then, using Wald formalism [27], Sen derived that the entropy S of the corresponding BH is given
by 2π times the Legendre transform of f ,
S = 2π
. (2.4)
Finally notice that the minimization procedure, can be taken only after S is defined. In this form
S is really an entropy function of (~u,~v, ~q, ~p), that after minimization equals the BH entropy as a
function of (~q, ~p) only.
In the rest of this section we will discuss the above formalism in a specific theory. We consider
the D1-D5-P supersymmetric solution of ten-dimensional type IIB supergravity, discussed in the
previous section, as the main example (this case was first analyzed in [17], at the level of supergravity
and for its higher order corrections). Our aim is to highlight the details of the application of Sen’s
formalism to this solution. This will provide a solid background to compare, in section 4, Sen’s
formalism with the Euclidean one developed in section 3.
From Appendix A.1 we know that the supersymmetric D1-D5-P metric, the RR two-form C(2)
and the dilaton Ψ are given by 6
ds2 =
[−dt2 + dy2 + Q
(dt− dy)2] +
H1H5(dr
2 + r2dΩ23) +
H1/H5
dz2i ,
C(2) = −
dt ∧ dy − Qbps5 cos
2 θdφ ∧ dψ , e2Ψ =
. (2.5)
whereH1 = (1+
), H5 = (1+
) and (Q
1 , Q
5 , Q
p ) are the D1,D5,P charges, respectively.
Then, it is easy to take the near horizon limit to obtain,
ds2 =
−ρ2dτ2 + dρ
dz +
dz2i ,
F(3) =
dρ ∧ dτ ∧ dz + 2Qbps5 ǫ3 , e
, (2.6)
where we used
t , ρ = r2 , z = y − t . (2.7)
6This is the string frame version of (A.4), and (A.7) and (A.8) with a1 = a2 = 0.
Note that, alternatively, all the information encoded in the near horizon structure (2.6) could
be extracted without knowing the full geometry, using Sen’s approach. Its application starts by
assuming that the near horizon metric is given in terms of the unknowns (~v, ~u,~e, ~p) as follows,
ds2 = v1
−ρ2dτ2 +
+ v2dΩ
3 + u1 (dz + e2ρdt)
dz2i ,
F(3) = e1 dρ ∧ dτ ∧ dz + 2Q
5 ǫ3 , e
2Φ = u23 . (2.8)
Having the Lagrangian (A.19) of type IIB at hand, one now follows the steps summarized in (2.2)-
(2.4) to find the on-shell expressions for (~v, ~u,~e). From (2.8) one has
−g̃ = u1/21 u22v1v
2 sin θ cos θ,
ρτz = e1 and F
θψφ = 2Q
5 sin θ cos θ. The entropy function, S(~u,~v, ~q, ~p) = 2π[q1e1 + q2e2 −
f(~u,~v,~e, ~p)] is then
S(~u,~v, ~q, ~p) = 2π
q1e1 + q2e2 −
Minimizing this entropy function with respect to ~u,~v,~e one finds the on-shell attractor values,
, ~u =
 , ~q =
1 , Q
, p = Q
One also finds that f(~q, ~p) = 0 on-shell. Plugging this information into the entropy function
S(~u,~v, ~q, ~p) we get
S(~q, ~p) = 2π [q1e1 + q2e2 − f ]on−shell
p , (2.9)
that is the well known result for this BH.
It will be relevant for section 4 to stress that the above analysis can be carried on in the case
where the magnetic field is replaced by its dual electric field. This electric field comes from the RR
seven-form field strength F(7), Poincaré dual of the magnetic part of F(3),
F(7) =
r3H25
dr ∧ dt ∧ dy ∧ dz1 ∧ dz2 ∧ dz3 ∧ dz4 . (2.10)
In the near horizon limit, i.e., after taking the change of coordinates (2.7), F(7) reduces to
F(7) =
dρ ∧ dτ ∧ dz ∧ dz1 ∧ dz2 ∧ dz3 ∧ dz4 . (2.11)
In the next lines we want to recover this near horizon attractor value for F(7), without making use
of the near horizon limit of the full geometry, i.e., using instead a Sen-like approach.
In this pure electric case, we first notice that there is an extra pair of conjugated variables
(e3, q3) and second, that f should be now calculated on a modified Lagrangian with the F(7) RR
field strength appropriately added. This is an effective “democratic” Lagrangian supplemented by
duality constraints imposed by hand 7. The motivation, limitations and formulation of this effective
Lagrangian are presented in detail in [18]. In this context, the string frame Lagrangian (A.19) of
the D1-D5-P system takes the form,
16πG10
R̃− 4∂µΨ∂µΨ
2 · 3!
F 2(3) −
2 · 7!
F 2(7)
, (2.12)
where the magnetic part of the original F(3) field is now encoded in the F(7) contribution. The
D1-branes and D5-branes source the electric F(3) and F(7) fields, respectively.
In the entropy function formalism, the function f(~u,~v,~e) is obtained by evaluating action (2.12)
at the horizon, i.e., by integrating along the S8 sphere. We use the near-horizon fields (2.8). So, the
metric determinant is
−g̃ = u1/21 u22v1v
2 sin θ cos θ, F
ρτz = e1 and F
ρτzz1···z4 = e3. The entropy
function, S(~u,~v, ~q) = 2π[q1e1 + q2e2 + q3e3 − f(~u,~v,~e)] is then
S(~u,~v, ~q) = 2π
q1e1 + q2e2 + q3e3 −
Minimizing this entropy function with respect to ~u,~v,~e one finds the on-shell attractor values
, ~u =
 , ~q =
1 , Q
p , Q
(2.13)
which are used to obtain the on-shell function: f(~q) = 1
p . Then, use of equation
(2.13) yields the on-shell entropy value,
S(~q) = 2π [q1e1 + q2e2 + q3e3 − f ]on−shell
p , (2.14)
that is, in this dual computation we indeed recover the value (2.9).
As commented in the introduction, the above approach was generalized to rotating BH in
ungauged and gauged supergravities [9, 6]. At the level of two derivative Lagrangian, rotating
BH in ungauged supergravity have their near horizon geometry fully determined by the entropy
functional only if they have no ergoregion. However, BH with ergoregion show only partial attractor
mechanism, since their entropy functional has flat directions [9, 26]. In this case, minimization does
not fix the value of all quantities in the near horizon geometry. There is some surviving dependance
7We should emphasize that the introduction of a RR p-form field strength with p > 5 doubles the number of degrees
of freedom. To get the right equations of motion from (2.12) we must then introduce by hand duality constraints
relating the lower- and higher-rank RR potentials. We ask the reader to see [18] for further details.
on the asymptotic value of the scalars, although it fixes the form of entropy itself and the electric
and angular fields.
Generalization to gauged supergravities includes AdS BH into the discussion. The resulting
picture is basically the same, where care has to be taken when evaluating f due to Chern-Simon
terms in the Lagrangian (see [6] for details). In these cases, the attractor mechanism is related to
a non trivial flow between fixed points at both boundaries of spacetime, the horizon AdS and the
asymptotic AdS at infinity.
3 Euclidean zero-temperature formalism: BPS black holes
In [12, 13] the “thermodynamics” or better “the statistical mechanics” of supersymmetric solitons in
gauged supergravity was studied in detail using an extension of standard Euclidean thermodynam-
ical methods to zero temperature systems. We call this approach the Euclidean zero-temperature
formalism. BPS BH can be studied as dual configurations of supersymmetric ensembles at zero
temperature but non-zero chemical potentials in the dual CFT. These potentials control the ex-
pectation value of the conjugated conserved charges carried by the BH, like e.g., angular momenta
and electric charge.
In these articles, the two main ideas are: First, there is a supersymmetric field theory dual to
the supergravity theory. Second, in this dual field theory the grand canonical partition function
over a given supersymmetric sector can be obtained as the zero temperature limit of the general
grand canonical partition function at finite temperature. This limit also fixes the values of several
chemical potentials of the system.
To make things more clear, recall that all supersymmetric states in a field theory saturate a
BPS inequality that translates into a series of constraints between the different physical charges.
For definiteness, let us consider a simple case where the BPS bound corresponds to the constraint8:
E = J . Then, defining the left and right variables E± = 1
(Eν ± Jν), β± = β(1 ± Ω) the grand
canonical partition function is given by
Z(β,Ω) =
e−(β+E++β−E−) . (3.1)
At this point, it is clear that taking the limit β− → ∞ while β+ → ω (constant), gives the correct
supersymmetric partition function. The above limiting procedure takes T to zero, but also scales
Ω in such a way that the new supersymmetric conjugated variable ω is finite and arbitrary. Note
that among all available states, only those that satisfy the BPS bound are not suppress in the sum,
resulting in the supersymmetric partition function
Z(ω) =
e−ωJ , (3.2)
where the sum is over all supersymmetric states (bps) with E = J . The above manipulations are
easy to implement in more complicated supersymmetric field theories like, e.g., N = 4 SYM theory
in four dimensions. What is less trivial is that amazingly it could also be implemented in the
8This type of BPS bound appears in two dimensional supersymmetric models like, e.g., the effective theory of 1/2
BPS chiral primaries of N = 4 SYM in R ⊗ S3 (see [19, 20, 21]).
dual supersymmetric configurations of gauged supergravity, since it means that these extreme BPS
solutions are somehow protected from higher string theory corrections.
Before we apply the Euclidean zero-temperature formalism to concrete black hole systems, it
is profitable to highlight its key steps. To study the statistical mechanics of supersymmetric black
holes we take the off-BPS BH solution and we send T → 0. In this limiting procedure, the angular
velocities and electric potentials at the horizon can be written as an expansion in powers of the
temperature. More concretely one has when T → 0,
β → ∞ ,
Ω → Ωbps −
+ O(β−2) ,
Φ → Φbps −
+ O(β−2) , (3.3)
where β is the inverse temperature; (Ω,Φ) are the angular velocities and electric potentials at the
horizon; the subscript bps stands for the values of these quantities in the on-shell BPS solution; and
(ω, φ) are what we call the supersymmetric conjugated potentials, i.e., the next to leading order
terms in the expansion. For all the systems studied, we find that the charges have the off-BPS
expansion,
E = Ebps +O
, Q = Qbps +O
, J = J
, (3.4)
where (E,Q, J) are the energy, charges and angular momenta of the BH. In supergravity, the grand
canonical partition function in the saddle point approximation is related to so called quantum
statistical relation (QSR) [14]
I(β,Φ,Ω) = βE − ΦQ−ΩJ − S , (3.5)
where S is the entropy, and (β,Φ,Ω) are interpreted as conjugated potentials to E,Q, J , respec-
tively. I is the Euclidean action (evaluated on the off-BPS BH solution) that, in this ensemble,
depends only on (β,Φ,Ω). It plays the role of free energy divided by the temperature. Inserting
(3.3) and (3.4) into (3.5) yields
I(β,Φ,Ω) = β(E
bps − ΦbpsQbps − ΩbpsJbps) + φQbps + ωJbps − Sbps +O
. (3.6)
Here, we observe that this action is still being evaluated off-BPS. Moreover, the term multiplying
β boils down to the BPS relation between the charges of the system and thus vanishes (this will
become explicitly clear in the several examples we will consider). This is an important feature,
since now we can finally take the β → ∞ limit yielding relation (1.2). With the present notation
it reads as
Ibps = φQ
bps + ωJbps − Sbps . (3.7)
It is important to stress that this zero temperature limiting procedure yields a finite, not diverging,
supersymmetric version of QSR, or shortly SQSR. Note that if we had evaluated the Euclidean
action (3.5) directly on-shell it would not be well defined, as is well-known. As a concrete realization,
we picked (and will do so along the paper) the SQSR to exemplify that the T → 0 limit yields
well-behaved supersymmetric relations. The reason being that this SQSR relation is the one that
will provide direct contact with Sen’s entropy functional formalism, which is the main aim of our
study. However, it also provides a suitable framework that extends to the study of the full statistical
mechanics of supersymmetric black holes.
Euclidean action and chemical potentials of BPS D1-D5-P black holes
As we pointed out in the introduction, due to the attractor mechanism, BH in ungauged super-
gravity have a dual CFT theory defined in the boundary of its near horizon geometry. Therefore,
and in a similar way as for asymptotic AdS spacetimes, these BH should be related to statistical en-
sembles in the dual CFT. As a direct consequence of this duality, we conclude that in the ungauged
case there should also exist a well defined zero temperature limit in the supergravity description
that yields the dual CFT chemical potentials.
In what follows, we apply the Euclidean T → 0 limit to the illuminating example of five-
dimensional three charged BH with two angular momenta that can be described as the D1-D5-P
system of type IIB supergravity 9. This solution can also be embedded as a solution of eleven-
dimensional supergravity, or as a solution of type IIA, where all these different descriptions are
related by dimensional reduction and U -dualities. A detailed review of the D1-D5-P BH solution
[22, 23] and its thermodynamic properties needed for our discussion can be found in Appendix A.
In type IIB, the ten-dimensional system can be compactified to five dimensions on T 4 × S1
with the D5-branes wrapping the full internal space and the D1-branes and KK-momentum on the
distinguished S1. The length of S1 is 2πR and the volume of T 4 is V . We will work in units such that
the five-dimensional Newton constant is G5 = G10/2πRV = π/4. The ten-dimensional solution
is characterized by six parameters: a mass parameter, M ; spin parameters in two orthogonal
planes, (a1, a2); and three boost parameters, (δ1, δ5, δp), which fix the D1-brane, D5-brane and
KK-momentum charges. The physical range of M is M ≥ 0. We assume without loss of generality
that δi ≥ 0 (i = 1, 5, p), and a1 ≥ a2 ≥ 0 (The solutions with a1a2 ≤ 0 are equivalent to the
a1a2 ≥ 0 ones due to the symmetries of the solution). We will use the notation ci ≡ cosh δi,
si ≡ sinh δi.
The BH charges are: ADM mass E, the angular momenta (Jφ, Jψ) and the gauge charges
(Q1, Q5, Qp) associated with the D1-branes, D5-branes and KK momentum. In terms of the pa-
rameters describing the solution they are given by
[cosh(2δ1) + cosh(2δ5) + cosh(2δp)] ,
Jφ = −M(a2c1c5cp − a1s1s5sp) ,
Jψ = −M(a1c1c5cp − a2s1s5sp) ,
Qi = Msici , i = 1, 5, p . (3.8)
Note that these quantities are invariant under interchange of the δi’s. This reflects the equivalence
of the several geometries obtained by U -dualities, that also interchange the several gauge charges.
Regarding the thermodynamical properties of these BH, it is convenient for future use to define
the left and right temperatures, TL and TR, through the relation β =
(βL + βR) (β = 1/T and
9We present this case as a main example, but include many others in the Appendix B.
βL,R = 1/TL,R). Then, using this relation together with (A.6) on (A.17) yields
2πM (c1c5cp − s1s5sp)
[M − (a2 − a1)2]1/2
, βR =
2πM (c1c5cp + s1s5sp)
[M − (a2 + a1)2]1/2
. (3.9)
The BH angular velocities Ωφ,ψ and electric potentials Φ(i) are computed in Appendix A. Here,
using (A.6), we rewrite them in terms of the parameters (M, δ1, δ5, δp, a1, a2)
Ωφ,ψ = −π
± a2 − a1
[M − (a2 − a1)2]1/2
a2 + a1
[M − (a2 + a1)2]1/2
, (3.10)
Φ(i) =
(tanh δi)c1c5cp − (coth δi)s1s5sp
[M − (a2 − a1)2]1/2
(tanh δi)c1c5cp + (coth δi)s1s5sp
[M − (a2 + a1)2]1/2
, (3.11)
while the expression for the entropy is
S = πM
c1c5cp + s1s5sp
[M − (a2 − a1)2]−1/2
c1c5cp − s1s5sp
[M − (a2 + a1)2]−1/2
. (3.12)
The BPS limit of the three charged BH is obtained by taking M → 0, δi → ∞, Jφ + Jψ → 0 while
keeping Qi fixed. In this supersymmetric regime, the charges satisfy the BPS constraints
Ebps = Q
p , J
= −Jbps
. (3.13)
As a first step to define the Euclidean T → 0 limit, we consider the near-BPS limit of this solution,
Jφ + Jψ → 0 ; M → 0 , δ1,5 → ∞ , Q1,5 fixed ; δp finite . (3.14)
That is, in the near-BPS limit we keep δp large but finite. This limit is also often called the dilute
gas regime since we are neglecting the interactions between left and right movers. Note that since
the three charges can be interchanged by U -dualities, it does not matter which one of the boosts
we keep finite. Given this equivalence we choose to keep δp finite, without any loss of generality.
Now, to take the T → 0 limit, we define the off-BPS parameter ε, that measures energy above
extremality, to be such that E ≡ Ebps + ε. In terms of the solution parameters it is given by
ε = Me−2δp/4. The details of the off-BPS expansion that we carry on in the sequel can be found
in Appendix A.2. Here we just quote the relevant results. We can expand the left and right
temperatures in terms of the off-BPS parameter ε yielding,
p − (Jbpsφ )2
, βR = π
. (3.15)
So the BPS limit corresponds to send the temperature T → 0 by sending βR → ∞ while keeping
βL finite (we are left with only left-movers). Hence, we find more appropriate to use βR as the
off-BPS parameter instead of ε . These two quantities are related by the second relation of (3.15).
10Expressions (3.9)-(3.11) agree with the ones first computed in [24] upon the notation identification a1 → −l2 and
a2 → −l1.
We can now expand all the thermodynamic quantities in terms of this off-BPS quantity β−1R .
For the angular velocities and electric potentials, the expansion yields
Ωφ,ψ = Ω
bps −
∓πJbps
p − (Jbpsφ )2
Φ(i) = Φ
p − (Jbpsφ )2
. (3.16)
where the BPS angular velocities and electric potentials are
bps = 0 ; Φ
bps = 1 . (3.17)
The expansion of the conserved charges yields
E = Ebps +O
, Jφ = J
, Jψ = −Jbpsφ +O
Q1 ≃ Qbps1 , Q5 ≃ Q
5 , Qp = Q
. (3.18)
Note that the BPS charges satisfy (3.13). They are written in terms of the parameters that describe
the system in (A.21). Finally, the expansion of the entropy yields
S = Sbps +O
, with Sbps = 2π
p − (J
. (3.19)
With the above off-BPS expansion, we are ready to define the BPS chemical potentials. Com-
paring (3.16) with (3.3) we obtain,
ωφ,ψ = ∓
p − (Jbpsφ )2
, φi =
p − (Jbpsφ )2
(3.20)
Notice that these chemical potentials only depend on the BPS conserved charges.
Now that all the BPS statistical mechanics conjugated pairs and entropy are defined, we are
ready to obtain the other thermodynamic functions. For example, consider the quantum statistical
relation,
I = βE − β
i=1,5,p
Φ(i)Qi − β
j=φ,ψ
ΩjJj − S . (3.21)
After the off-BPS expansion, i.e., using (3.18), (3.19) and (3.16) it yields
I = β
Ebps −
i=1,5,p
j=φ,ψ
i=1,5,p
j=φ,ψ
j − Sbps +O
(3.22)
The term in between brackets vanishes due to the BPS relations (3.13) and (3.17). Then, taking
β → ∞, we are left with the supersymmetric quantum statistical relation (SQSR) for the three-
charged BH,
Ibps = φ1Q
1 + φ5Q
5 + φpQ
p + 2ωφJ
− Sbps , (3.23)
where Ibps is the value of the Euclidean action in the supersymmetric limit of the D1-D5-P BH, and
we used J
= −Jbps
and ωψ = −ωφ. Notice that Ibps corresponds to the Legendre transformation
of the entropy with respect to all the BPS chemical potential and therefore should be interpreted
as the BH free energy.
The off-BPS expansion of the horizon angular velocities and electric potentials gives the su-
pergravity chemical potentials as the next to leading order term of the expansion around the BPS
solution. The motivation for this expansion analysis comes from the fact that BPS BHs can be
studied as dual configurations to supersymmetric ensembles at zero temperature but non-zero chem-
ical potentials in the dual CFT [12]. The supergravity conjugated potentials (3.20) are then the
strong coupling dual objects to the CFT chemical potentials. The SQSR relation (3.23) will be
connected to the well-known Sen’s entropy relation in the next section.
4 Euclidean zero-temperature and entropy functional formalisms
In previous sections we have described two apparently unrelated procedures to obtain the entropy of
supersymmetric BH that naturally contain the definitions of pairs of conjugated variables, related
to the BH charges. In this section we show that both procedures produce basically the same body
of final definitions, even though conceptually both approaches are rather different.
That both approaches produce the same final chemical potentials and definitions can be seen
in any of the examples at hand. As usual, the best way to illustrate our point is to pick a system
that captures the fundamental ingredients, while avoiding features that do not play a key role
and produce unnecessary distraction from the main point. In the present case, the appropriate
system is the non-rotating D1-D5-P BH (later, we will discuss the rotating case). Comparing the
thermodynamic relations (3.19), (3.20), and the Sen’s relations (2.13), (2.14), we can indeed confirm
that all the key quantities agree in the two formalisms. Explicitly we have that
φi = 2πei , Qi = qi , Ibps = 2πf . (4.1)
Nevertheless, that both frameworks are equivalent is a priori not at all obvious since they have
important differences. Sen’s approach relies completely on the structure of the near horizon geome-
try. In particular, the entropy is constructed analyzing Wald’s prescription and Einstein equations
in these spacetimes and all the analysis is carried on at the BPS bound i.e., when the solution is
extremal. In contrast, the zero temperature limit approach relies on the thermodynamical prop-
erties of BH and, in principle, uses the whole spacetime, not only the near horizon region. The
resulting thermodynamic definitions come as a limiting behavior of non-extremal BH and have a
nice straightforward interpretation in terms of the dual CFT thermodynamics.
4.1 Near-horizon and asymptotic contributions to the Euclidean action
To understand why the above close relations between the two formalisms hold, let us go back to
the calculation of the Euclidean action for general BH in the off-BPS regime. Inspired in ten
dimensional type II supergravity, we start with the general action11
(∂Ψ)2 − 1
eαΨF 2(n)
K , (4.2)
where Σ is the spacetime manifold, ∂Σ the boundary of that manifold and K is the extrinsic
curvature. In the BH case, once we have switched to Euclidean regime, it is necessary to com-
pactify the time direction to avoid a conical singularity. This compactification defines the Hawking
temperature as the inverse of the corresponding compactification radius.
To evaluate the Euclidean action on the BH solution, one of the methods to obtain a finite result,
i.e., to regularize and renormalize the action, consists of putting the BH in a box and subtract the
action of a background vacuum solution (g0,Ψ0, F 0). This procedures also defines the “zero” of all
the conserved charges. For asymptotic flat solutions we use Minkowski, while for asymptotic AdS
solutions we use AdS. Once in the box, the radial coordinate is restricted to the interval (r+, rb),
where r+ is the position of the horizon and rb corresponds to an arbitrary point which limits the
box and that at the end is sent to infinity. Another important ingredient is the boundary conditions
on the box. Basically, depending on which conditions we impose on the different fields, we will
have fixed charges or fixed potentials. If we do not add any boundary term to the above action,
we will be working with fixed potentials, i.e., we will work in the grand canonical ensemble [28].
The field equations are derived from a variational principle, where fields are kept constant at
the boundaries. In particular, the trace the of equation that comes from the variation of the metric
(for the D1-D5-P system, see equation (A.2)) implies that
(∂Ψ)2 = aeαΨF 2(n) , (4.3)
where a depends on the spacetime dimensions and n. Therefore, on-shell, the action reduces to12,
eαΨF 2(n) +
K −K0
, (4.4)
where b depends on the spacetime dimensions and n. The first term is a volume integral over Σ that
can easily be converted into a boundary integral over ∂Σ, once we recall that we are considering
electric fields only and hence F(n) = dC(n−1). Integrating by parts we get
eaΨF(n)C(n−1) +
K −K0
, (4.5)
where c depends on the spacetime dimensions and n. At this point, the on-shell Euclidean action
is completely recasted in two surface integrals terms, evaluated at r+ and rb. Consider first the
extrinsic curvature term. At rb, we get βEb, where Eb is the quasi-local energy. When rb is taken to
infinity, Eb reduces to the BH energy E and we recover usual term βE. At r+, only K contributes
and gives minus the area of the horizon divided by 4G, i.e., minus the Bekenstein-Hawking entropy
11For simplicity, the reasoning is done at the level of two derivative Lagrangian. Nevertheless, following Wald’s
approach for higher derivative actions, we notice that the BH action can always be recast as surface integrals.
Moreover, for definiteness, we anchor our discussion to type II action, but whenever needed we make comments to
extend our arguments to more general theories.
12Where we have used that the action of the background vacuum solution over Σ is zero.
S. Next consider the first term. Here the integral over time gives the factor β, while the integration
over the other directions (of the induced metric determinant at the boundary times eaΨF(n)) gives
the corresponding electric charge Q. Therefore, we get
eaΨF(n)Cn−1 = −βQ
C(n−1)(rb)− C(n−1)(r+)
. (4.6)
Then, we use the definition of the conjugated chemical potential φ as the difference of the gauge
potential at infinity and at the horizon,
Φ = C|∞ − C|r+ , (4.7)
and hence, when rb is sent to infinity, we recover the usual term −βQΦ. As a grand total we obtain
the QSR,
I = βE − βΦQ− S . (4.8)
Now, it is important to notice that the definition of Φ is gauge independent, and therefore we can
always choose a particular gauge that simplifies the picture depending on which physical concepts we
want to stress. Here, we choose the “natural gauge” adapted to the BPS limiting cases, C|∞ = Φbps,
where Φbps is usually 1 in natural units and for asymptotically flat BHs. Note that in this gauge one
has C|r+ = Φbps−Φ. This gauge choice is the one that makes direct contact between the Euclidean
zero temperature and entropy function formalisms for reasons that will become clear after (4.12).
At this point we are ready to rewrite the Euclidean action in two pieces, one evaluated in the
first boundary at r = r+, and the other in the second boundary at r = ∞,
eaΨF(n)C(n−1) +
eaΨF(n)C(n−1) +
K −K0
.(4.9)
Evaluating both terms as we did before but now in the adapted gauge we get,
I = β(Φbps − Φ)Q− S
︸ ︷︷ ︸
+ β(E − ΦbpsQ)
︸ ︷︷ ︸
. (4.10)
r = r+ r = ∞
Therefore we can always find a gauge in which the Euclidean action splits in two contributions,
one at the horizon and the other in the asymptotic region. It is perfectly adapted to understand
the near horizon regime. Equally interesting, this expression is also adapted to understand the
supersymmetric limit. In fact, from our discussion in section 3, it is easy to see that the first term
exactly reproduces the SQSR, i.e.,
BPS limit
β(Φbps − Φ)Q− S = φQbps − Sbps . (4.11)
On the other hand, the asymptotic term vanishes due to fact that Φbps = 1, and thus the lead-
ing term in the expansion is nothing else than the BPS relation Ebps = Qbps characteristic of
supersymmetric regimes, i.e.,13
BPS limit
β(E − ΦbpsQ) = lim
BPS limit
β(E −Q) = 0 . (4.12)
13This discussion is strictly valid for the asymptotically flat BHs where Φbps = 1. For asymptotically AdS BHs,
the normalization usually chosen in the literature yields in general Φbps 6= 1. However, in this case, the term inside
brackets in (4.12) still vanishes because it is exactly the BPS constraint on the charges. This follows by construction
and is explicitly confirmed for 5D gauged supergravity in [12, 15].
(Note that in the last equality we jump some steps that were already explained in detail after (3.6),
and that we do not repeat here. They guarantee that this term indeed vanishes and does not give an
indeterminacy of the type∞·0). We conclude that the Euclidean action of the BH at the BPS bound
is given exclusively from the near horizon part of the solution. This is another way to characterize
the attractor mechanism, since the physical properties of the solution are captured entirely by the
near horizon geometry. From the above result, it is easy to see why, for supersymmetric cases, I
is related to f . First, both are functionals of the near horizon geometry alone. Also, the time and
radial integrations are trivial and only integration on the other space directions actually contribute.
In fact, this is a way to understand why in the definition of f there is no integration in the AdS
part of the near horizon metric. Note also that in Sen’s approach the f function is defined as
the integral of the string frame Lagrangian evaluated at the near horizon geometry. Since in this
geometry the dilaton is a constant, the string frame and Einstein frame Lagrangians are related by
a trivial constant factor.
We now discuss the effects introduced by addition of rotation. Working in a coordinate system
in which the geometry is not rotating at infinity, the action can be splited as
I = β(Φbps − Φ)Q+ β(Ωbps − Ω)J − S
︸ ︷︷ ︸
+ β(E − ΦbpsQ−ΩbpsJ)
︸ ︷︷ ︸
. (4.13)
r = r+ r = ∞
By definition, the near horizon term contains all the information on the chemical potentials (once
the BPS limit is taken),
BPS limit
β(Φbps − Φ)Q+ β(Ωbps − Ω)J − S = φQbps + ωJbps − Sbps , (4.14)
while the asymptotic term is again the BPS constraint between the several charges and thus van-
ishes,
BPS limit
β(E − ΦbpsQ− ΩbpsJ) = 0 . (4.15)
For asymptotically flat BHs one always has Ωbps = 0 and (4.15) reduces to (4.12). The horizon of
flat BHs does not rotate (angular momentum comes from the Poynting vector of electromagnetic
fields) and this is one way to understand why the angular momenta does not appear in their BPS
constraint. On the other hand, the horizon velocity of asymptotically AdS BHs is, in general,
non-vanishing, and thus the angular momenta also contributes to the BPS constraint of these BHs.
4.2 Relation between chemical potentials in the two formalisms
At this point only reminds to understand the relation between the conjugated potentials in both
pictures. In Sen’s approach, the information about them is contained in the electric fields of
the near horizon geometry, while in the Euclidean zero temperature formalism this information is
encoded in the next to leading order term in an off-BPS expansion of the full geometry. Although
these definitions seem to be rather different at first sight, notice that in Sen’s approach the field
strength is just the radial derivative of the potential evaluated at the horizon. In the Euclidean zero
temperature case, the off-BPS expansion can be rewritten as an expansion in the radial position
of the horizon ρ+. Therefore, the next to leading order term in the off-BPS expansion of the
gauge potential at ρ+ is proportional to its derivative with respect to the radial position of the
horizon. Hence it is proportional to the field strength at the horizon. These words can be made
very precise by taking an example. Consider the D1-D5-P BH we have been working with (again
we do not include rotation in the analysis to avoid unnecessary non-insightful complications). In
the full geometry (2.5), where the zero temperature limit procedure is applied, we work with the t, r
coordinates. Sen’s approach uses instead the near-horizon fields (2.6) or (2.8) described in terms of
(τ, ρ) coordinates. The two set of coordinates are related by (2.7). Our purpose in the next lines is
to understand the first relation in (4.1). For definiteness we focus on the relation φ1 = 2πe1. From
(2.5), Cty = −Ms1c1/(ρ +Ms21), and one also has the relation between the gauge field written
in the two coordinate systems, Cty =
Cτy. In the near-horizon approach, the expression for e1
comes from the radial derivative of the potential evaluated at the BPS horizon (ρ
+ = 0):
∂ρCτy
. (4.16)
In the Euclidean zero temperature approach, the electric potential is obtained by contracting the
gauge field with the timelike Killing vector ξ = ∂t yielding: Φ
(1) = −Cty|ρ=ρ+ = −
Cτy|ρ=ρ+
(note that ρ+ = ρ
+ = 0 only in the BPS case). As is clear from (3.16), our conjugated potential
is defined as
φ1 = −
∂Φ(1)
∂β−1R
∂β−1R
∂Φ(1)
. (4.17)
Note the following key relations,14
∂Φ(1)
= −∂τ
Cτy|ρ=ρ+
∂ρCτy
. (4.18)
From (4.16)-(4.18), one finally has
∂β−1R
e1 = 2πe1 . (4.19)
The last equality follows from (2.7), and from ρ+ =M = 4π
R (see (3.15) and the
last statement of Appendix A.2). Physically we can understand it by noting that the near-horizon
coordinates are precisely the ones appropriate to find the value of the temperature, that avoids the
standard conical singularity in the Euclidean near-horizon geometry. An analysis along the lines
carried here for this specific case can be carried on for general cases and yield the relations (4.1)
between the conjugated potentials found using the two formalisms.
To summarize, we have seen that for supersymmetric BH, the Euclidean action and all the
chemical potentials are defined in the near horizon geometry. The asymptotic region would con-
tribute only in off-BPS cases. We have also shown why the chemical potentials are proportional
14The presence of the factor 1/2 in the last equality is due to a subtlety that occurs when we take ∂ρ+Φ (and thus
before sending ρ+ → ρbps+ ). In the large δ1 regime one has Ms21 ∼ Q
1 −M/2. Using this and ρ+ = M yields, in the
denominator of Φ, ρ+ +Ms
1 ∼ Qbps1 + ρ+/2. This is the 1/2 that appears when we further take the ρ+ derivative.
Note that this factor does not appear in the last derivative of (4.18), ∂ρCτy, because here we take the radial derivative
evaluated on the on-shell solution ρ = 0.
to the electric fields in the near horizon region, and ultimately, we have understood, from the BH
thermodynamics, the emergence of Sen’s entropy function as the extremal limit of the quantum
statistical relation or SQSR. As a bonus, we can now extend the statistical mechanics analysis like
the SQSR to BPS solutions with no off-BPS known extension, because we have learned how to
calculate the relevant chemical potentials directly in the BPS regime with no need of the limiting
procedure.
5 Extremal (non-BPS) black holes
So far we have seen that two completely different procedures, namely the Euclidean zero tempera-
ture formalism and Sen’s entropy formalism allow to compute the entropy and conjugated chemical
potentials of supersymmetric BHs. This is not an accident as proved in the previous section. Now,
as is well-known, Sen’s approach also allows to find the attractor values of non-BPS extreme BHs
[9, 26]. So a question that naturally raises is if whether or not the Euclidean zero temperature
approach is also able to deal successfully with these type of solutions. In this section we address
this issue.
It is straightforward to conclude that the Euclidean formalism indeed allows to find the chem-
ical potentials of non-BPS extreme configurations. This follows from an analysis similar to the
derivation presented in section 4, but this time slightly modified to account for the fact that the
extreme BH is not BPS. Choosing the gauge C|∞ = Φext (and thus C|r+ = Φext−Φ), the extreme
analogue of (4.14) is
I = β(Φext − Φ)Q+ β(Ωext − Ω)J − S
︸ ︷︷ ︸
+ β(E − ΦextQ− ΩextJ)
︸ ︷︷ ︸
. (5.1)
r = r+ r = ∞
where the first term boils down to the extreme counterpart of (4.14),
ext. limit
β(Φext − Φ)Q+ β(Ωext − Ω)J − S = φQext + ωJext − Sext , (5.2)
containing all the information on the chemical potentials.
On the other hand, for non-BPS extreme solutions, we find that the asymptotic term in (5.1),
ext. limit
β(E − ΦextQ− ΩextJ) , (5.3)
in general does not vanish, as oppose to its BPS cousin. However, we find the following important
feature, at least in the cases we studied: i) the cases where (5.3) does not vanish correspond
to extreme rotating solutions that have in common the presence of an ergoregion; (ii) rotating
extreme solutions without ergosphere and non-rotating extreme solutions have vanishing (5.3).
This occurs at least on the three-charged, four-charged and Kerr-Newman systems. In the cases
where it vanishes we again have that the Euclidean action of the BH at the extreme bound is given
exclusively from the near horizon part of the solution. The physical properties of the solution are
captured entirely by the near horizon geometry, which makes the attractor mechanism manifest 15.
15This discussion is at the level of two derivative Lagrangian. If corrections are added, we expect that the asymptotic
part vanishes producing a finite result, also for extreme BH with ergoregion.
In the above extremal non-BPS cases, we can explicitly verify that the two formalisms indeed
yield the same results. For this exercise and as an example, we will discuss below two extreme
three-charged BH (whose BPS cousin was studied in the previous sections). To emphasize that the
relation between the Euclidean and Sen’s formalism is universal and not restricted to the three-
charged system, in Appendices B.1 and B.2, we further extend the exercise to three other non-trivial
extreme solutions whose properties have been studied within Sen’s formalism.
5.1 Extreme three-charged black hole with ergoregion
In the D1-D5-P solution described by (A.4)-(A.8) we can take, instead of the BPS limit described
in section 3, a different limit that yields an extreme (but not BPS) BH with an ergoregion. This is
a case in which the system shows only partial attractor mechanism.
Concretely, we take the near-extreme limit
M → (a1 + a2)2 + ε , ε≪ 1 . (5.4)
When the off-extreme parameter ε vanishes, the temperature indeed vanishes since βR → ∞ in
(3.9). The off-extreme expansion of the conserved charges (3.8) around the corresponding extreme
values (obtained by replacing M by (a1 + a2)
2 in (3.8)) is straightforward, and the expansion of
the thermodynamic quantities (3.9)-(3.12) yields
βL = π (c1c5cp − s1s5sp)
(a1 + a2)
+O (ε) , βR = 2π(a1 + a2)2 (c1c5cp + s1s5sp)
S = Sext +O
, Ωφ,ψ = Ω
ext −
2ωφ,ψ
Φ(i) = Φ
ext −
, i = 1, 5, p , (5.5)
where the extreme values satisfy
Sext = 2π
a1a2(a1 + a2)
2 (c1c5cp + s1s5sp) ,
ext = Ω
ext = − [(a1 + a2)(c1c5cp + s1s5sp)]
ext =
(tanh δi)c1c5cp + (coth δi)s1s5sp
c1c5cp + s1s5sp
, i = 1, 5, p , (5.6)
and the conjugated potentials are
ωφ,ψ = −
a1 + a2√
c1c5cp − s1s5sp
c1c5cp + s1s5sp
± (a1 − a2)
, (5.7)
φi = −
[tanh δ1 tanh δ5 tanh δp − coth δ1 coth δ5 coth δp]−1 , i = 1, 5, p .
These expressions for the potentials could be rewritten only in terms of the conserved charges as
expected by the attractor mechanism. We avoid doing it because the expressions are long and
non-insightful.
The QSR for this system is
I = β
Eext −
i=1,5,p
i − Ω
ext − Ω
i=1,5,p
i + ωφJ
ext + ωψJ
ext − Sext +O
. (5.8)
In the supersymmetric system the analogue of the term in between brackets vanishes due to the
BPS constraint on the conserved charges. But, in general, for non-BPS extreme BHs it does not
vanish (see discussion associated with (5.3)). In the present case the factor in between brackets is
(a1+a2)
c1c5cp−s1s5sp
c1c5cp+s1s5sp
. Note that this quantity vanishes when rotation is absent (a1 = a2 = 0). When
it is present, the solution has an ergoregion and the non-vanishing contribution is associated with
its existence. Notice that in this case the Euclidean action is not well-defined but, nevertheless,
the chemical potentials (5.8) take finite values and are physically relevant.
5.2 Extreme three-charged black hole without ergoregion
The metric of the D1-D5-P system is also a solution of type I supergravity. A fundamental difference
between type IIB and type I theories is that the later theory has half of the supersymmetries of
type IIB. This feature implies that in type I, if we reverse the sign of the momentum in the
BPS D1-D5-P black hole, we get a distinct solution that is extreme but non-BPS. We study this
solution of type I in this subsection, as the main example of an extreme non-BPS solution without
ergoregion where attractor mechanism is fully manifest.
The near-extreme limit we now consider is similar to the near-BPS limit (3.14) in which we send
the boosts to infinity; the difference being that now we take one of the boosts to be negative (again,
by U -dualities it does not matter which one). The reason why these two limits are indeed different
and, in particular, why one of them yields a BPS BH and the other not is the following [25]. The
three-charged BH describes, in the supergravity approximation and after dualities, the F1-NS5-P
system that is a configuration of heterotic string theory compactified on T 4 × S1. We can describe
this system as an effective fundamental string with winding number n1n5 (where n1, n5 are the
numbers of F1 and NS5 constituents), and with momentum excitations traveling along it. Now,
heterotic string theory is chiral. Hence, the direction of the momentum along the fundamental string
sets if the solution is supersymmetric or not. In our conventions, the supersymmetric configuration
F1-NS5-P is the one with no right-movers. So, in the supergravity approximation, the BPS BH that
describes this system is obtained by taking δp → +∞. But we can also consider the heterotic string
configuration with only right-movers. Due to the chirality property, this F1-NS5-P̄ configuration
is then not supersymmetric. And the corresponding supergravity solution obtained by taking
δp → −∞ is not a BPS BH. Note that this solution is however extreme, i.e., it has zero temperature.
The reason being that there are no left-movers to collide with the right-movers and generate the
closed string emission that describes the Hawking radiation.
So we take the near-extreme limit (δ1,5 > 0; δp < 0, Qp < 0)
Jφ − Jψ → 0 ; M → 0 , δ1,5 → ∞ , Q1,5 fixed ; δp < 0 finite . (5.10)
The conserved charges of the non-extreme three-charged BH are listed in (3.8), and the temperature,
entropy, and angular velocities and potentials at the horizon are given in (3.9)-(3.11).
The charges in the extreme solution satisfy the constraint
Eext = Qext1 +Q
5 −Qextp , Jextψ = Jextφ , (5.11)
where we used Qextp = −Me−2δp/4.
The off-extreme parameter, ε = Me2δp/4, measures energy above extremality and is such that
E ≡ Ebps + ε. The expansion of the left and right temperatures in terms of the off-extreme
parameter ε yields,
βL = π
Qext1 Q
, βR = πQ
−Qext1 Qext5 Qextp − (Jextφ )2
]−1/2
. (5.12)
The extreme limit corresponds to send the temperature T → 0 by sending βL → ∞ while keeping
βR finite. In this limit there are no left-movers, only right-movers. The first relation in (5.12)
defines ε in terms of βL.
The expansion for the relevant thermodynamic quantities is
S = Sext +O
, Ωφ,ψ = Ω
ext −
2ωφ,ψ
Φ(i) = Φ
ext −
, i = 1, 5, p , (5.13)
where
Sext = 2π
−Qext1 Qext5 Qextp − (Jextφ )2
ext = 0 , Φ
(1,5)
ext = 1 , Φ
ext = −1 . (5.14)
The conjugated potentials are
ωφ,ψ = −
πJextφ
−Qext1 Qext5 Qextp − (Jextφ )2
φi = −
πQext1 Q
−Qext1 Qext5 Qextp − (Jextφ )2
, i = 1, 5, p . (5.15)
16 The rotation parameters in this limit go as
a1,2 = −
Jextφ
−Qext1 Qext5 Qextp
[1 +O (ε)] . (5.9)
For comparison, in the BPS limit a1,2 go instead as (A.24).
Although this is a non-BPS solution, it satisfies the extremal constraint (5.11) that is linear
in the charges. This, together with (5.14), has the consequence that (5.3) applied to this system
vanishes, and the QSR for this system simplifies to
Iext =
i=1,5,p
i + 2ωφJ
φ − Sext , (5.16)
where we used Jextψ = J
φ and ωψ = ωφ.
So, contrarily to the example of the previous subsection, where the system showed only partial
attractor mechanism due to the existence of an ergoregion, in the present system the ergoregion
is absent and the attractor mechanism is fully manifest, even though the extreme solution is not
6 Discussion
First of all, we would like to stress again the logic behind our approach: zero temperature limits to
reach extremal configurations are naturally defined in statistical analysis of quantum field theories.
The AdS/CFT correspondence then requires that there has to be a dual analysis for strings in
AdS. Supergravity is just the tree level part of the above theory, and thus we do not expect in
general a well defined zero temperature limit at this level. Here, by well defined we mean a limit
that generates a finite Euclidean action when T → 0. Nevertheless, we have found extremal BH
that seem to be protected, and therefore have a well defined zero temperature limit. In some of
these cases, the protection is based on supersymmetric arguments but, in other cases, we just have
extremal non-BPS BH where in fact it is not well understood why supergravity is giving the correct
answer.
In this article we have applied the Euclidean zero temperature formalism to supergravity so-
lutions where Sen’s formalism is well understood. In doing so, we have shown that this method
agrees with Sen’s entropy formalism, producing the same statistical mechanics functions like the
entropy and the chemical potentials. On the top of this, the Euclidean zero temperature formalism
has the key advantage of connecting the entropy functional with the statistical mechanics of the
dual CFT and with the more canonical BH thermodynamics.
More concretely, due to the attractor mechanism, we found that the Euclidean action is itself
given by the near horizon geometry alone, and therefore can be connected to Sen’s approach to
calculate the entropy. We showed how to relate all the different definitions in both approaches
and why they match. In particular, we are able to understand the CFT dual of Sen’s approach,
using the established map for the corresponding quantum statistical relation. For example, Sen’s
function f (evaluated on-shell) is nothing more than the BPS limit of the Euclidean action and
therefore is related to the dual CFT partition function. The above relation is relevant for the OSV
conjecture [3], since now Ibps or f naturally takes the place of free energy of the supersymmetric
We also worked out the extension to extremal but non-supersymmetric BH. Here, since we
are dealing with two derivative Lagrangians, we divide BH in two groups: those with ergoregion
and those without it. In all the cases with no ergoregion we have checked, the zero temperature
limit produces a well defined QSR at extremality, where all the chemical potentials, entropy and
the Euclidean action are related to Sen’s approach. This is not a triviality, since here there is
no supersymmetry to protect these tree-level results. This resuly seems to imply a protection
mechanism in the extremal case, as suggested in [25].
In the other case of extremal BH with ergoregions, we found an ill-defined limit, where the
asymptotic contribution to the Euclidean action diverges. Nevertheless, the near horizon contri-
bution is well behaved and produces the correct entropy and chemical potentials. These results
are in agrement with Sen’s approach since these geometries are not fully attracted. Therefore they
depend also on asymptotic values of the moduli. We interpret this result as a confirmation that
these geometries do receive corrections from string theory that in turn will modify the asymptotic
region, and thus asymptotic charges like the energy. In fact, in [29] rotating BH of this sort were
studied finding that for the ergoregion branch, the entropy, but not the energy, could be matched
with the microscopic CFT.
We would like to point out that although we worked with standard low-energy supergravity,
the inclusion of higher derivative terms should not spoil the results. In the Euclidean approach,
one now has to compute the Euclidean action with the modified Lagrangian and define the entropy
as its Legendre transform with respect to the BPS chemical potentials. This should give the same
entropy as defined by Wald (see [30]).
The zero temperature limit analysis of supersymmetric CFT ensembles motivated the corre-
sponding analysis in the dual supergravity system. In this paper our main goal was to make direct
contact between this formalism and Sen’s entropy function approach. The Euclidean zero tem-
perature formalism further allows to scan the phase structure of BH. A paradigm on the useful
information that this formalism allows to find about the CFT living on the boundary of a BH
geometry can be found in [12, 13]. It would be interesting to make a similar application, this time
to study the CFT of the BH systems discussed in this paper.
Acknowledgments
We acknowledge Roberto Emparan and Pau Figueras for a critical reading of the manuscript.
This work was partially funded by the Ministerio de Educacion y Ciencia under grant FPA2005-
02211 and by Fundacão para a Ciência e Tecnologia through project PTDC/FIS/64175/2006. OD
acknowledges financial support provided by the European Community through the Intra-European
Marie Curie contract MEIF-CT-2006-038924, and CENTRA for hospitality while part of this work
was done.
Appendices
A Three-charged black hole: solution and thermodynamics
A.1 The D1-D5-P black hole
In this section we describe the D1-D5-P BH and its thermodynamic properties that are used in
sections 3-5. The most general solution with arbitrary charges was originally constructed in [22]
(see also [23]). This solution generalizes the case with equal D1 and D5 charges found previously
in [32] and whose BPS limit yields the BMPV BH [33]. Here we follow the notation of [23, 31].
This three-charged BH is a solution of type IIB supergravity. The only IIB fields that are
turned on are the graviton gµν , the dilaton Ψ, and the RR 2-form C ≡ C(2). For the field strength
one has simply F(3) = dC(2) since the RR field C(0) and the NSNS field H(3) are absent. The type
IIB action, in the Einstein frame, reduces in these conditions to
16πG10
µΨ− 1
eΨF 2(3)
, (A.1)
where g is the determinant of the Einstein metric. The field equations that follow from variation
of action (A.1) are
Rµν −
gµνR−
∂µΨ∂νΨ−
gµν∂σΨ∂
3FµαβF
= 0 ,
−g gµν∂νΨ
eΨF 23 = 0 ,
−g eΨFµαβ
= 0 . (A.2)
Contraction of the graviton field equation yields for the Ricci scalar,
eΨF 23 . (A.3)
The graviton in the Einstein frame is (the relation between the parameters describing the solution
and the conserved charges is displayed in (3.8))
ds2 = −
(dt2 − dy2) +
(spdy − cpdt)2 (A.4)
r2dr2
(r2 + a21)(r
2 + a22)−Mr2
+ dθ2
H̃1H̃5 − (a22 − a21)(H̃1 + H̃5 − f) cos2 θ
cos2 θ dψ2
H̃1H̃5 + (a
2 − a21)(H̃1 + H̃5 − f) sin2 θ
sin2 θ dφ2 +
a1 cos
2 θdψ + a2 sin
2 θdφ
2M cos2 θ
(a1c1c5cp − a2s1s5sp)dt+ (a2s1s5cp − a1c1c5sp)dy
2M sin2 θ
(a2c1c5cp − a1s1s5sp)dt+ (a1s1s5cp − a2c1c5sp)dy
dφ+ H̃
dz2j ,
where y is the coordinate on S1, and zj ’s (j = 1, · · · , 4) are the coordinates on the torus T 4. We
use the notation ci ≡ cosh δi, si ≡ sinh δi, and
f(r) = r2 + a21 sin
2 θ + a22 cos
2 θ , H̃i(r) = f(r) +Ms
i , with i = 1, 5 ,
g(r) = (r2 + a21)(r
2 + a22)−Mr2 . (A.5)
The roots of g(r), r+ and r−, are given by
r2± =
(M − a21 − a22)±
(M − a21 − a22)2 − 4a21a22 , (A.6)
The system describes a regular BH17 when r2+ > 0, i.e., for M ≥ (a1 + a2)2. The ten-dimensional
determinant in the Einstein frame is
−g = r sin θ cos θH̃1/41 H̃
5 . The dilaton Ψ and 2-form RR
gauge potential C which support the D1-D5-P configuration are
e2Ψ =
, (A.7)
C(2) =
cos2 θ (atψdt+ ayψdy) ∧ dψ + sin2 θ (atφdt+ ayφdy) ∧ dφ
−s1c1dt ∧ dy − s5c5(r2 + a22 +Ms21) cos2 θ dψ ∧ dφ
, (A.8)
where we defined
atφ = a1c1s5cp − a2s1c5sp , atψ = a2c1s5cp − a1s1c5sp ,
ayφ = a2s1c5cp − a1c1s5sp , ayψ = a1s1c5cp − a2c1s5sp . (A.9)
By electric-magnetic duality18,
−gFµ1µ2µ3
ǫµ1µ2µ3ν1···ν7F
ν1···ν7 , (A.10)
our configuration can be equivalently described either by the 2-form C(2) in (A.8) or by the 6-form
C(6) that follows from (A.10). Using this equivalence, we rewrite (A.8) as
C(2) = −
s1c1dt+ ayφ sin
2 θdφ+ ayψ cos
2 θdψ
∧ dy ,
C(6) = −
s5c5dt+ atψ sin
2 θdφ+ atφ cos
2 θdψ
∧ dy ∧ dz1 ∧ dz2 ∧ dz3 ∧ dz4 .
(A.11)
The advantage of (A.11) is that we clearly identify the C(2) gauge potential sourced by the D1-
brane charges and the C(6) field sourced by the D5-brane charges. Thus, this expression will be
appropriate to find the electric potentials associated with the two type of D-branes. Note that
all the C
µν components contain the y-coordinate that parametrizes the S
1, while all the C
µναβγσ
components contain the y-coordinate and the zj ’s coordinates that parametrize the torus T
4. This
reflects the fact that D1-branes wrap S1 and the D5-branes wrap the full internal space T 4 × S1.
17For r2+ < 0, i.e., M ≤ (a1 − a2)2 the system can describe a smooth soliton without horizon [31, 34]. We will not
discuss this solution.
18We use the convention ǫtrθφψyz1z2z3z4 = 1, and the relation (A.10) is valid in the Einstein frame.
The general procedure to compute angular velocities when the geometry has several momenta
can be found in [35]. Applied to our case, the angular velocities at the horizon along the φ-plane,
Ωφ, the ψ-plane, Ωψ, and the velocity along y, Φ(p) are19
gty (gyφgψψ − gyψgφψ) + gtφ
g2yψ − gyygψψ
+ gtψ (gyygφψ − gyφgyψ)
gyygφφgψψ + 2gyφgyψgφψ − g2yψgφφ − g2yφgψψ − g2φψgyy
gty (gyψgφφ − gyφgφψ) + gtφ (gyygφψ − gyφgyψ) + gtψ
g2yφ − gyygφφ
gyygφφgψψ + 2gyφgyψgφψ − g2yψgφφ − g2yφgψψ − g2φψgyy
Φ(p) =
g2φψ − gφφgψψ
+ gtφ (gyφgψψ − gyψgφψ) + gtψ (gyψgφφ − gyφgφψ)
gyygφφgψψ + 2gyφgyψgφψ − g2yψgφφ − g2yφgψψ − g2φψgyy
(A.12)
which yields
Ωφ = −
r2+ + a
r2+c1c5cp + a1a2s1s5sp
Ωψ = −
r2+ + a
r2+c1c5cp + a1a2s1s5sp
Φ(p) =
r2+c1c5sp + a1a2s1s5cp
r2+c1c5cp + a1a2s1s5sp
. (A.13)
The horizon angular velocities are constant and, in particular have no angular dependence, as
required by Carter’s rigidity property of Killing horizons20. The electric potentials at the horizon
associated with the Q1 and Q5 gauge charges are computed using
Φ(i) = −Cµ{x(i)}ξ
Ct{x(i)} + Cφ{x(i)}Ω
φ +Cψ{x(i)}Ω
, i = 1, 5 , (A.14)
where,
ξ = ∂t +Ω
φ∂φ +Ω
ψ∂ψ , (A.15)
is the null Killing vector generator of the horizon (Ωφ,ψ are the horizon angular velocities). We
use the notation {x(1)} ≡ y, the coordinate of S1 wrapped by D1-branes, and {x(5)} ≡ yz1z2z3z4,
19We identify Ωy ≡ Φ(p) because the KK momentum plays effectively the role of a gauge charge with associated
electric potential.
20Note that the angular velocities can be more easily computed using the standard formulas valid for solutions
rotating along a single axis, as long as we evaluate them at a specific θ coordinate. More concretely, an inspection
of (A.12) concludes that the following relations are valid and provide the quickest computation of the corresponding
quantities:
φ ≡ gtφ
r=r+, θ=0
ψ ≡ gtψ
r=r+, θ=π/2
(p) ≡ gty
r=r+, θ=0
the coordinates of S1 × T 4 wrapped by D5-branes. Using the C gauge potential written in (A.11)
yields
Φ(i) =
r2+(tanh δi)c1c5cp + a1a2(coth δi)s1s5sp
r2+c1c5cp + a1a2s1s5sp
, i = 1, 5 . (A.16)
The temperature of the BH is T = κh/(2π) where the surface gravity of the horizon is κ
(∇µξν)(∇µξν)|r=r+ , and ξ
µ is the Killing vector horizon generator defined in (A.15). The inverse
temperature β = 1/T is then
r2+ + a
r2+ + a
r4+ − a21a22
r2+c1c5cp + a1a2s1s5sp
. (A.17)
The entropy S is just horizon area (in the Einstein frame) divided by 4G10,
r2+ + a
r2+ + a
r2+c1c5cp + a1a2s1s5sp
. (A.18)
To conclude this section, note that action (A.1) can be written in the string frame through the
Weyl rescaling of the metric, g̃AB = e
Ψ/2gAB , yielding
16πG10
R̃− 4∂µΨ∂µΨ
2 · 3!
F 2(3)
. (A.19)
A.2 The near-BPS limit of the D1-D5-P black hole
In this appendix we present the detailed computation of the near-BPS limit of the D1-D5-P BH,
and of the off-BPS construction that takes (3.9)-(3.11) into (3.15)-(3.23).
Using the trignometric properties
eδi + e−δi
, si =
eδi − e−δi
, (A.20)
the gauge charges and ADM mass (3.8) are, in the near-BPS regime (3.14),
Me2δp
Me−2δp
≡ Qbpsp − ε
Me2δ1
≡ Qbps1 , Q5 ≃
Me2δ5
≡ Qbps5 ,
+ ε = Ebps + ε , (A.21)
where the BPS constraint (3.13) was used. We can interpret the quantity Me
as the number of
left-movers, and ε = Me
as the number of right-movers (in the KK momentum sector). The
BPS configuration, ε = 0, is the one with no right-movers. In the D1 and D5 sectors there are
only left-movers since δ1,5 → ∞. From the last relation in (A.21), we conclude that ε is also an
off-BPS parameter that measures energy above extremality. We can also rewrite ε = Me
M = 4
ǫ, an expression that will be useful below.
The near-BPS limit (3.14) is completed with the angular momenta condition. It can be under-
stood as follows. Inversion of (3.8) yields
a1 = −
Jψc1c5cp + Jφs1s5sp
p − s21s25s2p
, a2 = −
Jφc1c5cp + Jψs1s5sp
p − s21s25s2p
. (A.22)
In the near-BPS limit (3.14) one has c21,5 ≃
, s21,5 ≃
, c2p =
, and
s2p =
. The expansion of a1,2 in the small M regime then gives
a1,2 = − (Jφ + Jψ)
(Jφ − Jψ)
, (A.23)
where we have defined η ≡ Qbps1 Q
ε, and γ ≡ Qbps1 Q
5 ε. Now, one must take appropriate limits of a1 and a2 such that they keep finite and the
angular momenta is kept fixed. But in (A.23) one sees that, for non-vanishing charges Q
i 6= 0
(i = 1, 5, p), a1,2 diverge as 1/
M when we takeM → 0. We can avoid this divergence by imposing
that Jφ + Jψ → 0 in the near-BPS limit. Note that as a consequence, in the limit ε→ 0, the BPS
solution must have angular momenta satisfying the relation (3.13)21. Under this condition, we can
now take a small ǫ expansion in (A.23) and get
a1,2 = ±
[1 +O (ε)] . (A.24)
Use of (A.20) and (A.24) in (3.9), (3.12) and (3.11) yields straightforwardly the near-BPS
expansions for the temperature, (3.15), for the entropy, (3.19), and for the angular velocities,
(3.16), respectively.
The off-BPS expansion of the electric potential Φ(p) leading to (3.16) is straightforward. How-
ever, the expansion of the D1 and D5 electric potentials is more subtle. Indeed, if in (3.11) we do
the most natural step, (tanh δ1)c1c5cp − (coth δ1)s1s5sp = s1c5cp − c1s5sp we just catch the BPS
value but not the next order term of the expansion. To capture the next order off-BPS contribu-
tion one has to introduce the parameter M that measures the energy above extremality. This is
consistently done with the following step: (tanh δ1)c1c5cp = c1c5cp
Ms1c1
M(1+s21)
(and similarly for the
term proportional to coth δ1). Then, use of Ms
1 ≃ Q
1 −M/2 and M = 4
ǫ allows to
finally write (tanh δ1)c1c5cp ≃ c1c5cp(1 − q
ε), where q is a ratio of BPS charges. The expansion
(3.16) for Φ(1), Φ(5) now follows naturally.
B Explicit agreement for other black hole systems
In this Appendix we will perform the Euclidean zero temperature limit and study the statistical
mechanics of some BHs that have not been considered in the main body of the text. The main
21Alternatively, note that we could relax this condition in the off-BPS regime. That is we could instead fix Jφ and
let Jψ arbitrary “during” the near-BPS approach, as long as in the BPS limit one ended with Jφ + Jψ = 0. Our final
result is independent of the particular off-BPS path choosen.
motivation to do this is two-folded. First, we explicitly verify that the relation between the Eu-
clidean zero temperature and Sen’s entropy formalisms is indeed general and not restricted to the
three-charged BH studied in the main body of the text. Second, we get a list of conjugated chemical
potentials for several BH systems. With these at hand we can also study the thermodynamics of
the dual CFT. We consider some relevant asymptotically flat systems that have been discussed
within Sen’s formalism context in [9], namely: the four-charged BH (subsection B.1), and the Kerr-
Newman BH (subsection B.2). The agreement between the two formalisms is also confirmed for
black holes of gauged supergravity elsewhere [12, 15].
B.1 Four-charged black holes
We study the statistical properties at zero temperature of the asymptotically flat four-charged BH
in four dimensions (4D). This system has three distinct extreme cases: the BPS BH (studied
in subsection B.1.1), the ergo-free branch family of BHs (subsection B.1.2), and the ergo-branch
family (subsection B.1.3). These last two are extreme but not BPS BHs and we are following the
nomenclature of [9].
The most general non-extremal rotating four-charged BH was first found in [37] as a solution
of heterotic string theory compactified on a six-torus. The four gauge fields of the solution were
however not explicitly given. This BH is also a solution of N = 2 supergravity coupled to three
vector multiplets, which in turn can be consistently embedded in N = 8 maximal supergravity
[37, 36, 38]. As first observed for the static non-extreme case [36], these theories can also be
obtained from compactification of type II supergravity on T 4 × S1 × S̃1. Therefore, from the 10D
viewpoint these BHs have a D-brane interpretation, e.g., they describe the D2-D6-NS5-P solution
of type IIA supergravity or the D1-D5-KK-P solution of type IIB supergravity (or any dual system
to these obtained by U -dualities).
Take N = 2 supergravity coupled to three vector multiplets. The field content of the theory
is: the graviton gµν , four gauge fields A(1)1,2 , Â1,2(1) , three dilatons ϕi and three axions χi (with
1 ≤ i ≤ 3). The full solution can be explicitly found in [38]. Compared with [38], we use the
parameters µ ≡ 4m and l ≡ 4a that avoid nasty factors of 4 in the thermodynamic quantities. The
horizons of the solution are at
µ2 − l2
, (B.1)
and thus the system has regular horizons when µ ≥ |l|. When l = 0 we recover the static solutions
found in [36].
The conserved mass E, angular momentum J , and gauge charges Qi’s of the BH are (we use
G4 ≡ 1/8 for this system)
cosh(2δi) , Jφ =
µl (c1c2c3c4 − s1s2s3s4) ,
Qi = µsici , i = 1, 2, 3, 4 , (B.2)
which are invariant under interchange of the δi’s, as expected from the U -duality relations.
The left and right movers inverse temperatures, the entropy, electric potentials and angular
velocity are [39],
βL = 2πµ (c1c2c3c4 − s1s2s3s4) , βR =
µ2 − l2
(c1c2c3c4 + s1s2s3s4) ,
S = πµ2 (c1c2c3c4 + s1s2s3s4) + πµ
µ2 − l2 (c1c2c3c4 − s1s2s3s4) ,
Φ(i) =
[(tanh δi)c1c2c3c4 − (coth δi)s1s2s3s4]
µ2 − l2
[(tanh δi)c1c2c3c4 + (coth δi)s1s2s3s4] , i = 1, 2, 3, 4 ,
µ2 − l2
. (B.3)
B.1.1 BPS black hole
The BPS limit of the four charged BH is obtained by taking µ→ 0, δi → ∞, while keeping Qi fixed
(i = 1, 2, 3, 4), and l → 0 at the same rate as µ, i.e., l/µ → 1. As a consequence J → 0 and the
BPS four-charged BH is non-rotating22. Therefore, the BPS charges satisfy the BPS constraints,
Ebps = Q
4 , J
bps = 0 , (B.4)
where Q
i = µe
2δi/4. To study the thermodynamics near the T = 0 BPS solution we work in the
near-BPS limit. We take
µ→ 0 , δ1,2,3 → ∞ , Q1,2,3 fixed ; δ4 finite; l → 0 (l/µ → 1) . (B.5)
Note that we take the four boosts to be positive and we choose to keep δ4 finite, without any loss
of generality (due to U -dualities).
Define the off-BPS parameter above extremality ε, to be ε = µe−2δ4/4 so that E ≡ Ebps + ε.
The procedure yielding the off-BPS expansion of the several thermodynamic quantities is quite
similar to the one done in the three-charged BH (see Appendix A.2). So we just quote the relevant
results.
Expanding the left and right temperatures in terms ε yields,
βL = π
, βR = π
. (B.6)
The BPS limit corresponds to send βR → ∞, and we now can use βR as the off-BPS parameter,
instead of ε.
The expansion in βR of the conserved charges is
E = Ebps +O
, J =
Q1,2,3 ≃ Qbps1,2,3 , Q4 = Q
. (B.7)
22The reason being that the roots that define the horizon are (B.1), and thus µ ≥ |l| must hold to have a regular
solution.
The remaining thermodynamic quantities have the expansion,
S = Sbps +O
, Ω =
Φ(i) = Φ
bps −
, i = 1, 2, 3, 4 , (B.8)
where
Sbps = 2π
= 1 , φi =
, i = 1, 2, 3, 4 . (B.9)
The last relation gives the key quantities, namely the conjugated potentials φi’s of the solution that
have an important role in the dual CFT. The expressions of the BPS entropy Sbps, and conjugated
potentials φi’s agree with the corresponding quantities computed in [9] using Sen’s entropy function
formalism23.
The SQSR for the four-charge BH is then
Ibps = φ1Q
1 + φ2Q
2 + φ3Q
3 + φ4Q
4 − Sbps . (B.10)
B.1.2 Extreme (non-BPS) black hole: ergo-free solution
In the four-charged system we can take an extremal limit that yields a rotating BH without ergo-
sphere. For this reason, this BH was dubbed ergo-free solution in [9].
This limit is similar to the BPS regime token in the previous Appendix B.1.1 in which we send
the boosts to infinity; the difference being that we take an odd number (one, for definiteness, but
it could as well be three) of boosts to be negative. As explained in a similar context in section 5,
this limit yields an extreme, but not BPS, BH.
Concretely, take the near-extremal limit (δ1,2,3 > 0; δ4 < 0, Q4 < 0):
µ→ 0 , δ1,2,3 → ∞ , Q1,2,3 fixed ; δ4 < 0 finite ;
−Q1Q2Q3Q4
. (B.11)
The charges in the extreme solution satisfy the constraint
Eext = Qext1 +Q
3 −Qext4 , (B.12)
where Q
1,2,3 = µe
2δ1,2,3/4, Qext4 = −µe−2δ4/4, and Jext is arbitrary. Using the off-extremality
parameter, ε = µe2δ4/4 = π2Qext1 Q
L (so the extremal limit is obtained by sending βL →
∞), we get the following expansion for the relevant thermodynamic quantities:
S = Sext +O
, Ω = Ωext −
Φ(i) = Φ
ext −
, i = 1, 2, 3, 4 , (B.13)
23Once we match the notation ω ≡ 2πα and φi ≡ 2πei and we take into consideration that we use G4 ≡ 1/8, while
[9] uses G4 ≡ 1/(16π)).
where
Sext = 2π
−Qext1 Qext2 Qext3 Qext4 − (Jext)2
Ωext = 0 , Φ
(1,2,3)
ext = 1 , Φ
ext = −1 . (B.14)
The conjugated potentials are
ω = −
2πJext
[−Qext1 Qext2 Qext3 Qext4 − (Jext)2]
φi = −
πQext1 Q
Qexti [−Qext1 Qext2 Qext3 Qext4 − (Jext)2]
, i = 1, 2, 3, 4 . (B.15)
Again, these expressions for Sext, ω and φi’s match the ones found in [9] using Sen’s entropy function
formalism (see footnote 23 for normalization conventions).
Although this is a non-BPS solution, it satisfies the extremal constraint (B.12) that is linear in
the charges. Using in addition (B.14), we find that (5.3), applied to this system, vanishes and the
QSR for this system simplifies to
Iext =
i + ωJ
ext − Sext . (B.16)
This is an example of a rotating extreme solution without ergosphere. It has a finite on-shell action.
B.1.3 Extreme (non-BPS) black hole: ergo-branch solution
This time we take the limit µ→ l. This yields an extreme BH with an ergosphere that was coined as
ergo-branch solution in [9] (This is the four-charged counterpart of the solution studied in Section
5.1).
We take the near-extreme limit
µ→ l + ε , ε≪ 1 . (B.17)
When the off-extreme parameter ε vanishes, the temperature indeed vanishes since βR → ∞ in
(B.3). The off-extreme expansion of the conserved charges (B.2) around the corresponding ex-
treme values (obtained by replacing µ by l in (B.2)) is straightforward, and the expansion of the
thermodynamic quantities (B.3) yields24
βL = 2πl (c1c2c3c4 − s1s2s3s4) +O (ε) , βR =
2πl3/2 (c1c2c3c4 + s1s2s3s4)
S = Sext +O
, Ω = Ωext −
Φ(i) = Φ
ext −
, i = 1, 2, 3, 4 , (B.18)
24We use the relation
µ2 − l2 ≃ 2πµ2(c1c2c3c4 + s1s2s3s4)/βR
where the extreme values satisfy
Sext = 2π
Qext1 Q
4 + (J
ext)2
, Ωext = 2l
−1 (c1c2c3c4 + s1s2s3s4)
ext =
(tanh δi)c1c2c3c4 + (coth δi)s1s2s3s4
c1c2c3c4 + s1s2s3s4
, i = 1, 2, 3, 4 , (B.19)
and the conjugated potentials are
2πJext
[Qext1 Q
4 + (J
ext)2]
Qexti
s1c1s2c2s3c3s4c4
c1c2c3c4 + s1s2s3s4
, i = 1, 2, 3, 4 . (B.20)
Note that in the last expression could be rewritten only in terms of the conserved charges as
expected by the attractor mechanism. We do not do it here because the expression is too long.
The expressions of the extremal entropy Sext, and conjugated potentials ω and φi’s agree with the
corresponding quantities computed in [9] using Sen’s entropy function formalism (see footnote 23
for normalization conventions).
The QSR for this system is
I = β
Eext −
ΦextQ
i − ΩextJext
i + ω J
ext − Sext +O
(B.21)
In the supersymmetric system the analogue of the first term vanishes due to the BPS constraint on
the conserved charges. But, in general, for non-BPS extreme BHs it does not vanish (see also discus-
sion associated with (5.3)). In the present case the factor in between brackets is − l
c1c2c3c4−s1s2s3s4
c1c2c3c4+s1s2s3s4
Note that this quantity vanishes when rotation is absent. When it is present, the solution has an er-
gosphere and the non-vanishing contribution seems to be associated with its existence, as discussed
in section 5.
B.2 Extreme Kerr-Newman black hole
In this section we take the near-extreme limit of the Kerr-Newman BH with ADM mass M , ADM
charge Q and ADM angular momentum J = aM that is a solution of the Einstein-Maxwell action
I = 1
R− F 2
(so, we set G4 ≡ 1). In the extreme state the charges satisfy the
constraint M2 = a2 + Q2, the horizons coincide, r± = M , and one also has the useful relation
M2 + a2 = 2
J2 +Q4/4. Define the off-extremality parameter ε such that M = Me + ε which
implies that r+ ∼Me+
ε (the subscript e stands for the on-shell extreme solution). In terms
of the inverse temperature β =
2π(r2++a
r+−M it is given by
2π(M2e+a
2Me β
. Using the expressions
S = π(r2+ + a
2), Ω = a/(r2+ + a
2) and Φ = Qr+/(r
+ + a
2) one gets the expansion:
S = Se +O
, Se = 2π
J2e +Q
e/4 ;
Ω = Ωe −
, Ωe =
J2e +Q
, ω =
J2e +Q
Φ = Φe −
, Φe =
J2e +Q
, φ =
J2e +Q
The extremal entropy Se, and conjugated potentials ω and φ agree with the corresponding quantities
computed in [9] using Sen’s entropy function formalism25.
The QSR for this system is
I = β (Me − ΦeQe − ΩeJe) + φQe + ω Je − Se +O
(B.22)
The first term does not vanish, a feature that seems to be common to non-BPS extreme black
holes with ergosphere. The factor in between brackets is Me(M
e −Q2e)/(M2e + a2e). If rotation is
absent, a = 0, one has Me = Qe and the above term vanishes. When it is present, the solution
has an ergosphere and the non-vanishing contribution seems to be associated with its existence, as
discussed in section 5.
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	Introduction
	Attractor mechanism and entropy functional formalism
	Zero temperature limit and chemical potentials
	Entropy functional formalism from an Euclidean perspective
	Main results and structure of the paper
	Entropy functional formalism revisited
	Euclidean zero-temperature formalism: BPS black holes
	Euclidean zero-temperature and entropy functional formalisms
	Near-horizon and asymptotic contributions to the Euclidean action
	Relation between chemical potentials in the two formalisms
	Extremal (non-BPS) black holes
	Extreme three-charged black hole with ergoregion
	Extreme three-charged black hole without ergoregion 
	Discussion
	 Three-charged black hole: solution and thermodynamics
	 The D1-D5-P black hole
	 The near-BPS limit of the D1-D5-P black hole
	Explicit agreement for other black hole systems
	Four-charged black holes
	BPS black hole
	Extreme (non-BPS) black hole: ergo-free solution
	Extreme (non-BPS) black hole: ergo-branch solution
	Extreme Kerr-Newman black hole
