7 BPS Black Holes
Bernard de Wita
aInstitute for Theoretical Physics & Spinoza Institute,
Utrecht University, Utrecht, The Netherlands
The entropy of BPS black holes in four space-time dimensions is discussed from both macroscopic and micro-
scopic points of view.
1. INTRODUCTION
Classical black holes are solutions of Einstein’s
equations of general relativity that exhibit an
event horizon. From inside this horizon, noth-
ing (and in particular, no light) can escape. The
region inside the horizon is therefore not in the
backward lightcone of future timelike infinity.
However, since the discovery of Hawking radia-
tion [1], it has become clear that many of the
classical features of black holes will be subject to
change.
In these lecture we consider static, spherically
symmetric black holes in four space-time dimen-
sions that carry electric and/or magnetic charges
with a flat space-time geometry at spatial in-
finity. Such solutions exist in Einstein-Maxwell
theory, the classical field theory of gravity and
electromagnetism. The most general static black
holes of this type correspond to the Reissner-
Nordstrom solutions and are characterized by a
charge Q and a mass M . In the presence of mag-
netic charges, Q is replaced by
q2 + p2 in most
formulae, where q and p denote the electric and
the magnetic charge, respectively. Hence there
is no need to distinguish between the two types
of charges. For zero charges one is dealing with
Schwarzschild black holes.
Two quantities associated with the black hole
horizon are its area A and the surface gravity κs.
The area is simply the area of the two-sphere de-
fined by the horizon. The surface gravity, which
is constant on the horizon, is related to the force
(measured at spatial infinity) that holds a unit
test mass in place. On the other hand, the mass
M and charge Q of the black hole are not primar-
ily associated with the horizon and are defined in
terms of appropriate surface integrals at spatial
infinity.
As is well known, there exists a striking cor-
respondence between the laws of thermodynam-
ics and the laws of black hole mechanics [2]. Of
particular importance is the first law, which, for
thermodynamics, states that the variation of the
total energy is equal to the temperature times
the variation of the entropy, modulo work terms,
for instance proportional to a change of the vol-
ume. The corresponding formula for black holes
expresses how the variation of the black hole mass
is related to the variation of the horizon area, up
to work terms proportional to the variation of the
angular momentum. In addition there can also be
a term proportional to a variation of the charge,
multiplied by the electric/magnetic potential µ at
the horizon. Specifically, the first law of thermo-
dynamics, δE = T δS − p δV , translates into
+ µ δQ+Ω δJ . (1)
The reason for factorizing the first term on the
right-hand side in this particular form, is that
κs/2π equals the Hawking temperature [1]. This
then leads to the identification of the black hole
entropy in terms of the horizon area,
Smacro = 14A , (2)
a result that is known as the area law [3]. In
these equations the various quantities have been
defined in Planck units, meaning that they have
http://arxiv.org/abs/0704.1452v1
2 B. de Wit
been made dimensionless by multiplication with
an appropriate power of Newton’s constant (we
will set ~ = c = 1). This constant appears
in the Einstein-Hilbert Lagrangian according to
LEH = −(16πGN)−1
|g|R. With this normal-
ization the quantities appearing in the first law
are independent of the scale of the metric.
Einstein-Maxwell theory can be naturally em-
bedded into N = 2 supergravity. This super-
gravity theory has possible extensions with sev-
eral abelian gauge fields and a related number of
massless scalar fields (often called ‘moduli’ fields,
for reasons that will become clear later on). At
spatial infinity these moduli fields will tend to a
constant, and the black hole mass will depend
on these constants, thus introducing additional
terms on the right-hand side of (1).
For Schwarzschild black holes the only relevant
parameter is the mass M and we note the follow-
ing relations,
A = 16πM2 , κs =
, (3)
consistent with (1). For the Reissner-Nordstrom
black hole, the situation is more subtle. Here one
distinguishes three different cases. For M > Q
one has non-extremal solutions, which exhibit two
horizons, an exterior event horizon and an interior
horizon. When M = Q one is dealing with an
extremal black hole, for which the two horizons
coalesce and the surface gravity vanishes. In that
case one has
A = 4πM2 , κs = 0 , µ = Q
. (4)
It is straightforward to verify that this result is
consistent with (1) for variations δM = δQ and
κs = 0. Because the surface gravity vanishes, one
might expect the entropy to vanish as well, as is
suggested by the third law of thermodynamics.
However, this is not the case because the hori-
zon area remains finite for zero surface gravity.
Finally, solutions with M < Q are not regarded
as physically acceptable. Their total energy is
less than the electromagnetic energy alone and
they no longer have an event horizon but exhibit
a naked singularity. Hence extremal black holes
saturate the bound M ≥ Q for physically accept-
able black hole solutions.
When embedding Einstein-Maxwell theory into
a complete supergravity theory, the above classifi-
cation has an interpretation in terms of the super-
symmetry algebra. This algebra has a central ex-
tension proportional to the black hole charge(s).
Unitary representations of the supersymmetry
algebra must necessarily have masses that are
larger than or equal to the charge. When this
bound is saturated, one is dealing with so-called
BPS supermultiplets. Such supermultiplets are
smaller than the generic massiveN = 2 supermul-
tiplets and have a different spin content. Because
of this, BPS states are stable under (adiabatic)
changes of the coupling constants and the rela-
tion between charge and mass remains preserved.
This important feature of BPS states will be rele-
vant for what follows. In these lectures BPS black
hole solutions are defined by the fact that they
have some residual supersymmetry, so that they
saturate a bound implied by the supersymmetry
algebra.
So far we did not refer to the explicit
Schwarzschild or Reissner-Nordstrom black hole
solutions, which can be found in many places
in the literature. One feature that should be
stressed, concerns the near-horizon geometry. For
extremal, static and spherically symmetric black
holes, this geometry is restricted to the product of
the sphere S2 and an anti-de Sitter space AdS2,
corresponding to the line element,
ds2 = −r2dt2 + dr
2 + r2(dθ2 + sin2 θ dϕ2)
. (5)
In these coordinates the horizon is located at
r = 0, where the timelike Killing vector K = ∂t
turns lightlike. Such a horizon is called a Killing
horizon.
In these notes we discuss various aspects of the
relation between black hole solutions and corre-
sponding microscopic descriptions. Section 2 gen-
erally describes the macroscopic (field theoretic)
and microscopic (statistical) approach to black
hole entropy and indicates why they are related.
Section 3 summarizes the calculation of the black
hole entropy based on a fivebrane wrapping on
a Calabi-Yau four-cycle in a compactification of
BPS Black Holes 3
M-Theory on the product space of a Calabi-Yau
threefold and a circle. In section 4 the attractor
equations are discussed for extremal black holes
on the basis of a variational principle defined for
a generic gravitational theory. Section 5 contains
a brief review of N = 2 supergravity and the su-
perconformal multiplet calculus. This material is
used in the description of the BPS attractor equa-
tions in N = 2 supergravity presented in section
6. In this case there also exists is a corresponding
formulation in terms of a BPS entropy function.
Finally section 7 discusses the relation between
this entropy function and the black hole partition
function.
We try to concentrate as much as possible on
the conceptual basis of the underlying ideas, but
these notes cannot do justice to all aspects rele-
vant for the study of black holes in string theory,
such as use of the AdS3/CFT2 correspondence,
black holes or related objects in other than four
space-time dimensions, the relation with indices
for BPS states, and the like. We refer to some
recent reviews where some of these topics have
been discussed [4,5,6,7].
2. DUAL PERSPECTIVES
A central question in black hole thermodynam-
cis concerns the statistical interpretation of the
black hole entropy. String theory has provided
new insights here [8], which enable the identi-
fication of the black hole entropy as the loga-
rithm of the degeneracy of states d(Q) of charge
Q belonging to a certain system of microstates.
In string theory these microstates are provided
by the states of wrapped brane configurations of
given momentum and winding. When calculating
the black hole solutions in the orresponding ef-
fective field theory with the charges specified by
the brane configuration, one discovers that the
black hole area is equal to the logarithm of the
brane state degeneracy, at least in the limit of
large charges. We will be reviewing some aspects
of this remarkable correspondence here.
The horizon area, which is expected to be pro-
portional to the macroscopic entropy according
to the Bekenstein-Hawking area law, turns out
to grow quadratically with the charges Q. After
converting to string units the radius of a black
hole is therefore proportional to
Rhorizon
lstring
∼ gs Q , (6)
where lstring and gs are the string length and cou-
pling constant, respectively. Since we will be as-
suming that the charges are large, the black holes
are generically much larger than the string scale.
Consequently these black holes are called large
and can be identified with the macroscopic black
holes we have been discussing earlier. However,
there are also situations where the leading contri-
bution to the area is only linear in the charges Q.
In that case (6) is replaced by
Rhorizon
lstring
Q , (7)
Moreover, in that case the string coupling (in-
versely proportional to the dilaton field) cannot
be taken constant, but tends to zero for large
charges according to gs ∼ Q−1/2, so that the ra-
dius of the black hole remains comparable to the
string scale. These black holes are called small.
Their corresponding classical supergravity solu-
tions exhibit a vanishing horizon area and a dila-
ton field that diverges at the horizon. To reliably
compute the right-hand side of (7) therefore re-
quires to include appropriate terms in the effec-
tive action of higher order in space-time deriva-
tives. We return to this issue in subsection 6.2.
To further understand the relation between a
field-theoretic description and a microscopic de-
scription it is relevant that strings live in more
that four space-time dimensions. In most situ-
ations the extra dimensions are compactified on
some internal manifold X and one is dealing with
the standard Kaluza-Klein scenario leading to ef-
fective field theories in four dimensions, describ-
ing low-mass modes of the fields associated with
appropriate eigenfunctions on the internal mani-
fold. Locally, the original space-time is a product
M4×X , where M4 denotes the four-dimensional
space-time that we experience in daily life. In
this situation there exists a corresponding space
X at every point xµ ofM4, whose size is such that
it will not be directly observable. However, this
4 B. de Wit
space X does not have to be the same at every
point in M4, and moving through M4 one may
encounter various spaces X that are not necessar-
ily equivalent. In principle they belong to some
well-defined class of fixed topology parametrized
by certain moduli. These moduli will appear as
fields in the four-dimensional effective field the-
ory. For instance, suppose that the spaces X are
n-dimensional tori T n. The metric of T n will ap-
pear as a field in the four-dimensional theory and
is related to the torus moduli. Hence, when deal-
ing with a solution of the four-dimensional theory
that is not constant in M4, each patch in M4 has
a non-trivial image in the space of moduli that
parametrize the internal spaces X .
Let us now return to a black hole solution,
viewed in this higher-dimensional perspective.
The fields, and in particular the four-dimensional
space-time metric, will vary nontrivially overM4,
and so will the internal spaces X . When moving
to the center of the black hole the gravitational
fields will become strong and the local product
structure intoM4×X could break down. Conven-
tional Kaluza-Klein theory does not have much
to say about what happens, beyond the fact that
the four-dimensional solution can be lifted to the
higher-dimensional one, at least in principle.
However, there is a feature of string theory that
is absent in a purely field-theoretic approach. In
the effective field-theoretic context only the lo-
cal degrees of freedom of strings and branes are
captured. But extended objects may also carry
global degrees of freedom, as they can also wrap
themselves around non-trivial cycles of the inter-
nal space X . This wrapping tends to take place
at a particular position inM4, so in the context of
the four-dimensional effective field theory this will
reflect itself as a pointlike object. The wrapped
object is the string theory representation of the
black hole!
We are thus dealing with two complementary
pictures of the black hole. One is based on general
relativity where a point mass generates a global
solution of space-time with strongly varying grav-
itational fields, which we shall refer to as the
macroscopic description. The other one, based
on the internal space where an extended object is
entangled in one of its cycles, does not immedi-
ately involve gravitational fields and can easily be
described in flat space-time. This description will
be refered to as microscopic. To understand how
these two descriptions are related is far from easy,
but a connection must exist in view of the fact
that gravitons are closed string states which inter-
act with the wrapped branes. These interactions
are governed by the string coupling constant gs
and we are thus confronted with an interpolation
in that coupling constant. In principle, such an
interpolation is very difficult to carry out, so that
a realistic comparison between microscopic and
macroscopic results is usually impossible. How-
ever, reliable predictions are possible for extremal
black holes that are BPS! As we indicated earlier,
in that situation there are reasons to trust such
interpolations. Indeed, it has been shown that
the predictions based on these two alternative de-
scriptions can be successfully compared and new
insights about black holes can be obtained.
But how do the wrapped strings and branes
represent themselves in the effective action de-
scription and what governs their interactions with
the low-mass fields? Here it is important to real-
ize that the massless four-dimensional fields are
associated with harmonic forms on X . Harmonic
forms are in one-to-one correspondence with so-
called cohomology groups consisting of equiva-
lence classes of forms that are closed but not
exact. The number of independent harmonic
forms of a given degree is given by the so-called
Betti numbers, which are fixed by the topology
of the spaces X . When expanding fields in a
Kaluza-Klein scenario, the number of correspond-
ing massless fields can be deduced from an expan-
sion in terms of tensors onX corresponding to the
various harmonic forms. The higher-dimensional
fields Φ(x, y) thus decompose into the massless
fields φA(x) according to (schematically),
Φ(x, y) = φA(x) ωA(y) , (8)
where ωA(y) denotes the independent harmonic
forms on X . The above expression, when sub-
stituted into the action of the higher-dimensional
theory, leads to interactions of the fields φA pro-
portional to the ‘coupling constants’,
CABC··· ∝
ωA ∧ ωB ∧ ωC · · · . (9)
BPS Black Holes 5
These constants are known as intersection num-
bers, for reasons that will become clear shortly.
We already mentioned that the Betti numbers
depend on the topology of X . This is related to
Poincaré duality, according to which cohomology
classes are related to homology classes. The latter
consist of submanifolds of X without boundary
that are themselves not a boundary of some other
submanifold of X . This is precisely relevant for
wrapped branes which indeed cover submanifolds
of X , but are not themselves the boundary of a
submanifold because otherwise the brane could
collapse to a point. Without going into detail,
this implies that there exists a dual relationship
between harmonic p-forms ω and (dX −p)-cycles,
where dX denotes the dimension of X . We can
therefore choose a homology basis for the (dX−p)-
cycles dual to the basis adopted for the p-forms.
Denoting this basis by ΩA the wrapping of an ex-
tended object can now be characterized by spec-
ifying its corresponding cycle P in terms of the
homology basis,
P = pAΩA . (10)
The integers pA count how many times the ex-
tended object is wrapped around the correspond-
ing cycle, so we are actually dealing with integer-
valued cohomology and homology. The wrap-
ping numbers pA reflect themselves as magnetic
charges in the effective action. The electric
charges are already an integer part of the effective
action, because they are associated with gauge
transformations that usually originate from the
higher-dimensional theory.
Owing to Poincaré duality it is thus very nat-
ural that the winding numbers interact with the
massless modes in the form of magnetic charges,
so that they can be incorporated in the effective
action. Before closing this section, we note that,
by Poincaré duality, we can express the number
of intersections by
P · P · P · · · = CABC··· pApBpC · · · . (11)
This is a topological characterization of the wrap-
ping, which will appear in later formulae.
3. BLACK HOLES IN M/STRING THE-
ORY – AN EXAMPLE
As an example we now discuss the black hole
entropy derived from microscopic arguments in
a special case. In a later section we will con-
sider the corresponding expression for the macro-
scopic entropy. We start from M-theory, which is
the strong coupling limit of type-IIA string the-
ory. Its massless states are described by eleven-
dimensional supergravity. The latter is invariant
under 32 supersymmetries. Seven of the eleven
space-time dimensions are compactified on an in-
ternal space which is the product of a Calabi-Yau
threefold (a Ricci flat three-dimensional complex
manifold, henceforth denoted by CY3) times a cir-
cle S1. Such a space induces a partial breaking of
supersymmetry which leaves 8 supersymmetries
unaffected. In the context of the four-dimensional
space-time M4, these 8 supersymmetries are en-
coded into two independent Lorentz spinors and
for that reason this symmetry is referred to as
N = 2 supersymmetry. Hence the effective four-
dimensional field theory will be some version of
N = 2 supergravity.
M-theory contains a five-brane and this is the
microscopic object that is responsible for the
black holes that we consider: the five-brane has
wrapped itself on a 4-cycleP of the CY 3 space [9].
Alternatively one may study this class of black
holes in type-IIA string theory, with a 4-brane
wrapping the 4-cycle [10]. The 4-cycle is subject
to certain requirements which will be mentioned
in due course.
The massless modes captured by the effective
field theory correspond to harmonic forms on the
CY3 space; they do not depend on the S
1 co-
ordinate. The 2-forms are of particular interest.
In the effective theory they give rise to vector
gauge fields Aµ
A, which originate from the rank-
three tensor gauge field in eleven dimensions. In
addition there is an extra vector field Aµ
0 cor-
responding to a 0-form which is related to the
graviphoton associated with S1. This field will
couple to the electric charge q0 associated with
momentum modes on S1 in the standard Kaluza-
Klein fashion. The 2-forms are dual to 4-cyles
and the wrapping of the five-brane is encoded in
6 B. de Wit
terms of the wrapping numbers pA, which appear
in the effective field theory as magnetic charges
coupling to the gauge fields Aµ
A. Here we see
Poincaré duality at work, as the magnetic charges
couple nicely to the corresponding gauge fields.
In view of the fact that the product of three 2-
forms defines a 6-form that can be integrated over
the CY3 space, there exist non-trivial triple inter-
section numbers CABC . These numbers will ap-
pear in the three-point couplings of the effective
field theory. There is a subtle topological fea-
ture that we have not explained before, which is
typical for complex manifolds containing 4-cycles,
namely the existence of another quantity of topo-
logical interest known as the second Chern class.
The second Chern class is a 4-form whose inte-
gral over a four-dimensional Euclidean space de-
fines the instanton number. The 4-form can be
integrated over the 4-cycle P and yields c2A pA,
where the c2A are integers.
Let us now turn to the microscopic counting of
degrees of freedom [9]. These degrees of freedom
are associated with the massless excitations of the
wrapped five-brane characterized by the wrap-
ping numbers pA on the 4-cycle. The 4-cycle P
must correspond to a holomorphically embedded
complex submanifold in order to preserve 4 super-
symmetries. The massless excitations of the five-
brane are then described by a (1+1)-dimensional
superconformal field theory (the reader may also
consult [11]). Because we have compactified the
spatial dimension on S1, we are dealing with a
closed string with left- and right-moving states.
The 4 supersymmetries of the conformal field the-
ory reside in one of these two sectors, say the
right-handed one. Conformal theories in 1+1 di-
mensions are characterized by a central charge,
and in this case there is a central charge for the
right- and for the left-moving sector separately.
The two central charges are expressible in terms
of the wrapping numbers pA and depend on the
intersection numbers and the second Chern class,
according to
cL = CABC p
ApBpC + c2A p
cR = CABC p
ApBpC + 1
c2A p
Here we should stress that the above result is far
from obvious and holds only under the condition
that the pA are large. In that case every generic
deformation of P will be smooth. Under these cir-
cumstances it is possible to relate the topological
properties of the 4-cycle to the topological data
of the Calabi-Yau space.
We can now choose a state of given momentum
q0 which is supersymmetric in the right-moving
sector. From rather general arguments it fol-
lows that such states exist. The corresponding
states in the left-moving sector have no bearing
on the supersymmetry and these states have a
certain degeneracy depending on the value of q0.
In this way we have a tower of degenerate BPS
states invariant under 4 supersymmetries, built
on a supersymmetric state in the right-moving
sector. We can then use Cardy’s formula, which
states that the degeneracy of states for fixed but
large momentum (large as compared to cL) equals
exp[2π
|q0| cL/6]. This leads to the following ex-
pression for the entropy,
Smicro(p, q) =
|q̂0|(CABC pApBpC + c2A pA) ,
where q0 has been shifted according to
q̂0 = q0 +
CABqAqB . (14)
Here CAB is the inverse of CAB = CABCp
C . This
modification is related to the fact that the electric
charges associated with the gauge fields Aµ
A will
interact with the M-theory two-brane [9]. The
existence of this interaction can be inferred from
the fact that the two-brane interacts with the
rank-three tensor field in eleven dimensions, from
which the vector gauge fields Aµ
A originate.
We stress that the above results apply in the
limit of large charges. The first term propor-
tional to the triple intersection number is obvi-
ously the leading contribution whereas the terms
proportional to the second Chern class are sub-
leading. The importance of the subleading terms
will become more clear in later sections. Having
obtained a microscopic representation of a BPS
black hole, it now remains to make contact with
it by deriving the corresponding black hole solu-
tion directly in the N = 2 supergravity theory.
This is discussed in some detail in section 6.
BPS Black Holes 7
4. ATTRACTOR EQUATIONS
The microscopic expression for the black hole
entropy depends only on the charges, whereas a
field-theoretic calculation can in principle depend
on other quantities, such as the values of the mod-
uli fields at the horizon. To establish any agree-
ment between these two approaches, the moduli
(as well as any other relevant fields that enter the
calculation) must take fixed values at the horizon
which may depend on the charges. As it turns out
this is indeed the case for extremal black hole so-
lutions, as was first demonstrated in the context
of N = 2 supersymmetric black holes [12,13,14].
The values taken by the fields at the horizon are
independent of their asymptotic values at spatial
infinity. This fixed point behaviour is encoded in
so-called attractor equations.
In the presence of higher-derivative interac-
tions it is very difficult to explicitly construct
black hole solutions and to exhibit the attractor
phenomenon. However, by concentrating on the
near-horizon region one can usually determine the
fixed-point values directly without considering
the interpolation between the horizon and spa-
tial infinity. Provided the symmetry of the near-
horizon region is restrictive enough, the attrac-
tor phenomenon can be described conveniently in
terms of a variational principle for a so-called en-
tropy function. The stationarity of this entropy
function then yields the attractor equations and
its value at the attractor point equals the macro-
scopic entropy. The purpose of the present sec-
tion is to explain this phenomenon.
For N = 2 BPS black holes with higher-
derivative interactions the attractor equations fol-
low from classifying possible solutions with full
supersymmetry [15]. As it turns out supersym-
metry determines the near-horizon geometry (and
thus the horizon area), the values of the moduli
fields in terms of the charges and the value of the
entropy as defined by the Noether charge defini-
tion of Wald [16]. This result can also be de-
scribed in terms of a variational principle [17,18],
as we shall explain in section 6.
Let us now turn to the more generic case of ex-
tremal, static, spherically symmetric black holes
that are not necessarily BPS, following the ap-
proach of [19] which is based on an action prin-
ciple. Similar approaches can be found elsewhere
in the literature, for instance, in [20,21,22]. When
dealing with spherically symmetric solutions, one
integrates out the spherical degrees of freedom
and obtains a reduced action for a 1 + 1 dimen-
sional field theory which fully describes the black
hole solutions. Here we consider a general system
of abelian vector gauge fields, scalar and other
matter fields coupled to gravity. The geometry
is thus restricted to the product of the sphere S2
and a 1 + 1 dimensional space-time, and the de-
pendence of the fields on the S2 coordinates θ
and ϕ is fixed by symmetry arguments. For the
moment we will not make any assumption regard-
ing the dependence on the remaining two cooor-
dinates r and t. Consequently we write the gen-
eral field configuration consistent with the various
isometries as
ds2(4) = gµνdx
= ds2(2) + v2
dθ2 + sin2 θ dϕ2
I = eI , Fθϕ
sin θ .
The Fµν
I denote the field strengths associated
with a number of abelian gauge fields. The θ-
dependence of Fθϕ
I is fixed by rotational invari-
ance and the pI denote the magnetic charges. The
latter are constant by virtue of the Bianchi iden-
tity, but all other fields are still functions of r
and t. As we shall see in a moment the fields eI
are dual to the electric charges. The radius of
S2 is defined by the field v2. The line element of
the 1+1 dimensional space-time will be expressed
in terms of the two-dimensional metric ḡij , whose
determinant will be related to a field v1 according
|ḡ| . (16)
Eventually ḡij will be taken proportional to an
AdS2 metric,
ds2(2) = ḡij dx
idxj = v1
− r2 dt2 + dr
. (17)
In addition to the fields eI , v1 and v2, there may
be a number of other fields which for the moment
we denote collectively by uα.
8 B. de Wit
As is well known theories based on abelian vec-
tor fields are subject to electric/magnetic dual-
ity, because their equations of motion expressed
in terms of the dual field strengths,1
GµνI =
|g| εµνρσ
∂FρσI
, (18)
take the same form as the Bianchi identities for
the field strengths Fµν
I . Adopting the conven-
tions where xµ = (t, r, θ, ϕ) and εtrθϕ = 1, and
the signature of the space-time metric equals
(−,+,+,+), which is consistent with (17), it fol-
lows that, in the background (15),
Gθϕ I = −v1v2 sin θ
Grt I = −4π v1v2
These two tensors can be written as qI sin θ/(4π)
and fI . The quantities qI and fI are conjugate
to pI and eI , respectively, and can be written as
qI(e, p, v, u) = −4π v1v2
fI(e, p, v, u) = −4π v1v2
They depend on the constants pI and on the fields
eI , v1,2 and uα, and possibly their t and r deriva-
tives, but no longer on the S2 coordinates θ and
ϕ. Upon imposing the field equations it follows
that the qI are constant and correspond to the
electric charges. Our aim is to obtain a descrip-
tion in terms of the charges pI and qI , rather than
in terms of the pI and eI .
Electric/magnetic duality transformation are
induced by rotating the tensors Fµν
I andGµν I by
a constant transformation, so that the new linear
combinations are all subject to Bianchi identities.
Half of them are then selected as the new field
strengths defined in terms of new gauge fields,
while the Bianchi identities on the remaining lin-
ear combinations are regarded as field equations
belonging to a new Lagrangian defined in terms
1We assume that the Lagrangian is a function of the
abelian field strengths and does not depend explicitly on
the gauge fields.
of the new field strengths. In order that this du-
alization can be effected the rotation of the ten-
sors must belong to Sp(2n + 2;R), where n + 1
denotes the number of independent gauge fields
[23]. Naturally the duality leads to new quan-
tities (p̃I , q̃I) and (ẽ
I , f̃I), related to the original
ones by the same Sp(2n+2;R) rotation. Since the
charges are not continuous but will take values in
an integer-valued lattice, this group should even-
tually be restricted to an appropriate arithmetic
subgroup.
Subsequently we define the reduced Lagrangian
by the integral of the full Lagrangian over S2,
F(e, p, v, u) =
dθ dϕ
|g| L , (21)
We note that the definition of the conjugate quan-
tities qI and fI takes the form,
qI = −
, fI = −
. (22)
It is known that a Lagrangian does not trans-
form as a function under electric/magnetic dual-
ities (see, e.g. [24]), but one can generally show
that the following combination does [25],
E(q, p, v, u) = −F(e, p, v, u)− eIqI . (23)
More precisely, this quantity transforms un-
der electric/magnetic duality according to
Ẽ(q̃, p̃, v, u) = E(q, p, v, u). In view of the first
equation (22), the definition (23) takes the form
of a Legendre transform. Furthermore the field
equations imply that the qI are constant and that
the action,
dtdr E , is stationary under varia-
tions of the fields v and u, while keeping the pI
and qI fixed. This is to be expected as E is in fact
minus the Hamiltonian density associated with
the reduced Lagrangian density (21), at least as
far as the vector fields are concerned.
In the near-horizon background (17), assuming
fields that are invariant under the AdS2 isome-
tries, the generally covariant derivatives of the
fields vanish and the function E depends only on
the fields which no longer depend on the coordi-
nates. The equations of motion then imply that
the values of the fields v1,2 and uα are determined
BPS Black Holes 9
by demanding E to be stationary under variations
of v and u,
= 0 , qI = constant . (24)
The function 2π E(q, p, v, u) then coincides with
the entropy function proposed by Sen [19]. The
first two equations of (24) are interpreted as the
attractor equations. At the attractor point one
may prove
attractor
dθ dϕ
|g|Rrtrt
∂Rrtrt
, (25)
where the right-hand side is evaluated in the near-
horizon geometry. This leads to the expression
attractor
∂Rµνρσ
εµνερσ , (26)
where Σhor denotes a spacelike cross section of
the Killing horizon, and εµν the normal bivector
which acts in the space normal to Σhor. This is
precisely the expression for the Wald entropy [16]
(applied to this particular case). For the Einstein-
Hilbert action, (26) equals a quarter of the hori-
zon area in Planck units. For more general La-
grangians (26) may lead to deviations from the
area law, as we will see in due course. Note that
the entropy function does not necessarily depend
on all fields at the horizon. The values of some
of the fields will then be left unconstrained, but
those will not appear in the expression for the
Wald entropy either.
The above derivation of the entropy function
applies to any gauge and general coordinate in-
variant Lagrangian, including Lagrangians with
higher-derivative interactions. In the absence of
higher-derivative terms, the reduced Lagrangian
F is at most quadratic in eI and pI and the Leg-
endre transform (23) can easily be carried out.
The results coincide with corresponding terms in
the so-called black hole potential discussed in e.g.
[20,21,22].
5. N=2 SUPERGRAVITY
In the previous section the symmetry of the
near-horizon geometry played a crucial role. For
BPS black holes the supersymmetry enhancement
at the horizon is the crucial input that constrains
both some of the fields at the horizon as well as
the near-horizon geometry itself. The black hole
solutions that we will be considering have a resid-
ual N = 1 supersymmetry (so that they are BPS)
and are solitonic interpolations between N = 2
configurations at the horizon and at spatial infin-
ity. Obviously to describe BPS black holes one
needs to consider extended supergravity theories.
In view of the application described in section 3,
N = 2 supergravity is relevant. Moreover, N = 2
supergravity has off-shell formulations (meaning
that supersymmetry transformations realize the
supersymmetry algebra without the need for im-
posing field equations associated with a specific
Lagrangian) and this facilitates the calculations
in an essential way. This aspect is especially im-
portant because we will be considering supersym-
metric Lagrangians with interactions containing
more than two derivatives.
In the following subsections we present a brief
introduction to N = 2 supergravity. Supermul-
tiplets are introduced in subsection 5.1 and su-
persymmetric actions in subsection 5.2. Finally,
in subsection 5.3, we elucidate the use of compen-
sating fields and corresponding supermultiplets to
familiarize the reader with the principles under-
lying the superconformal multiplet calculus. Fur-
ther details can be found in the literature [26,27].
5.1. Supermultiplets
In this subsection we briefly introduce the su-
permultiplets that play a role in the following. Of
particular interest are the vector and the Weyl su-
permultiplet. Other multiplets are the tensor su-
permultiplets and the hypermultiplets, but those
will not be discussed as they only play an ancil-
lary role.
The covariant fields and field strengths of the
various gauge fields belonging to the vector or to
the Weyl supermultiplet comprise a chiral mul-
tiplet. Such multiplets are described in super-
space by chiral superfields.At this point it is con-
venient to systematize our discussion by using su-
perspace notions, although we do not intend to
make an essential use of superfields. Scalar chiral
fields in N = 2 superspace have 16 + 16 bosonic
+ fermionic field components, but there exists a
10 B. de Wit
constraint which reduces the superfield to only
8+8 field components. This constraint expresses
higher-θ components of the superfield in terms
of lower-θ components or space-time derivatives
thereof. The vector supermultiplet and the Weyl
supermultiplet are both related to reduced chiral
multiplets. Note, however, that products of re-
duced chiral superfields constitute general chiral
fields.
The vector supermultiplet is described by a
scalar reduced chiral superfields, whose lowest-
θ component is a complex field which we de-
note by X . Then there is a doublet of chiral
fermions Ωi, where i is an SU(2) R-symmetry
doublet index. The position of the index i in-
dicates the chirality of the spinor field: Ωi car-
ries positive, and Ωi negative chirality. The fields
X̄ and Ωi appear as lowest-θ components in the
anti-chiral superfield that one obtains by complex
conjugation of the chiral superfield. We recall
that the so-called R-symmetry group is defined as
the maximal group that rotates the supercharges
in a way that commutes with Lorentz symmetry
and is compatible with the supersymmetry alge-
bra. For N = 2 supersymmetry this group equals
SU(2)×U(1), which acts chirally on the spinors.
In spite of its name, R-symmetry does not nec-
essarily consitute an invariance of supersymmet-
ric Lagrangians. Finally, at the θ2-level, we en-
counter the field strength Fµν of the gauge field
and an auxiliary field which we write as a sym-
metric real tensor, Yij = Yji = εikεjlY
kl. Here
we note that complex conjugation will often be
indicated by rasing and/or lowering of SU(2) in-
dices. One can easily verify that in this way we
have precisely 8+8 independent field components
(which we will refer to as off-shell degrees of free-
dom, as we did not impose any field equations).
Note the difference with on-shell degrees of free-
dom. The conventional Lagrangian for the vector
supermultiplet describes 4 + 4 physical massless
bosonic and fermionic states: 2 scalars associated
with X and X̄, the 2 helicities associated with
the vector gauge field, and 2 Majorana fermions,
each carrying 2 helicities.
The Weyl supermultiplet has a rather similar
decomposition, but in this case the reduced chi-
ral superfield is not a scalar but an anti-selfdual
Lorentz tensor. For extended supersymmetry the
Weyl superfield is also assigned to the antisym-
metric representation of the R-symmetry group,
so that its lowest-order θ-component is denoted
by Tab
ij . Its complex conjugate belongs to the
corresponding anti-chiral superfield and its corre-
sponding tensor is selfdual and denoted by Tabij .
Here the indices a, b, . . . denote the components of
space-time tangent space tensors. In view of its
tensorial character the Weyl supermultiplet com-
prises 24 + 24 off-shell degrees of freedom. The
covariant components of the Weyl multiplet are as
follows. Linear in θ one has the fermions decom-
posing into the field strength of the gravitini and
a doublet spinor χi. The gravitino field strengths
comprise 16 degrees of freedom so that together
with the spinors χi we count 24 off-shell degrees of
freedom. At the θ2-level we have the Weyl tensor,
the field strengths belonging to the gauge fields
associated with R-symmetry, and a real scalar
field denoted by D, comprising 5, 4× 3 and 1 off-
shell degrees of freedom, respectively. Together
with the 6 degrees of freedom belonging to Tab
we count 24 bosonic degrees of freedom.
The Weyl multiplet contains the fields ofN = 2
conformal supergravity and an invariant action
can be written down that is quadratic in its
components. Although Tab
ij is not subject to a
Bianchi identity, it is often called the “gravipho-
ton field strength”. The reason for this misnomer
is that the gravitini transform into Tab
ij and in lo-
cally supersymmetric Lagrangians of vector mul-
tiplets that are at most quadratic in space-time
derivatives, Tab
ij acts as an auxiliary field and
couples to a field-dependent linear combination
of the vector multiplet field strengths. For this
class of Lagrangians, all the fields of the Weyl
multiplet with the exception of the graviton and
the gravitini fields, act as auxiliary fields.
In this subsection we mainly describe linearized
results. Ultimately we are interested in con-
structing a theory of local supersymmetry. This
means that the vector multiplets must first be
formulated in a supergravity background. This
leads to additional terms in the supersymmetry
transformation rules and in the superfield com-
ponents which depend on the supergravity back-
ground. Some of these terms correspond to re-
BPS Black Holes 11
placing ordinary space-time derivatives by covari-
ant ones. However, we consider only vector mul-
tiplets and the Weyl multiplet here and the latter
describes the supermultiplet of conformal super-
gravity. Consistency therefore requires that we
formulate the vector supermultiplet in a super-
conformal background, and, indeed, the vector
supermultiplet is a representation of the full su-
perconformal algebra. Therefore all the super-
conformal symmetries can be realized as local
symmetries. Naturally the Weyl multiplet itself
will also acquire non-linear corrections but those
do not involve the vector multiplet fields. The
reader may wonder why we are interested in con-
formal supergravity here, but this will be clarified
shortly. Other than that, the situation is concep-
tually the same as when considering multiplets of
matter fields coupled to a nonabelian gauge the-
ory, or to gravity.
5.2. Supersymmetric actions
In view of the fact that both the vector and
Weyl supermultiplets are described by chiral su-
perfields, even beyond their linearized form, it
is clear how to construct supersymmetric ac-
tions. Namely one takes some function of the
vector superfield (actually we will need several
of these superfields, which we label by indices
I = 0, 1, 2, . . . , n) and the square of the Weyl
superfield. Taking the square implies no loss of
generality because we are interested in Lorentz
invariant couplings. When expanding the super-
fields in θ components, one generates multiple
derivatives of this function which depend on the
lowest-θ components, X and
A = (Tab
ijεij)
2 . (27)
Because the function is holomorphic, i.e., it de-
pends on X and A, but not on their complex
conjugates, we take the imaginary part of the re-
sulting expression. However, in order that the
action is superconformally invariant, the function
F (X,A) must be holomorphic and homogeneous,
F (λX, λ2A) = λ2F (X,A) . (28)
We refrain from giving full results. In principle
they are derived straightforwardly, but the for-
mulae are often lengthy and require extra defini-
tions. Therefore we discuss only a few character-
istic terms.
First of all, let us consider the scalar kinetic
terms. They are accompanied by a coupling to
the Ricci scalar and the scalar field D of the Weyl
multiplet in the following way,
L ∝ i(∂µFI ∂µX̄I − ∂µF̄I ∂µXI)
− i(1
R−D)(FI X̄I − F̄I XI) ,
where FI denotes the derivative of F with respect
to XI . Observe that when the function F de-
pends on Tab
ij this will generate interactions be-
tween the kinetic term for the vector multiplet
scalars and the tensor field of the Weyl multiplet.
Of course, this pattern continues for other terms.
A second term concerns the kinetic term of the
vector fields, which is proportional to the second
derivative of the function F ,
L ∝ 1
iFIJ (F
I − 1
X̄ITab
ijεij)
× (F−abJ − 1
X̄JT abklεkl) + h.c. .
A third term involves the square of the Weyl
tensor, contained in the tensor R(M),
L ∝ 16 iFA[2R(M)cdab R(M)−cd
− 16T abij DaDcTcbij ] + h.c. .
Here Da denotes a superconformally covariant
derivative (which also contains terms propor-
tional to the Ricci tensor). We refrain from giving
further details at this point and refer to the liter-
ature.
5.3. Compensator multiplets
The theories discussed so far are invariant un-
der the local symmetries of the superconformal
algebra. This high degree of symmetry seems un-
necessary for, or even an obstacle to, practical
applications. The purpose of this section is to
explain that this is not the case.
Let us start with a simple example, namely
massive SU(N) Yang-Mills theory, with La-
grangian,
L = 1
Tr[Fµν(W )F
µν(W )]
M2Tr[Wµ W
12 B. de Wit
where we use a Lie-algebra valued notation and
the definition Fµν = 2 ∂[µWν] − [Wµ,Wν ]. In-
troduce now a matrix-valued field Φ ∈ SU(N)
transforming under SU(N) gauge transforma-
tions from the left and substitute Wµ = Φ
−1DµΦ
into the Lagrangian, where the covariant deriva-
tive reads DµΦ = (∂µ − Aµ)Φ. The first term is
not affected by this transformation which takes
the form of a field-dependent gauge transforma-
tion. But the mass term changes, and we find the
following Lagrangian,
L = 1
Tr[Fµν(A)F
µν(A)]
M2 Tr[DµΦD
µΦ−1] ,
which is manifestly gauge invariant. Clearly, this
is a massless gauge theory coupled to N2 − 1
scalars. However, this formulation is gauge equiv-
alent to (32), as one readily verifies by imposing
the gauge condition Φ = .
What do we achieve by rewriting (32) into the
form (33)? Both Lagrangians describe the same
number of physical states and are based on the
same number of off-shell degrees of freedom. In
(32) the degrees of freedom are contained in a sin-
gle field, Wµ, carrying 4 components per genera-
tor. In (33), however, the degrees of freedom are
distributed over two fields in a local and Lorentz
invariant way, namely 3 components per gener-
ator in Aµ (we subtracted the gauge degrees of
freedom) and 1 component per generator in Φ.
Hence the second formulation can be regarded as
reducible, and this reducibility has been achieved
by introducing extra gauge invariance.
A similar situation exists for gravity, as is
shown by the Lagrangian,
|g|L ∝
gµν∂µφ∂νφ− 16Rφ
, (34)
which is invariant under local scale transforma-
tion with parameter Λ(x): δφ = Λφ, δgµν =
−2Λ gµν. This Lagrangian is gauge equivalent
to the Einstein-Hilbert Lagrangian. To see this
one either rewrites it in terms of a scale invariant
metric φ2gµν , or one simply imposes the gauge
condition and sets φ equal to a constant (which
will then be related to Newton’s constant). Again
the Einstein-Hilbert Lagrangian and (34) contain
the same number of off-shell degrees of freedom,
but the latter field configuration is reducible: gµν
describes only 5 degrees of freedom (in view of
the local scale invariance) and the sixth one can
be assigned to the scalar field φ.
Fields such as Φ and φ are called compensat-
ing fields, because they can be used to convert any
quantity that transforms under the gauge symme-
try into a gauge invariant one. Often the gauge
equivalent formulation, based on the introduction
of compensator fields and gauge symmetries at
the same time, is exploited for reasons of renor-
malizability as one can use the gauge freedom to
choose a different gauge that leads to better short-
distance behaviour. This is not the issue here but
the crucial point is that the compensating degrees
of freedom must be contained in full supermulti-
plets. By keeping the gauge invariance manifest
one realizes a higher degree of symmetry which
facilitates the construction of Lagrangians and
clarifies the geometrical features of the resulting
supergravity theories. In this way, pureN = 2 su-
pergravity is, for instance, constructed from two
compensating supermultiplets, one of which is a
vector multiplet and the other one can be a ten-
sor multiplet or a hypermultipet. Of these two
supermultiplet only the vector field will describe
physical degrees of freedom (namely, those corre-
sponding to the graviphoton). The other compo-
nents play the role of either compensating fields
(associated with local scale, R-symmetry and spe-
cial conformal supersymmetry transformations),
or are constrained by the field equations, either
by Lagrange multipliers, or because they are aux-
iliary.
6. BPS ATTRACTORS
As we already discussed in section 4 the BPS
attractor equations follow from the requirement
of full N = 2 supersymmetry at the horizon. In
the context of an off-shell representation of su-
perymmetry, the corresponding equations can be
derived in a way that is rather independent of
the action. As explained in the previous section,
N = 2 vector multiplets contain complex physical
scalar fields which we denoted by XI . In the con-
text of the superconformal framework these fields
are defined projectively, in view of the invariance
BPS Black Holes 13
under local scale and U(1) transformations. The
action for vector multiplets with additional inter-
actions involving the square of the Weyl tensor
is encoded in a holomorphic function F (X,A),
which is homogenous of second degree (c.f. (28)).
Here A is quadratic in the anti-selfdual field Tab
as shown in (27). The normalizations of the La-
grangian and of the charges adopted below differ
from the normalizations used in section 4.
Another issue that we should explain concerns
electric/magnetic duality, which in principle per-
tains to the gauge fields. Straightforward appli-
cation of this duality to an N = 2 supersymmet-
ric Lagrangian with vector multiplets, leads to a
new Lagrangian that no longer takes the canoni-
cal form in terms of a function F (X,A). In order
to bring it into that form one must simultane-
ously apply a field redefinition to the scalar and
spinor fields. On the scalar fields, this redefinition
follows from the observation that (XI , FI(X,A))
transforms as a sympletic vector analogous to the
tensors (Fµν
I , GµνI) discussed previously. Hence
the (2n + 2)-dimensional vector (XI , FI(X,A))
is rotated into a new vector (X̃I , F̃I). When
this rotation belongs to Sp(2n + 2;R) then F̃I
can be written as the derivative of a new func-
tion F̃ (X̃, A) with respect to X̃I . The new func-
tion then encodes the dual action. The need for
this additional field redefinition follows from the
fact that the gauge fields, and thus their field
strengths Fµν
I , and the fields XI have a well-
defined relation encoded in the supersymmetry
transformation rules. Therefore, the transforma-
tions of (XI , FI(X,A)) must be taken into ac-
count when considering electric/magnetic dual-
ity. We refer to [24,28] for further details and a
convenient list of formulae.
To determine the BPS attractor equations one
classifies all possible N = 2 supersymmetric so-
lutions. This is done by studying the supersym-
metry variations of the fermions in an arbitrary
bosonic background. Requiring that these varia-
tions vanish then imposes strong restrictions on
this background. The analysis was performed in
[15] for N = 2 supergravity with an arbitrary
number of vector multiplets and hypermultiplets,
including higher-order derivative couplings pro-
portional to the square of the Weyl tensor. Also
the interpolating BPS solutions were studied in
considerable detail. It was found that N = 2 su-
persymmetric solutions are unique and depend on
a harmonic function with a single center. Hence
the horizon geometry and the values of the rel-
evant fields are fully determined in terms of the
charges. The hypermultiplet scalar fields are co-
variantly constant but otherwise arbitrary. How-
ever, the horizon and the entropy do not depend
on these fields, so that they can be ignored. In the
absence of charges one is left with flat Minkowski
space-time with arbitrary constant moduli and
ij = 0.
As it turns out the attractor equations have
a universal form. Before commenting further
on their derivation, let us present the equa-
tions, which are manifestly covariant under elec-
tric/magnetic duality,
PI = 0 , QI = 0 , Υ = −64 , (35)
where
PI ≡ pI + i(Y I − Ȳ I) ,
QI ≡ qI + i(FI − F̄I) .
Here the Y I and Υ are related to the XI and A,
respectively, by a uniform rescaling and FI and
FΥ will denote the derivatives of F (Y,Υ) with
respect to Y I and Υ. To explain the details of
the rescaling, we introduce the complex quantity
Z, sometimes refered to as the ’holomorphic BPS
mass’, which equals the central charge associated
with the vector supermultiplets. In terms of the
original variables XI it is defined as
Z = exp[K/2] (pIFI(X,A)− qIXI) , (37)
where
e−K = i (X̄IFI(X,A)− F̄I(X̄, Ā)XI) . (38)
At the horizon the variables Y I and Υ are de-
fined by
Y I = exp[K/2] Z̄ XI ,
Υ = exp[K] Z̄2 A .
Note that Y I and Υ are invariant under arbitrary
complex rescalings of the underlying variables XI
14 B. de Wit
and A. The reader may easily verify that for fields
satisfying the attractor equations (35), one estab-
lishes that
|Z|2 ≡ pIFI − qIY I , (40)
which is obviously real and positive, is equal to
i(Ȳ IFI − Y I F̄I).
Finally we wish to draw attention to just one
aspect of the derivation of the attractor equations
(35). Consider the spinor fields belonging to the
vector supermultiplets, and concentrate on their
supersymmetry variation in terms of a two-rank
tensor,
I = 1
I − 1
εklTab
kl X̄I
. (41)
This particular linear combination of the field
strength F−ab
I and the field X̄I arises because the
symmetry transformations are evaluated in a su-
perconformal background. Full supersymmetry
therefore implies that the right-hand side of (41)
vanishes, so that
I = 1
εklTab
kl X̄I . (42)
An extension of this argument gives a similar re-
sult for the conjugate field strengths,
G−abI =
εklTab
kl F̄I . (43)
Note that these two equations are consistent with
respect to electric/magnetic duality. Given the
fact that the field strengths Fab
I and GabI satisfy
the Maxwell equations and therefore become pro-
portional to the magnetic and electric charges, pI
and qI , it is not surprising that one finds the at-
tractor equations (35). For further details of the
analysis we refer to [15].
6.1. The BPS entropy function
The BPS attractor equations follow also from
a variational principle based on the entropy func-
tion [17,18],
Σ(Y, Ȳ , p, q) = F(Y, Ȳ ,Υ, Ῡ)
− qI(Y I + Ȳ I) + pI(FI + F̄I) ,
where pI and qI couple to the correspond-
ing magneto- and electrostatic potentials at the
horizon (c.f. [15]) in a way that is consistent
with electric/magnetic duality. The quantity
F(Y, Ȳ ,Υ, Ῡ), which will be regarded as a ‘free
energy’ in what follows, is defined by
F(Y, Ȳ ,Υ, Ῡ) = −i
Ȳ IFI − Y I F̄I
ΥFΥ − ῩF̄Υ
where FΥ = ∂F/∂Υ. Just as the entropy function
discussed in section 4, the entropy function (44)
transforms as a function under electric/magnetic
duality [28]. Varying this entropy function with
respect to the Y I , while keeping the charges and
Υ fixed, yields the result,
δΣ = PI δ(FI + F̄I)−QI δ(Y I + Ȳ I) . (46)
Here we made use of the homogeneity of the func-
tion F (Y,Υ). Under the mild assumption that
the matrix
NIJ = i(F̄IJ − FIJ), (47)
is non-degenerate, it thus follows that station-
ary points of Σ satisfy the attractor equations.
The macroscopic entropy is equal to the entropy
function taken at the attractor point. This im-
plies that the macroscopic entropy is the Legen-
dre transform of the free energy (45). An explicit
calculation yields the entropy formula [29,30,31],
Smacro(p, q) = πΣ
attractor
|Z|2 − 256 ImFΥ
Υ=−64
Here the first term represents a quarter of the
horizon area (in Planck units) so that the second
term defines the deviation from the Bekenstein-
Hawking area law. The entropy coincides pre-
cisely with the Wald entropy [16] as given by
the right-hand side of (26). In fact, the original
derivation of (48) was not based on an entropy
function and made direct use of the expression
(26).
In the absence of Υ-dependent terms, the ho-
mogeneity of the function F (Y ) implies that the
area scales quadratically with the charges, as was
discussed already at the beginning of section 2.
However, in view of the fact that Υ takes a fixed
BPS Black Holes 15
value, the second term will be subleading in the
limit of large charges.2 Note, however, that also
the area will contain subleading terms, as it de-
pends on Υ.
The entropy equation (48) has been confronted
with the result of microstate counting, for in-
stance, in the situation described in section 3.
In that case the effective supergravity action is
known and based on the function
F (Y,Υ) = −1
CABC Y
AY BY C
− c2A
24 · 64
Substituting this result into (48) and imposing
the attractor equations (35) with p0 = 0, one in-
deed derives the result (13) for the macroscopic
entropy [29]. The entropy formula (48) has also
been put to a test in other cases. Some of them
will be discussed in due course.
The relation between (44) and the entropy
function introduced in section 4 was discussed
in [25], where it was established that both en-
tropy functions lead to identical results for BPS
black holes. When the black holes are not BPS
(i.e. have no residual supersymmetry), then the
entropy function (44) is simply not applicable.
In this connection the question arises whether
other, independent, higher-derivative interactions
associated with matter multiplets will not con-
tribute to the entropy either. For instance, La-
grangians for tensor supermultiplets that con-
tain interactions of fourth order in space-time
derivatives, lead to terms quadratic in the Ricci
scalar that will in principle contribute to the
Wald entropy [32]. Indeed, for non-BPS black
holes these terms yield finite contributions to
the entropy, but for BPS black holes these cor-
rections vanish. A comprehensive treatment of
higher-derivative interactions is yet to be given
for N = 2 supergravity, but it seems that this
result is generic. At any rate, this observation
seems in line with more recent findings [33,34]
based on heterotic string α′-corrections encoded
in a higher-derivative effective action in higher
2 Here one usually assumes that F (Y,Υ) can be expanded
in positive powers of Υ.
dimensions. In four dimensions this action leads
to additional matter-coupled higher-derivative in-
teractions. When these are taken into account,
the matching of the macroscopic entropy with the
microscopic result is established [34].
Another modification concerns possible non-
holomorphic corrections to the function F (X,A).
This holomorphic function leads to a supersym-
metric action that corresponds to the so-called
effective Wilsonian action, based on integrating
out the massive degrees of freedom. The Wilso-
nian action describes the correct physics for en-
ergies between appropriately chosen infrared and
ultraviolet cutoffs. However, this action does not
reflect all the physical symmetries. To preserve
those symmetries non-holomorphic contributions
should be included associated with integrating
out massless degrees of freedom. In the special
case of heterotic black holes in N = 4 supersym-
metric compactifications, the requirement of ex-
plicit S-duality invariance of the entropy and the
attractor equations allows one to determine the
contribution from these non-holomorphic correc-
tions, as was first demonstrated in [31] for BPS
black holes. In [35,18] it was established that
non-holomorphic corrections to the BPS entropy
function (44) can be encoded into a real function
Ω(Y, Ȳ ,Υ, Ῡ) which is homogeneous of second
degree. The modifications to the entropy func-
tion are then effected by substituting F (Y,Υ) →
F (Y,Υ) + 2iΩ(Y, Ȳ ,Υ, Ῡ). There are good rea-
sons to expect that this same substitution should
be applied to the more general entropy function
based on (23) [25]. Note that when Ω is harmonic,
i.e., when it satisfies ∂2Ω/∂Y I ∂Ȳ J = 0, it can
simply be absorbed into the original holomorphic
funtion F (Y,Υ).
Finally we point out the existence of a formu-
lation in terms of real, rather than the complex
fields (Y I ,Υ), that we used before. This for-
mulation is manifestly covariant with respect to
electric/magnetic duality. We first decompose Y I
and FI into their real and imaginary parts,
Y I = xI + iuI , FI = yI + ivI , (50)
where FI = FI(Y,Υ). The real parametriza-
tion is obtained by taking (xI , yI ,Υ, Ῡ) instead
of (Y I , Ȳ I ,Υ, Ῡ) as the independent variables.
16 B. de Wit
This reparametrization is only well defined pro-
vided det(NIJ) 6= 0. Subsequently one defines the
Hesse potential, the real analogue of the Kähler
potential, which equals twice the Legendre trans-
form of the imaginary part of the prepotential
with respect to uI = ImY I ,
H(x, y,Υ, Ῡ) = 2 ImF (x+ iu,Υ, Ῡ)− 2 yI uI ,
Owing to the homogeneity of the function
F (Y,Υ) one can show that the free energy (45)
equals twice the Hesse potential. The entropy
function (44) is now replaced by
Σ(x, y, p, q) = 2H(x, y,Υ, Ῡ)− 2 qIxI + 2 pIyI ,
and it is straightforward to show that the ex-
tremization equations are just the attractor equa-
tions (35), expressed in terms of the new variables
(xI , yI). The value of Σ(x, y, p, q) at the attrator
point coincides again with the macroscopic en-
tropy.
6.2. Partial Legendre transforms
It is, of course, possible to define the macro-
scopic entropy as a Legendre transform with re-
spect to only a subset of the fields, by substitut-
ing part of the attractor equations such that the
variational principle remains valid. These par-
tial Legendre transforms constitute a hierarchy of
Legendre transforms for the black hole entropy.
Here we discuss two relevant examples, namely,
the one proposed in [36] where all the magnetic
attractor equations are imposed, and the dila-
tonic one for heterotic black holes, where only two
real potentials are left which together constitute
the complex dilaton field [31]. A possible dis-
advantage of considering partial Legendre trans-
forms is that certain invariances may no longer
be manifest. As it turns out, the dilatonic for-
mulation does not suffer from this shortcoming.
Apart from that, there is no reason to prefer one
version over the other. This will change in sec-
tion 7 when we discuss corresponding partition
functions and inverse Laplace transforms for the
microscopic degeneracies in semiclassical approx-
imation.
Let us first impose the magnetic attractor equa-
tions so that only the real parts of the Y I will re-
main as independent variables. Hence one makes
the substitution,
Y I = 1
(φI + ipI) . (53)
The entropy function (44) then takes the form,
Σ(φ, p, q) = FE(p, φ,Υ, Ῡ)− qI φI , (54)
where the corresponding free energy FE(p, φ)
equals
FE(p, φ) =
ImF (Y,Υ) + Ω(Y, Ȳ ,Υ, Ῡ)
Y I=(φI+ipI)/2
To derive this result one makes use of the
homogeneity of the functions F (Y,Υ) and
Ω(Y, Ȳ ,Υ, Ῡ). The latter function may contain
possible non-holomorphic terms. When extrem-
izing (55) with respect to φI we obtain the at-
tractor equations qI = ∂FE/∂φI . This shows
that the macroscopic entropy is a Legendre trans-
form of FE(p, φ) subject to Υ = −64, as was first
noted in [36] in the absence of Ω. In the latter
case this Legendre transform led to the conjecture
that there is a relation with topological strings, in
view of the fact that exp[FE] equals the modulus
square of the topological string partition function
[37]. We return to this in subsection 7.1.
Along the same line one can now proceed and
eliminate some of the φI as well. A specific exam-
ple of this is the dilatonic formulation heterotic
black holes, where we eliminate all the φI with
the exception of two of them which parametrize
the complex dilaton field. This leads to an en-
tropy function that depends only on the charges
and on the dilaton field [31,35]. Here it is conve-
nient to include all the Υ-dependent terms into
Ω, which also contains the non-holomorphic cor-
rections. The heterotic classical function F (Y ) is
given by
F (Y ) = −Y
1 Y aηabY
, a = 2, . . . , n, (56)
with real constants ηab. In the application that we
will be considering the function Ω depends only
BPS Black Holes 17
linearly on Υ and Ῡ, as well as on the dilaton
field S = −iY 1/Y 0 and its complex conjugate
S̄. The result for the BPS entropy function then
takes the form,
Σ(S, S̄, p, q) = −q
2 − ip · q (S − S̄) + p2 |S|2
S + S̄
+ 4Ω(S, S̄,Υ, Ῡ) ,
where q2, p2 and p · q are T-duality invariant bi-
linears of the various charges, defined by
q2 = 2q0p
1 − 1
abqb ,
p2 = −2p0q1 − 2paηabpb ,
q · p = q0p0 − q1p1 + qapa .
Note that these bilinears are not positive defi-
nite. Furthermore, Ω captures the Υ-dependent
corrections to the classical result (56). Its form
was derived for N = 4 heterotic string compact-
ifications by requiring S-duality of the attractor
equations and of the entropy [31,35],
Ω(Y, Ȳ ,Υ, Ῡ) =
Υ log η24(S) + Ῡ log η24(S̄)
(Υ + Ῡ) log(S + S̄)12
where η(S) denotes the Dedekind function. Note
the presence of the last term which is non-
holomorphic. This term is in accord with the
result for the effective action obtained from
five-brane instantons [38]. The attractor equa-
tions associated with the dilaton take the form
∂SΣ(S, S̄, p, q) = 0.
It is interesting to consider the consequences
of (57) in the classical case (Ω = 0). Then the
attractor equations yield the following values for
the real part of the dilaton field and the macro-
scopic entropy,
S + S̄ = − 2
p2q2 − (p · q)2 ,
Smacro = π
p2q2 − (p · q)2 .
Obviously there are two types of black holes, de-
pending on whether p2q2 − (p · q)2 is positive or
zero. In the context of N = 4 heterotic string
compactifications these correspond to 1/4- and
1/2-BPS black holes, respectively. The 1/4-BPS
states are dyonic, so that they necessarily carry
both electric and magnetic charges. The 1/2-BPS
states can be purely electric. We derived these
results from the BPS entropy function, but they
have also been obtained directly from the full su-
pergravity solutions [39,40,41].
Clearly the 1/4-BPS black holes are large black
holes as their area (entropy) is nonzero and scales
quadratically with the charges. Note that, de-
pending on the choice of the charges, the com-
plex dilaton field can remain finite in the limit
of large charges. This is relevant when studying
the asymptotic growth of the dyonic degeneracy
of 1/4-BPS dyons in heterotic string theory com-
pactified on a six-torus and in the related class of
heterotic CHL models [42]. These degeneracies
are encoded in automorphic forms Φk(ρ, σ, υ) of
weight k under Sp(2,Z), or an appropriate sub-
group thereof [43,44]. The torus compactifica-
tion corresponds to k = 10 and the CHL mod-
els to k = 1, 2, 4, 6. The three modular param-
eters, ρ, σ, υ, parametrize the period matrix of
an auxiliary genus-two Riemann surface, which
takes the form of a complex, symmetric, two-by-
two matrix. The microscopic degeneracy of 1/4-
BPS dyons is expressed as an integral over an
appropriate 3-cycle,
dk(p, q) =
dρ dσ dυ
eiπ[ρ p
2+σ q2+(2υ−1) p·q]
Φk(ρ, σ, υ)
Since Φk has zeros in the interior of the Siegel
half-space in addition to the zeros at the cusps,
the value of the integral (61) depends sensitively
on the choice of the integration 3-cycles. The
charges are in general integer, with the exception
of q1 which equals a multiple of N , and p
1 which
is fractional and quantized in units of 1/N . Here
N and k are related by (k+2)(N +1) = 24. The
inverse of the modular form Φk takes the form
of a Fourier sum with integer powers of exp[2πiρ]
and exp[2πiυ] and fractional powers of exp[2πiσ]
which are multiples of 1/N . The 3-cycle is then
defined by choosing integration contours where
the real parts of ρ and υ take values in the interval
(0, 1) and the real part of σ takes values in the
18 B. de Wit
interval (0, N). The formula (61) is manifestly
invariant under T-duality (the integrand depends
on the three T-duality invariant bilinears (58)),
as well as under S-duality, which is a subgroup of
the full modular group.
The integral (61) can be evaluated in saddle-
point approximation which yields the leading and
subleading contribtions to dk(p, q) [35,44]. As it
turns out these contributions are precisely en-
coded in (57) and (59), including the Dedekind
eta-functions and the non-holomorphic terms.
The presence of the non-holomorphic terms is
not surprising in view of the S-duality invariance.
Note that the expression (59) refers to k = 10 and
that there exist similar formulae for k = 1, 2, 4, 6.
The 1/2-BPS black holes are small black holes
as their area scales linearly with the charges. Ac-
cording to (60) their entropy (and horizon area)
vanishes while the dilaton field diverges at the
horizon, because we have p2 = p · q = 0. To de-
scribe the situation more accurately one retains
the leading term of (59). In that case one ob-
tains the following result (we restrict ourselves to
k = 10) [31],
S + S̄ ≈
|q2|/2 ,
Smacro ≈ 4 π
|q2|/2− 6 log |q2| ,
where the logarithmic term is due to the non-
holomorphic contribution. Because the dilaton
is large in this case, all the exponentials in the
Dedekind eta-function are suppressed and we are
at weak string coupling gs ∝ (S + S̄)−1/2.
We already stressed that small black holes have
a size of the order of the string scale and, in-
deed, these states are precisely generated by per-
turbative heterotic string states arising in N = 4
supersymmetric compactifications to four space-
time dimensions. In the supersymmetric right-
moving sector these states carry only momen-
tum and winding and contain no oscillations,
whereas in the left-moving sector oscillations are
allowed that satisfy the string matching condi-
tion. The oscillator number is then linearly re-
lated to q2. These perturbative states received
quite some attention in the past [45]. Because
the higher-mass string states are expected to be
within their Schwarzschild radius, it was conjec-
tured that they should have an interpretation as
black holes.3 Their calculable level density, pro-
portional to the exponent of 4π
|q2|/2, implies a
nonzero microscopic entropy for these black holes
[46].
This result was confronted with explicit black
hole solutions [47,48,49] based on standard super-
gravity Lagrangians that are at most quadratic
in derivatives, which have a vanishing horizon
area. Based on the area law one thus obtains
a vanishing macroscopic entropy. The fact that
(48), which takes into account higher-derivative
interactions, can nicely account for the discrep-
ancies encountered in the classical description of
the 1/2-BPS black holes, was first emphasized in
[50,51]. Note also that, since the electric states
correspond to perturbative heterotic string states,
their degeneracy is known from string theory and
given by
d(q) =
eiπσq
η24(σ)
≈ exp
|q2|/2− 27
log |q2|
where the integration contour encircles the point
exp(2πiσ) = 0. This large-|q2| approximation is
based on a standard saddle-point approximation.
Obviously the leading term of (63) is in agreement
with (62, but beyond that there is a disagreement
as the logarithmic corrections carry different co-
efficients. This discrepancy may be regarded as a
first indication that small black holes are not well
understood (for a disucssion, see, for instance,
[18]). Therefore we will mainly concentrate on
large black holes in the next chapter.
7. PARTITION FUNCTIONS AND IN-
VERSE LAPLACE TRANSFORMS
To again make the connection with microstate
degeneracies, we conjecture, in the spirit of [36],
that the Legendre transforms of the entropy are
indicative of a thermodynamic origin of the vari-
ous entropy functions. It is then natural to as-
sume that the corresponding free energies are
3 The idea that elementary particles, or string states, are
behaving like black holes, has been around for quite some
time.
BPS Black Holes 19
related to black hole partition functions corre-
sponding to suitable ensembles of black hole mi-
crostates. Following [18], we define
Z(φ, χ) =
{p,q}
d(p, q) eπ[qIφ
I−pIχI ] , (64)
where d(p, q) denotes the microscopic degenera-
cies of the black hole microstates with black hole
charges pI and qI . This is the partition sum
over a canonical ensemble, which is invariant un-
der the various duality symmetries, provided that
the electro- and magnetostatic potentials (φI , χI)
transform as a symplectic vector. Identifying a
free energy with the logarithm of Z(φ, χ), it is
clear that it should, perhaps in an appropriate
limit, be related to the macroscopic free energy
introduced earlier. On the other hand, viewing
Z(φ, χ) as an analytic function in φI and χI , the
degeneracies d(p, q) can be retrieved by an inverse
Laplace transform,
d(p, q) ∝
dχI dφ
I Z(φ, χ) eπ[−qIφ
I+pIχI ] ,
where the integration contours run, for instance,
over the intervals (φ − i, φ + i) and (χ − i, χ + i)
(we are assuming an integer-valued charge lat-
tice). Obviously, this makes sense as Z(φ, χ) is
formally periodic under shifts of φ and χ by mul-
tiples of 2i.
These arguments suggest to identify Z(φ, χ)
with the Hesse potential (51),
{p,q}
d(p, q) eπ[qIφ
I−pIχI ] ∼
shifts
e2πH(φ/2,χ/2,Υ,Ῡ) ,
where Υ is equal to its attractor value and where
we suppressed possible non-holomorphic contri-
butions for simplicity. However, the Hesse po-
tential is a macroscopic quantity which does not
in general exhibit the periodicity that is charac-
teristic for the partition function. Therefore, the
right-hand side of (66) requires an explicit peri-
odicity sum over discrete imaginary shifts of the φ
and χ.4 When substituting 2πH into the inverse
Laplace transform, we expect that the periodic-
ity sum can be incorporated into the integration
contour.
It is in general difficult to find an explicit repre-
sentation for the Hesse potential, as the relation
(50) between the complex variables Y I and the
real variables xI and yI is complicated. Therefore
we rewrite (66) in terms of the original variables
Y I and Ȳ I , where explicit results are known,
{p,q}
d(p, q) eπ[qI (Y+Ȳ )
I−pI(F̂+ ˆ̄F )I ] ∼
shifts
eπF(Y,Ȳ ,Υ,Ῡ) .
Here F equals the free energy (45) suitably mod-
ified with possible non-holomorphic corrections.
The latter requires that FI is changed into F̂I =
FI + 2iΩI , as was demonstrated in [18]. It is im-
portant to note that both sides of (67) (as well as
of (66)) are manifestly consistent with duality.
Again, it is possible to formally invert (67) by
means of an inverse Laplace transform,
d(p, q) ∝
d(Y + Ȳ )I d(F̂ + ˆ̄F )I e
πΣ(Y,Ȳ ,p,q)
dY dȲ ∆−(Y, Ȳ ) eπΣ(Y,Ȳ ,p,q) ,
where ∆−(Y, Ȳ ) is an integration measure whose
form depends on F̂I+
ˆ̄F I . The expression for ∆
as well as for a related determinant ∆+, reads as
follows,
∆±(Y, Ȳ ) =
ImFKL + 2Re(ΩKL ± ΩKL̄)
As before, FIJ and FI refer to Y -derivatives
of the holomorphic function F (Y,Υ) whereas
ΩIJ and ΩIJ̄ denote the holomorphic and mixed
holomorphic-antiholomorphic second derivatives
4In case that the Hesse potential exhibits a periodicity
with a different periodicity interval, then the sum over the
imaginary shifts will have to be modded out appropriately
such as to avoid overcounting.
20 B. de Wit
of Ω, respectively. In the absence of non-
holomorphic corrections ∆+ = ∆−.
A priori it is not clear whether the integral (68)
is well-defined and we refer to [18] for a discus-
sion. Note that the periodicity sum in (68) is
defined in terms of the variables φ and χ, which
should have some bearing on the integration con-
tour in (68). Leaving aside these subtle points
one may consider a saddle-point approximation of
the intergral representation (68). In view of the
previous results it is clear that the saddle point
coincides with the attractor point. Subsequently
one evaluates the semiclassical Gaussian integral
that emerges when expanding the exponent in the
integrand to second order in δY I and δȲ I about
the attractor point. When Y I − Ȳ I = ipI , the
resulting determinant factorizes into the square
roots of two subdeterminants,
∆+ and
Here the plus (minus) sign refers to the contribu-
tion of integrating over the real (imaginary) part
of δY I . Consequently, the result of a saddle-point
approximation applied to (68) yields,
d(p, q) =
∆−(Y, Ȳ )
∆+(Y, Ȳ )
attractor
eSmacro(p,q) . (70)
In the absence of non-holomorphic corrections the
ratio of the two determinants is equal to unity
and one recovers precisely the macroscopic en-
tropy. In the presence of non-holomorphic terms,
the deviations from unity are usually suppressed
in the limit of large charges, and one recovers the
leading and subleading corrections to the entropy
[18]. Of course, this is only the case when the
saddle-point approximation is appropriate.
Alternatively one may choose to integrate only
over the imaginary values of the fluctuations δY I
in saddle-point approximation. The saddle point
then occurs in the subspace defined by the mag-
netic attractor equations, so that one obtains a
modified version of the OSV integral [36],
d(p, q) ∝
∆−(p, φ) eπ[FE(p,φ)−qIφ
I ] ,
where FE(p, φ) was defined in (55) and ∆−(p, φ)
is defined in (69) with the Y I given by (53).
Hence this integral contains a non-trivial integra-
tion measure factor
∆− in order to remain con-
sistent with electric/magnetic duality. Without
the integral measure this is the integral conjec-
tured in [36]. In view of the original setting in
terms of the Hesse potential, we expect that the
integration contours in (71) should be taken along
the imaginary axes.
Inverting (71) to a partition sum over a mixed
ensemble, one finds,
Z(p, φ) =
d(p, q) eπ qIφ
shifts
∆−(p, φ) eπFE(p,φ) .
It should be noted that this expression and the
preceding one is less general than (68) because
it involves a saddle-point approximation. More-
over the function FE is not duality invariant and
the invariance is only recaptured when complet-
ing the saddle-point approximation with respect
to the fields φI . Therefore an evaluation of (71)
beyond the saddle-point approximation will most
likely give rise to a violation of (some of) the du-
ality symmetries again.
7.1. The integration measure and the
mixed partition function
As mentioned earlier, the partition function
Z(p, φ) was originally conjectured to be equal to
the modulus square of the partition function of
the topological string [36]. Soon thereafter, how-
ever, it was realized that this relationship must
be more subtle. The arguments in [18], based
on electric/magnetic duality clearly indicate some
of the missing ingredients, resulting in the mea-
sure factor
∆− and in the presence of the non-
holomorphic corrections. In fact, it was already
clear from an early analysis of small heterotic
black holes that T-duality was not conserved
when straightforwardly applying these ideas [52].
Although the presence of the measure factor cor-
rects for the lack of duality invariance, the semi-
classical results for small black holes seem to re-
main inconsistent with the analysis of microstate
counting, a fact we already alluded to at the end
of subsection 6.2.
BPS Black Holes 21
It is possible to test the result (72) in the
context of the 1/4-BPS states of the heterotic
N = 4 supersymmetric string compactifications,
by making use of the degeneracy formula (61).
Such a calculation was first performed in [53] and
it was subsequently generalized in [18] for more
general charge configurations and for CHL mod-
els. Using the degeneracies (61) one calculates
the mixed partition sum on the left-hand side of
(72). As it turns out, the resulting expression is
indeed proportional to the square of the partition
function of the topological string,
|Ztop(p, φ)|2 = e−2πi[F (Y,Υ)−F̄(Ȳ ,Ῡ] , (73)
where F (Y,Υ) is now the holomorphic part of
(56) and (59),
F (Y,Υ) = −Y
1 Y aηabY
log η24(−iY 1/Y 0) .
Here Υ = −64 and the Y I are given by (53).5
However, there is a non-trivial proportionality
factor, which, up to subleading contributions, we
expect to coincide with
∆−(p, φ) exp[4πΩnonholo] . (75)
The expression for Ωnonholo follows from the last
term in (59), and is thus equal to
exp[4π Ωnonholo] = (S + S̄)
−12 , (76)
where
S + S̄ =
2(p1φ0 − p0φ1)
(φ0)2 + (p0)2
. (77)
The factor (S + S̄)−12 cancels against a similar
factor in
∆− and, up to subleading terms, (75)
becomes
∆−(p, φ) exp[4πΩnonholo] ≈
i(Ȳ I F̂I − Y I F̂I)
2 |Y 0|2
e−K(Y,Ȳ ,Υ,Ῡ)
2 |Y 0|2
Indeed this result coincides with the result found
in [53,18]. The expression for e−K was already de-
fined in (38) up to non-holomorphic corrections.
5For convenience, we only refer to the case k = 10.
The latter can be dropped as they are subleading.
Note that e−K has been rescaled by replacing the
XI and A by Y I and Υ, respectively. This is to
be expected in view of the fact that the right-
hand side of (78) must be invariant under such
rescalings.
However, we should discuss a subtlety related
to the fact that we derived the expression for√
∆− in the context of N = 2 supergravity,
whereas the evaluation based on (61) is based on
N = 4 supersymmetric compactifications. This
means that the number of scalar moduli (related
to the number of N = 2 vector multiplets) is not
obviously the same. In the case ofN = 4 the rank
of the gauge group is equal to 28 (for simplicity
we restrict ourselves to the case k = 10). Of the
28 abelian vector gauge fields, 6 will correspond
to the graviphotons of pure N = 4 supergrav-
ity, and 22 will each belong to an N = 4 vector
supermultiplet. In the reduction to N = 2 super-
gravity, one of the graviphotons will be contained
in an additional N = 2 vector supermultiplet and
another one will play the role of the gravipho-
ton of N = 2 supergravity. There are thus 24
abelian vector gauge fields, of which 23 are asso-
ciated with N = 2 vector multiplets and one is
associated with the N = 2 graviphoton. The re-
maining 4 graviphotons are associated with two
N = 2 gravitino supermultipets, and these vector
fields seem to have no place in the N = 2 de-
scription. Therefore it is often assumed that the
effective rank of the gauge group in the N = 2
description should be taken equal to 24, rather
than to 28.
Nevertheless, in the calculation of
∆− leading
to (78) we assumed that the N = 2 description
is based on 28 vector fields, corresponding to 27
vector supermultiplets and one graviphoton field.
Only in that case, the factor (S + S̄)−12 cancels
so that one obtains the proportionality constant
noted in (78) based on the N = 2 expression for√
∆−. This somewhat confusing issue seems en-
tirely due to the fact that the N = 2 description
of an N = 4 theory is not fully understood, as the
corresponding description of N = 4 supergravity
is indeed based on 28 electrostatic potentials φ
which are contained in 28, rather than 24, ana-
logues of the N = 2 quantities Y I .
22 B. de Wit
In a recent series of papers [54,55,56] progress
was made towards a further understanding of the
relation between the mixed black hole partition
function and the partition function of the topo-
logical string. Unfortunately no evidence for the
presence of the integration measure factor in (71)
and (72) was presented. However, a more de-
tailed analysis for compact Calabi-Yau [57] mod-
els subsequently revealed the presence of the mea-
sure factor. Based on an extensive analysis of
the factorization formula for BPS indices, it is
shown that the partition function does not com-
pletely factorize into a holomophic and an anti-
holomorphic sector and the measure that is found
agrees (for p0 = 0) with (78). The power of |Y 0|
depends on whether one is discussing N = 2 or
N = 4 black holes. These results seem to be
in line with what was discussed in this section,
although there are many subtleties. Obviously
more work is needed to fully explore their conse-
quences.
The work described in these lectures is based on
various collaborations with Gabriel Lopes Car-
doso, Jürg Käppeli, Swapna Mahapatra, Thomas
Mohaupt and Frank Saueressig. I thank Gabriel
Lopes Cardoso for discussion and valuable com-
ments on the manuscript.
This work is partly supported by NWO
grant 047017015, EU contracts MRTN-CT-2004-
005104 and MRTN-CT-2004-512194, and INTAS
contract 03-51-6346.
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	INTRODUCTION
	DUAL PERSPECTIVES
	BLACK HOLES IN M/STRING THEORY – AN EXAMPLE
	ATTRACTOR EQUATIONS
	N=2 SUPERGRAVITY
	Supermultiplets
	Supersymmetric actions
	Compensator multiplets
	BPS ATTRACTORS
	The BPS entropy function
	Partial Legendre transforms
	PARTITION FUNCTIONS AND INVERSE LAPLACE TRANSFORMS
	The integration measure and the mixed partition function

  The entropy of BPS black holes in four space-time dimensions is discussed
from both macroscopic and microscopic points of view.

7 BPS Black Holes
Bernard de Wita
aInstitute for Theoretical Physics & Spinoza Institute,
Utrecht University, Utrecht, The Netherlands
The entropy of BPS black holes in four space-time dimensions is discussed from both macroscopic and micro-
scopic points of view.
1. INTRODUCTION
Classical black holes are solutions of Einstein’s
equations of general relativity that exhibit an
event horizon. From inside this horizon, noth-
ing (and in particular, no light) can escape. The
region inside the horizon is therefore not in the
backward lightcone of future timelike infinity.
However, since the discovery of Hawking radia-
tion [1], it has become clear that many of the
classical features of black holes will be subject to
change.
In these lecture we consider static, spherically
symmetric black holes in four space-time dimen-
sions that carry electric and/or magnetic charges
with a flat space-time geometry at spatial in-
finity. Such solutions exist in Einstein-Maxwell
theory, the classical field theory of gravity and
electromagnetism. The most general static black
holes of this type correspond to the Reissner-
Nordstrom solutions and are characterized by a
charge Q and a mass M . In the presence of mag-
netic charges, Q is replaced by
q2 + p2 in most
formulae, where q and p denote the electric and
the magnetic charge, respectively. Hence there
is no need to distinguish between the two types
of charges. For zero charges one is dealing with
Schwarzschild black holes.
Two quantities associated with the black hole
horizon are its area A and the surface gravity κs.
The area is simply the area of the two-sphere de-
fined by the horizon. The surface gravity, which
is constant on the horizon, is related to the force
(measured at spatial infinity) that holds a unit
test mass in place. On the other hand, the mass
M and charge Q of the black hole are not primar-
ily associated with the horizon and are defined in
terms of appropriate surface integrals at spatial
infinity.
As is well known, there exists a striking cor-
respondence between the laws of thermodynam-
ics and the laws of black hole mechanics [2]. Of
particular importance is the first law, which, for
thermodynamics, states that the variation of the
total energy is equal to the temperature times
the variation of the entropy, modulo work terms,
for instance proportional to a change of the vol-
ume. The corresponding formula for black holes
expresses how the variation of the black hole mass
is related to the variation of the horizon area, up
to work terms proportional to the variation of the
angular momentum. In addition there can also be
a term proportional to a variation of the charge,
multiplied by the electric/magnetic potential µ at
the horizon. Specifically, the first law of thermo-
dynamics, δE = T δS − p δV , translates into
+ µ δQ+Ω δJ . (1)
The reason for factorizing the first term on the
right-hand side in this particular form, is that
κs/2π equals the Hawking temperature [1]. This
then leads to the identification of the black hole
entropy in terms of the horizon area,
Smacro = 14A , (2)
a result that is known as the area law [3]. In
these equations the various quantities have been
defined in Planck units, meaning that they have
http://arxiv.org/abs/0704.1452v1
2 B. de Wit
been made dimensionless by multiplication with
an appropriate power of Newton’s constant (we
will set ~ = c = 1). This constant appears
in the Einstein-Hilbert Lagrangian according to
LEH = −(16πGN)−1
|g|R. With this normal-
ization the quantities appearing in the first law
are independent of the scale of the metric.
Einstein-Maxwell theory can be naturally em-
bedded into N = 2 supergravity. This super-
gravity theory has possible extensions with sev-
eral abelian gauge fields and a related number of
massless scalar fields (often called ‘moduli’ fields,
for reasons that will become clear later on). At
spatial infinity these moduli fields will tend to a
constant, and the black hole mass will depend
on these constants, thus introducing additional
terms on the right-hand side of (1).
For Schwarzschild black holes the only relevant
parameter is the mass M and we note the follow-
ing relations,
A = 16πM2 , κs =
, (3)
consistent with (1). For the Reissner-Nordstrom
black hole, the situation is more subtle. Here one
distinguishes three different cases. For M > Q
one has non-extremal solutions, which exhibit two
horizons, an exterior event horizon and an interior
horizon. When M = Q one is dealing with an
extremal black hole, for which the two horizons
coalesce and the surface gravity vanishes. In that
case one has
A = 4πM2 , κs = 0 , µ = Q
. (4)
It is straightforward to verify that this result is
consistent with (1) for variations δM = δQ and
κs = 0. Because the surface gravity vanishes, one
might expect the entropy to vanish as well, as is
suggested by the third law of thermodynamics.
However, this is not the case because the hori-
zon area remains finite for zero surface gravity.
Finally, solutions with M < Q are not regarded
as physically acceptable. Their total energy is
less than the electromagnetic energy alone and
they no longer have an event horizon but exhibit
a naked singularity. Hence extremal black holes
saturate the bound M ≥ Q for physically accept-
able black hole solutions.
When embedding Einstein-Maxwell theory into
a complete supergravity theory, the above classifi-
cation has an interpretation in terms of the super-
symmetry algebra. This algebra has a central ex-
tension proportional to the black hole charge(s).
Unitary representations of the supersymmetry
algebra must necessarily have masses that are
larger than or equal to the charge. When this
bound is saturated, one is dealing with so-called
BPS supermultiplets. Such supermultiplets are
smaller than the generic massiveN = 2 supermul-
tiplets and have a different spin content. Because
of this, BPS states are stable under (adiabatic)
changes of the coupling constants and the rela-
tion between charge and mass remains preserved.
This important feature of BPS states will be rele-
vant for what follows. In these lectures BPS black
hole solutions are defined by the fact that they
have some residual supersymmetry, so that they
saturate a bound implied by the supersymmetry
algebra.
So far we did not refer to the explicit
Schwarzschild or Reissner-Nordstrom black hole
solutions, which can be found in many places
in the literature. One feature that should be
stressed, concerns the near-horizon geometry. For
extremal, static and spherically symmetric black
holes, this geometry is restricted to the product of
the sphere S2 and an anti-de Sitter space AdS2,
corresponding to the line element,
ds2 = −r2dt2 + dr
2 + r2(dθ2 + sin2 θ dϕ2)
. (5)
In these coordinates the horizon is located at
r = 0, where the timelike Killing vector K = ∂t
turns lightlike. Such a horizon is called a Killing
horizon.
In these notes we discuss various aspects of the
relation between black hole solutions and corre-
sponding microscopic descriptions. Section 2 gen-
erally describes the macroscopic (field theoretic)
and microscopic (statistical) approach to black
hole entropy and indicates why they are related.
Section 3 summarizes the calculation of the black
hole entropy based on a fivebrane wrapping on
a Calabi-Yau four-cycle in a compactification of
BPS Black Holes 3
M-Theory on the product space of a Calabi-Yau
threefold and a circle. In section 4 the attractor
equations are discussed for extremal black holes
on the basis of a variational principle defined for
a generic gravitational theory. Section 5 contains
a brief review of N = 2 supergravity and the su-
perconformal multiplet calculus. This material is
used in the description of the BPS attractor equa-
tions in N = 2 supergravity presented in section
6. In this case there also exists is a corresponding
formulation in terms of a BPS entropy function.
Finally section 7 discusses the relation between
this entropy function and the black hole partition
function.
We try to concentrate as much as possible on
the conceptual basis of the underlying ideas, but
these notes cannot do justice to all aspects rele-
vant for the study of black holes in string theory,
such as use of the AdS3/CFT2 correspondence,
black holes or related objects in other than four
space-time dimensions, the relation with indices
for BPS states, and the like. We refer to some
recent reviews where some of these topics have
been discussed [4,5,6,7].
2. DUAL PERSPECTIVES
A central question in black hole thermodynam-
cis concerns the statistical interpretation of the
black hole entropy. String theory has provided
new insights here [8], which enable the identi-
fication of the black hole entropy as the loga-
rithm of the degeneracy of states d(Q) of charge
Q belonging to a certain system of microstates.
In string theory these microstates are provided
by the states of wrapped brane configurations of
given momentum and winding. When calculating
the black hole solutions in the orresponding ef-
fective field theory with the charges specified by
the brane configuration, one discovers that the
black hole area is equal to the logarithm of the
brane state degeneracy, at least in the limit of
large charges. We will be reviewing some aspects
of this remarkable correspondence here.
The horizon area, which is expected to be pro-
portional to the macroscopic entropy according
to the Bekenstein-Hawking area law, turns out
to grow quadratically with the charges Q. After
converting to string units the radius of a black
hole is therefore proportional to
Rhorizon
lstring
∼ gs Q , (6)
where lstring and gs are the string length and cou-
pling constant, respectively. Since we will be as-
suming that the charges are large, the black holes
are generically much larger than the string scale.
Consequently these black holes are called large
and can be identified with the macroscopic black
holes we have been discussing earlier. However,
there are also situations where the leading contri-
bution to the area is only linear in the charges Q.
In that case (6) is replaced by
Rhorizon
lstring
Q , (7)
Moreover, in that case the string coupling (in-
versely proportional to the dilaton field) cannot
be taken constant, but tends to zero for large
charges according to gs ∼ Q−1/2, so that the ra-
dius of the black hole remains comparable to the
string scale. These black holes are called small.
Their corresponding classical supergravity solu-
tions exhibit a vanishing horizon area and a dila-
ton field that diverges at the horizon. To reliably
compute the right-hand side of (7) therefore re-
quires to include appropriate terms in the effec-
tive action of higher order in space-time deriva-
tives. We return to this issue in subsection 6.2.
To further understand the relation between a
field-theoretic description and a microscopic de-
scription it is relevant that strings live in more
that four space-time dimensions. In most situ-
ations the extra dimensions are compactified on
some internal manifold X and one is dealing with
the standard Kaluza-Klein scenario leading to ef-
fective field theories in four dimensions, describ-
ing low-mass modes of the fields associated with
appropriate eigenfunctions on the internal mani-
fold. Locally, the original space-time is a product
M4×X , where M4 denotes the four-dimensional
space-time that we experience in daily life. In
this situation there exists a corresponding space
X at every point xµ ofM4, whose size is such that
it will not be directly observable. However, this
4 B. de Wit
space X does not have to be the same at every
point in M4, and moving through M4 one may
encounter various spaces X that are not necessar-
ily equivalent. In principle they belong to some
well-defined class of fixed topology parametrized
by certain moduli. These moduli will appear as
fields in the four-dimensional effective field the-
ory. For instance, suppose that the spaces X are
n-dimensional tori T n. The metric of T n will ap-
pear as a field in the four-dimensional theory and
is related to the torus moduli. Hence, when deal-
ing with a solution of the four-dimensional theory
that is not constant in M4, each patch in M4 has
a non-trivial image in the space of moduli that
parametrize the internal spaces X .
Let us now return to a black hole solution,
viewed in this higher-dimensional perspective.
The fields, and in particular the four-dimensional
space-time metric, will vary nontrivially overM4,
and so will the internal spaces X . When moving
to the center of the black hole the gravitational
fields will become strong and the local product
structure intoM4×X could break down. Conven-
tional Kaluza-Klein theory does not have much
to say about what happens, beyond the fact that
the four-dimensional solution can be lifted to the
higher-dimensional one, at least in principle.
However, there is a feature of string theory that
is absent in a purely field-theoretic approach. In
the effective field-theoretic context only the lo-
cal degrees of freedom of strings and branes are
captured. But extended objects may also carry
global degrees of freedom, as they can also wrap
themselves around non-trivial cycles of the inter-
nal space X . This wrapping tends to take place
at a particular position inM4, so in the context of
the four-dimensional effective field theory this will
reflect itself as a pointlike object. The wrapped
object is the string theory representation of the
black hole!
We are thus dealing with two complementary
pictures of the black hole. One is based on general
relativity where a point mass generates a global
solution of space-time with strongly varying grav-
itational fields, which we shall refer to as the
macroscopic description. The other one, based
on the internal space where an extended object is
entangled in one of its cycles, does not immedi-
ately involve gravitational fields and can easily be
described in flat space-time. This description will
be refered to as microscopic. To understand how
these two descriptions are related is far from easy,
but a connection must exist in view of the fact
that gravitons are closed string states which inter-
act with the wrapped branes. These interactions
are governed by the string coupling constant gs
and we are thus confronted with an interpolation
in that coupling constant. In principle, such an
interpolation is very difficult to carry out, so that
a realistic comparison between microscopic and
macroscopic results is usually impossible. How-
ever, reliable predictions are possible for extremal
black holes that are BPS! As we indicated earlier,
in that situation there are reasons to trust such
interpolations. Indeed, it has been shown that
the predictions based on these two alternative de-
scriptions can be successfully compared and new
insights about black holes can be obtained.
But how do the wrapped strings and branes
represent themselves in the effective action de-
scription and what governs their interactions with
the low-mass fields? Here it is important to real-
ize that the massless four-dimensional fields are
associated with harmonic forms on X . Harmonic
forms are in one-to-one correspondence with so-
called cohomology groups consisting of equiva-
lence classes of forms that are closed but not
exact. The number of independent harmonic
forms of a given degree is given by the so-called
Betti numbers, which are fixed by the topology
of the spaces X . When expanding fields in a
Kaluza-Klein scenario, the number of correspond-
ing massless fields can be deduced from an expan-
sion in terms of tensors onX corresponding to the
various harmonic forms. The higher-dimensional
fields Φ(x, y) thus decompose into the massless
fields φA(x) according to (schematically),
Φ(x, y) = φA(x) ωA(y) , (8)
where ωA(y) denotes the independent harmonic
forms on X . The above expression, when sub-
stituted into the action of the higher-dimensional
theory, leads to interactions of the fields φA pro-
portional to the ‘coupling constants’,
CABC··· ∝
ωA ∧ ωB ∧ ωC · · · . (9)
BPS Black Holes 5
These constants are known as intersection num-
bers, for reasons that will become clear shortly.
We already mentioned that the Betti numbers
depend on the topology of X . This is related to
Poincaré duality, according to which cohomology
classes are related to homology classes. The latter
consist of submanifolds of X without boundary
that are themselves not a boundary of some other
submanifold of X . This is precisely relevant for
wrapped branes which indeed cover submanifolds
of X , but are not themselves the boundary of a
submanifold because otherwise the brane could
collapse to a point. Without going into detail,
this implies that there exists a dual relationship
between harmonic p-forms ω and (dX −p)-cycles,
where dX denotes the dimension of X . We can
therefore choose a homology basis for the (dX−p)-
cycles dual to the basis adopted for the p-forms.
Denoting this basis by ΩA the wrapping of an ex-
tended object can now be characterized by spec-
ifying its corresponding cycle P in terms of the
homology basis,
P = pAΩA . (10)
The integers pA count how many times the ex-
tended object is wrapped around the correspond-
ing cycle, so we are actually dealing with integer-
valued cohomology and homology. The wrap-
ping numbers pA reflect themselves as magnetic
charges in the effective action. The electric
charges are already an integer part of the effective
action, because they are associated with gauge
transformations that usually originate from the
higher-dimensional theory.
Owing to Poincaré duality it is thus very nat-
ural that the winding numbers interact with the
massless modes in the form of magnetic charges,
so that they can be incorporated in the effective
action. Before closing this section, we note that,
by Poincaré duality, we can express the number
of intersections by
P · P · P · · · = CABC··· pApBpC · · · . (11)
This is a topological characterization of the wrap-
ping, which will appear in later formulae.
3. BLACK HOLES IN M/STRING THE-
ORY – AN EXAMPLE
As an example we now discuss the black hole
entropy derived from microscopic arguments in
a special case. In a later section we will con-
sider the corresponding expression for the macro-
scopic entropy. We start from M-theory, which is
the strong coupling limit of type-IIA string the-
ory. Its massless states are described by eleven-
dimensional supergravity. The latter is invariant
under 32 supersymmetries. Seven of the eleven
space-time dimensions are compactified on an in-
ternal space which is the product of a Calabi-Yau
threefold (a Ricci flat three-dimensional complex
manifold, henceforth denoted by CY3) times a cir-
cle S1. Such a space induces a partial breaking of
supersymmetry which leaves 8 supersymmetries
unaffected. In the context of the four-dimensional
space-time M4, these 8 supersymmetries are en-
coded into two independent Lorentz spinors and
for that reason this symmetry is referred to as
N = 2 supersymmetry. Hence the effective four-
dimensional field theory will be some version of
N = 2 supergravity.
M-theory contains a five-brane and this is the
microscopic object that is responsible for the
black holes that we consider: the five-brane has
wrapped itself on a 4-cycleP of the CY 3 space [9].
Alternatively one may study this class of black
holes in type-IIA string theory, with a 4-brane
wrapping the 4-cycle [10]. The 4-cycle is subject
to certain requirements which will be mentioned
in due course.
The massless modes captured by the effective
field theory correspond to harmonic forms on the
CY3 space; they do not depend on the S
1 co-
ordinate. The 2-forms are of particular interest.
In the effective theory they give rise to vector
gauge fields Aµ
A, which originate from the rank-
three tensor gauge field in eleven dimensions. In
addition there is an extra vector field Aµ
0 cor-
responding to a 0-form which is related to the
graviphoton associated with S1. This field will
couple to the electric charge q0 associated with
momentum modes on S1 in the standard Kaluza-
Klein fashion. The 2-forms are dual to 4-cyles
and the wrapping of the five-brane is encoded in
6 B. de Wit
terms of the wrapping numbers pA, which appear
in the effective field theory as magnetic charges
coupling to the gauge fields Aµ
A. Here we see
Poincaré duality at work, as the magnetic charges
couple nicely to the corresponding gauge fields.
In view of the fact that the product of three 2-
forms defines a 6-form that can be integrated over
the CY3 space, there exist non-trivial triple inter-
section numbers CABC . These numbers will ap-
pear in the three-point couplings of the effective
field theory. There is a subtle topological fea-
ture that we have not explained before, which is
typical for complex manifolds containing 4-cycles,
namely the existence of another quantity of topo-
logical interest known as the second Chern class.
The second Chern class is a 4-form whose inte-
gral over a four-dimensional Euclidean space de-
fines the instanton number. The 4-form can be
integrated over the 4-cycle P and yields c2A pA,
where the c2A are integers.
Let us now turn to the microscopic counting of
degrees of freedom [9]. These degrees of freedom
are associated with the massless excitations of the
wrapped five-brane characterized by the wrap-
ping numbers pA on the 4-cycle. The 4-cycle P
must correspond to a holomorphically embedded
complex submanifold in order to preserve 4 super-
symmetries. The massless excitations of the five-
brane are then described by a (1+1)-dimensional
superconformal field theory (the reader may also
consult [11]). Because we have compactified the
spatial dimension on S1, we are dealing with a
closed string with left- and right-moving states.
The 4 supersymmetries of the conformal field the-
ory reside in one of these two sectors, say the
right-handed one. Conformal theories in 1+1 di-
mensions are characterized by a central charge,
and in this case there is a central charge for the
right- and for the left-moving sector separately.
The two central charges are expressible in terms
of the wrapping numbers pA and depend on the
intersection numbers and the second Chern class,
according to
cL = CABC p
ApBpC + c2A p
cR = CABC p
ApBpC + 1
c2A p
Here we should stress that the above result is far
from obvious and holds only under the condition
that the pA are large. In that case every generic
deformation of P will be smooth. Under these cir-
cumstances it is possible to relate the topological
properties of the 4-cycle to the topological data
of the Calabi-Yau space.
We can now choose a state of given momentum
q0 which is supersymmetric in the right-moving
sector. From rather general arguments it fol-
lows that such states exist. The corresponding
states in the left-moving sector have no bearing
on the supersymmetry and these states have a
certain degeneracy depending on the value of q0.
In this way we have a tower of degenerate BPS
states invariant under 4 supersymmetries, built
on a supersymmetric state in the right-moving
sector. We can then use Cardy’s formula, which
states that the degeneracy of states for fixed but
large momentum (large as compared to cL) equals
exp[2π
|q0| cL/6]. This leads to the following ex-
pression for the entropy,
Smicro(p, q) =
|q̂0|(CABC pApBpC + c2A pA) ,
where q0 has been shifted according to
q̂0 = q0 +
CABqAqB . (14)
Here CAB is the inverse of CAB = CABCp
C . This
modification is related to the fact that the electric
charges associated with the gauge fields Aµ
A will
interact with the M-theory two-brane [9]. The
existence of this interaction can be inferred from
the fact that the two-brane interacts with the
rank-three tensor field in eleven dimensions, from
which the vector gauge fields Aµ
A originate.
We stress that the above results apply in the
limit of large charges. The first term propor-
tional to the triple intersection number is obvi-
ously the leading contribution whereas the terms
proportional to the second Chern class are sub-
leading. The importance of the subleading terms
will become more clear in later sections. Having
obtained a microscopic representation of a BPS
black hole, it now remains to make contact with
it by deriving the corresponding black hole solu-
tion directly in the N = 2 supergravity theory.
This is discussed in some detail in section 6.
BPS Black Holes 7
4. ATTRACTOR EQUATIONS
The microscopic expression for the black hole
entropy depends only on the charges, whereas a
field-theoretic calculation can in principle depend
on other quantities, such as the values of the mod-
uli fields at the horizon. To establish any agree-
ment between these two approaches, the moduli
(as well as any other relevant fields that enter the
calculation) must take fixed values at the horizon
which may depend on the charges. As it turns out
this is indeed the case for extremal black hole so-
lutions, as was first demonstrated in the context
of N = 2 supersymmetric black holes [12,13,14].
The values taken by the fields at the horizon are
independent of their asymptotic values at spatial
infinity. This fixed point behaviour is encoded in
so-called attractor equations.
In the presence of higher-derivative interac-
tions it is very difficult to explicitly construct
black hole solutions and to exhibit the attractor
phenomenon. However, by concentrating on the
near-horizon region one can usually determine the
fixed-point values directly without considering
the interpolation between the horizon and spa-
tial infinity. Provided the symmetry of the near-
horizon region is restrictive enough, the attrac-
tor phenomenon can be described conveniently in
terms of a variational principle for a so-called en-
tropy function. The stationarity of this entropy
function then yields the attractor equations and
its value at the attractor point equals the macro-
scopic entropy. The purpose of the present sec-
tion is to explain this phenomenon.
For N = 2 BPS black holes with higher-
derivative interactions the attractor equations fol-
low from classifying possible solutions with full
supersymmetry [15]. As it turns out supersym-
metry determines the near-horizon geometry (and
thus the horizon area), the values of the moduli
fields in terms of the charges and the value of the
entropy as defined by the Noether charge defini-
tion of Wald [16]. This result can also be de-
scribed in terms of a variational principle [17,18],
as we shall explain in section 6.
Let us now turn to the more generic case of ex-
tremal, static, spherically symmetric black holes
that are not necessarily BPS, following the ap-
proach of [19] which is based on an action prin-
ciple. Similar approaches can be found elsewhere
in the literature, for instance, in [20,21,22]. When
dealing with spherically symmetric solutions, one
integrates out the spherical degrees of freedom
and obtains a reduced action for a 1 + 1 dimen-
sional field theory which fully describes the black
hole solutions. Here we consider a general system
of abelian vector gauge fields, scalar and other
matter fields coupled to gravity. The geometry
is thus restricted to the product of the sphere S2
and a 1 + 1 dimensional space-time, and the de-
pendence of the fields on the S2 coordinates θ
and ϕ is fixed by symmetry arguments. For the
moment we will not make any assumption regard-
ing the dependence on the remaining two cooor-
dinates r and t. Consequently we write the gen-
eral field configuration consistent with the various
isometries as
ds2(4) = gµνdx
= ds2(2) + v2
dθ2 + sin2 θ dϕ2
I = eI , Fθϕ
sin θ .
The Fµν
I denote the field strengths associated
with a number of abelian gauge fields. The θ-
dependence of Fθϕ
I is fixed by rotational invari-
ance and the pI denote the magnetic charges. The
latter are constant by virtue of the Bianchi iden-
tity, but all other fields are still functions of r
and t. As we shall see in a moment the fields eI
are dual to the electric charges. The radius of
S2 is defined by the field v2. The line element of
the 1+1 dimensional space-time will be expressed
in terms of the two-dimensional metric ḡij , whose
determinant will be related to a field v1 according
|ḡ| . (16)
Eventually ḡij will be taken proportional to an
AdS2 metric,
ds2(2) = ḡij dx
idxj = v1
− r2 dt2 + dr
. (17)
In addition to the fields eI , v1 and v2, there may
be a number of other fields which for the moment
we denote collectively by uα.
8 B. de Wit
As is well known theories based on abelian vec-
tor fields are subject to electric/magnetic dual-
ity, because their equations of motion expressed
in terms of the dual field strengths,1
GµνI =
|g| εµνρσ
∂FρσI
, (18)
take the same form as the Bianchi identities for
the field strengths Fµν
I . Adopting the conven-
tions where xµ = (t, r, θ, ϕ) and εtrθϕ = 1, and
the signature of the space-time metric equals
(−,+,+,+), which is consistent with (17), it fol-
lows that, in the background (15),
Gθϕ I = −v1v2 sin θ
Grt I = −4π v1v2
These two tensors can be written as qI sin θ/(4π)
and fI . The quantities qI and fI are conjugate
to pI and eI , respectively, and can be written as
qI(e, p, v, u) = −4π v1v2
fI(e, p, v, u) = −4π v1v2
They depend on the constants pI and on the fields
eI , v1,2 and uα, and possibly their t and r deriva-
tives, but no longer on the S2 coordinates θ and
ϕ. Upon imposing the field equations it follows
that the qI are constant and correspond to the
electric charges. Our aim is to obtain a descrip-
tion in terms of the charges pI and qI , rather than
in terms of the pI and eI .
Electric/magnetic duality transformation are
induced by rotating the tensors Fµν
I andGµν I by
a constant transformation, so that the new linear
combinations are all subject to Bianchi identities.
Half of them are then selected as the new field
strengths defined in terms of new gauge fields,
while the Bianchi identities on the remaining lin-
ear combinations are regarded as field equations
belonging to a new Lagrangian defined in terms
1We assume that the Lagrangian is a function of the
abelian field strengths and does not depend explicitly on
the gauge fields.
of the new field strengths. In order that this du-
alization can be effected the rotation of the ten-
sors must belong to Sp(2n + 2;R), where n + 1
denotes the number of independent gauge fields
[23]. Naturally the duality leads to new quan-
tities (p̃I , q̃I) and (ẽ
I , f̃I), related to the original
ones by the same Sp(2n+2;R) rotation. Since the
charges are not continuous but will take values in
an integer-valued lattice, this group should even-
tually be restricted to an appropriate arithmetic
subgroup.
Subsequently we define the reduced Lagrangian
by the integral of the full Lagrangian over S2,
F(e, p, v, u) =
dθ dϕ
|g| L , (21)
We note that the definition of the conjugate quan-
tities qI and fI takes the form,
qI = −
, fI = −
. (22)
It is known that a Lagrangian does not trans-
form as a function under electric/magnetic dual-
ities (see, e.g. [24]), but one can generally show
that the following combination does [25],
E(q, p, v, u) = −F(e, p, v, u)− eIqI . (23)
More precisely, this quantity transforms un-
der electric/magnetic duality according to
Ẽ(q̃, p̃, v, u) = E(q, p, v, u). In view of the first
equation (22), the definition (23) takes the form
of a Legendre transform. Furthermore the field
equations imply that the qI are constant and that
the action,
dtdr E , is stationary under varia-
tions of the fields v and u, while keeping the pI
and qI fixed. This is to be expected as E is in fact
minus the Hamiltonian density associated with
the reduced Lagrangian density (21), at least as
far as the vector fields are concerned.
In the near-horizon background (17), assuming
fields that are invariant under the AdS2 isome-
tries, the generally covariant derivatives of the
fields vanish and the function E depends only on
the fields which no longer depend on the coordi-
nates. The equations of motion then imply that
the values of the fields v1,2 and uα are determined
BPS Black Holes 9
by demanding E to be stationary under variations
of v and u,
= 0 , qI = constant . (24)
The function 2π E(q, p, v, u) then coincides with
the entropy function proposed by Sen [19]. The
first two equations of (24) are interpreted as the
attractor equations. At the attractor point one
may prove
attractor
dθ dϕ
|g|Rrtrt
∂Rrtrt
, (25)
where the right-hand side is evaluated in the near-
horizon geometry. This leads to the expression
attractor
∂Rµνρσ
εµνερσ , (26)
where Σhor denotes a spacelike cross section of
the Killing horizon, and εµν the normal bivector
which acts in the space normal to Σhor. This is
precisely the expression for the Wald entropy [16]
(applied to this particular case). For the Einstein-
Hilbert action, (26) equals a quarter of the hori-
zon area in Planck units. For more general La-
grangians (26) may lead to deviations from the
area law, as we will see in due course. Note that
the entropy function does not necessarily depend
on all fields at the horizon. The values of some
of the fields will then be left unconstrained, but
those will not appear in the expression for the
Wald entropy either.
The above derivation of the entropy function
applies to any gauge and general coordinate in-
variant Lagrangian, including Lagrangians with
higher-derivative interactions. In the absence of
higher-derivative terms, the reduced Lagrangian
F is at most quadratic in eI and pI and the Leg-
endre transform (23) can easily be carried out.
The results coincide with corresponding terms in
the so-called black hole potential discussed in e.g.
[20,21,22].
5. N=2 SUPERGRAVITY
In the previous section the symmetry of the
near-horizon geometry played a crucial role. For
BPS black holes the supersymmetry enhancement
at the horizon is the crucial input that constrains
both some of the fields at the horizon as well as
the near-horizon geometry itself. The black hole
solutions that we will be considering have a resid-
ual N = 1 supersymmetry (so that they are BPS)
and are solitonic interpolations between N = 2
configurations at the horizon and at spatial infin-
ity. Obviously to describe BPS black holes one
needs to consider extended supergravity theories.
In view of the application described in section 3,
N = 2 supergravity is relevant. Moreover, N = 2
supergravity has off-shell formulations (meaning
that supersymmetry transformations realize the
supersymmetry algebra without the need for im-
posing field equations associated with a specific
Lagrangian) and this facilitates the calculations
in an essential way. This aspect is especially im-
portant because we will be considering supersym-
metric Lagrangians with interactions containing
more than two derivatives.
In the following subsections we present a brief
introduction to N = 2 supergravity. Supermul-
tiplets are introduced in subsection 5.1 and su-
persymmetric actions in subsection 5.2. Finally,
in subsection 5.3, we elucidate the use of compen-
sating fields and corresponding supermultiplets to
familiarize the reader with the principles under-
lying the superconformal multiplet calculus. Fur-
ther details can be found in the literature [26,27].
5.1. Supermultiplets
In this subsection we briefly introduce the su-
permultiplets that play a role in the following. Of
particular interest are the vector and the Weyl su-
permultiplet. Other multiplets are the tensor su-
permultiplets and the hypermultiplets, but those
will not be discussed as they only play an ancil-
lary role.
The covariant fields and field strengths of the
various gauge fields belonging to the vector or to
the Weyl supermultiplet comprise a chiral mul-
tiplet. Such multiplets are described in super-
space by chiral superfields.At this point it is con-
venient to systematize our discussion by using su-
perspace notions, although we do not intend to
make an essential use of superfields. Scalar chiral
fields in N = 2 superspace have 16 + 16 bosonic
+ fermionic field components, but there exists a
10 B. de Wit
constraint which reduces the superfield to only
8+8 field components. This constraint expresses
higher-θ components of the superfield in terms
of lower-θ components or space-time derivatives
thereof. The vector supermultiplet and the Weyl
supermultiplet are both related to reduced chiral
multiplets. Note, however, that products of re-
duced chiral superfields constitute general chiral
fields.
The vector supermultiplet is described by a
scalar reduced chiral superfields, whose lowest-
θ component is a complex field which we de-
note by X . Then there is a doublet of chiral
fermions Ωi, where i is an SU(2) R-symmetry
doublet index. The position of the index i in-
dicates the chirality of the spinor field: Ωi car-
ries positive, and Ωi negative chirality. The fields
X̄ and Ωi appear as lowest-θ components in the
anti-chiral superfield that one obtains by complex
conjugation of the chiral superfield. We recall
that the so-called R-symmetry group is defined as
the maximal group that rotates the supercharges
in a way that commutes with Lorentz symmetry
and is compatible with the supersymmetry alge-
bra. For N = 2 supersymmetry this group equals
SU(2)×U(1), which acts chirally on the spinors.
In spite of its name, R-symmetry does not nec-
essarily consitute an invariance of supersymmet-
ric Lagrangians. Finally, at the θ2-level, we en-
counter the field strength Fµν of the gauge field
and an auxiliary field which we write as a sym-
metric real tensor, Yij = Yji = εikεjlY
kl. Here
we note that complex conjugation will often be
indicated by rasing and/or lowering of SU(2) in-
dices. One can easily verify that in this way we
have precisely 8+8 independent field components
(which we will refer to as off-shell degrees of free-
dom, as we did not impose any field equations).
Note the difference with on-shell degrees of free-
dom. The conventional Lagrangian for the vector
supermultiplet describes 4 + 4 physical massless
bosonic and fermionic states: 2 scalars associated
with X and X̄, the 2 helicities associated with
the vector gauge field, and 2 Majorana fermions,
each carrying 2 helicities.
The Weyl supermultiplet has a rather similar
decomposition, but in this case the reduced chi-
ral superfield is not a scalar but an anti-selfdual
Lorentz tensor. For extended supersymmetry the
Weyl superfield is also assigned to the antisym-
metric representation of the R-symmetry group,
so that its lowest-order θ-component is denoted
by Tab
ij . Its complex conjugate belongs to the
corresponding anti-chiral superfield and its corre-
sponding tensor is selfdual and denoted by Tabij .
Here the indices a, b, . . . denote the components of
space-time tangent space tensors. In view of its
tensorial character the Weyl supermultiplet com-
prises 24 + 24 off-shell degrees of freedom. The
covariant components of the Weyl multiplet are as
follows. Linear in θ one has the fermions decom-
posing into the field strength of the gravitini and
a doublet spinor χi. The gravitino field strengths
comprise 16 degrees of freedom so that together
with the spinors χi we count 24 off-shell degrees of
freedom. At the θ2-level we have the Weyl tensor,
the field strengths belonging to the gauge fields
associated with R-symmetry, and a real scalar
field denoted by D, comprising 5, 4× 3 and 1 off-
shell degrees of freedom, respectively. Together
with the 6 degrees of freedom belonging to Tab
we count 24 bosonic degrees of freedom.
The Weyl multiplet contains the fields ofN = 2
conformal supergravity and an invariant action
can be written down that is quadratic in its
components. Although Tab
ij is not subject to a
Bianchi identity, it is often called the “gravipho-
ton field strength”. The reason for this misnomer
is that the gravitini transform into Tab
ij and in lo-
cally supersymmetric Lagrangians of vector mul-
tiplets that are at most quadratic in space-time
derivatives, Tab
ij acts as an auxiliary field and
couples to a field-dependent linear combination
of the vector multiplet field strengths. For this
class of Lagrangians, all the fields of the Weyl
multiplet with the exception of the graviton and
the gravitini fields, act as auxiliary fields.
In this subsection we mainly describe linearized
results. Ultimately we are interested in con-
structing a theory of local supersymmetry. This
means that the vector multiplets must first be
formulated in a supergravity background. This
leads to additional terms in the supersymmetry
transformation rules and in the superfield com-
ponents which depend on the supergravity back-
ground. Some of these terms correspond to re-
BPS Black Holes 11
placing ordinary space-time derivatives by covari-
ant ones. However, we consider only vector mul-
tiplets and the Weyl multiplet here and the latter
describes the supermultiplet of conformal super-
gravity. Consistency therefore requires that we
formulate the vector supermultiplet in a super-
conformal background, and, indeed, the vector
supermultiplet is a representation of the full su-
perconformal algebra. Therefore all the super-
conformal symmetries can be realized as local
symmetries. Naturally the Weyl multiplet itself
will also acquire non-linear corrections but those
do not involve the vector multiplet fields. The
reader may wonder why we are interested in con-
formal supergravity here, but this will be clarified
shortly. Other than that, the situation is concep-
tually the same as when considering multiplets of
matter fields coupled to a nonabelian gauge the-
ory, or to gravity.
5.2. Supersymmetric actions
In view of the fact that both the vector and
Weyl supermultiplets are described by chiral su-
perfields, even beyond their linearized form, it
is clear how to construct supersymmetric ac-
tions. Namely one takes some function of the
vector superfield (actually we will need several
of these superfields, which we label by indices
I = 0, 1, 2, . . . , n) and the square of the Weyl
superfield. Taking the square implies no loss of
generality because we are interested in Lorentz
invariant couplings. When expanding the super-
fields in θ components, one generates multiple
derivatives of this function which depend on the
lowest-θ components, X and
A = (Tab
ijεij)
2 . (27)
Because the function is holomorphic, i.e., it de-
pends on X and A, but not on their complex
conjugates, we take the imaginary part of the re-
sulting expression. However, in order that the
action is superconformally invariant, the function
F (X,A) must be holomorphic and homogeneous,
F (λX, λ2A) = λ2F (X,A) . (28)
We refrain from giving full results. In principle
they are derived straightforwardly, but the for-
mulae are often lengthy and require extra defini-
tions. Therefore we discuss only a few character-
istic terms.
First of all, let us consider the scalar kinetic
terms. They are accompanied by a coupling to
the Ricci scalar and the scalar field D of the Weyl
multiplet in the following way,
L ∝ i(∂µFI ∂µX̄I − ∂µF̄I ∂µXI)
− i(1
R−D)(FI X̄I − F̄I XI) ,
where FI denotes the derivative of F with respect
to XI . Observe that when the function F de-
pends on Tab
ij this will generate interactions be-
tween the kinetic term for the vector multiplet
scalars and the tensor field of the Weyl multiplet.
Of course, this pattern continues for other terms.
A second term concerns the kinetic term of the
vector fields, which is proportional to the second
derivative of the function F ,
L ∝ 1
iFIJ (F
I − 1
X̄ITab
ijεij)
× (F−abJ − 1
X̄JT abklεkl) + h.c. .
A third term involves the square of the Weyl
tensor, contained in the tensor R(M),
L ∝ 16 iFA[2R(M)cdab R(M)−cd
− 16T abij DaDcTcbij ] + h.c. .
Here Da denotes a superconformally covariant
derivative (which also contains terms propor-
tional to the Ricci tensor). We refrain from giving
further details at this point and refer to the liter-
ature.
5.3. Compensator multiplets
The theories discussed so far are invariant un-
der the local symmetries of the superconformal
algebra. This high degree of symmetry seems un-
necessary for, or even an obstacle to, practical
applications. The purpose of this section is to
explain that this is not the case.
Let us start with a simple example, namely
massive SU(N) Yang-Mills theory, with La-
grangian,
L = 1
Tr[Fµν(W )F
µν(W )]
M2Tr[Wµ W
12 B. de Wit
where we use a Lie-algebra valued notation and
the definition Fµν = 2 ∂[µWν] − [Wµ,Wν ]. In-
troduce now a matrix-valued field Φ ∈ SU(N)
transforming under SU(N) gauge transforma-
tions from the left and substitute Wµ = Φ
−1DµΦ
into the Lagrangian, where the covariant deriva-
tive reads DµΦ = (∂µ − Aµ)Φ. The first term is
not affected by this transformation which takes
the form of a field-dependent gauge transforma-
tion. But the mass term changes, and we find the
following Lagrangian,
L = 1
Tr[Fµν(A)F
µν(A)]
M2 Tr[DµΦD
µΦ−1] ,
which is manifestly gauge invariant. Clearly, this
is a massless gauge theory coupled to N2 − 1
scalars. However, this formulation is gauge equiv-
alent to (32), as one readily verifies by imposing
the gauge condition Φ = .
What do we achieve by rewriting (32) into the
form (33)? Both Lagrangians describe the same
number of physical states and are based on the
same number of off-shell degrees of freedom. In
(32) the degrees of freedom are contained in a sin-
gle field, Wµ, carrying 4 components per genera-
tor. In (33), however, the degrees of freedom are
distributed over two fields in a local and Lorentz
invariant way, namely 3 components per gener-
ator in Aµ (we subtracted the gauge degrees of
freedom) and 1 component per generator in Φ.
Hence the second formulation can be regarded as
reducible, and this reducibility has been achieved
by introducing extra gauge invariance.
A similar situation exists for gravity, as is
shown by the Lagrangian,
|g|L ∝
gµν∂µφ∂νφ− 16Rφ
, (34)
which is invariant under local scale transforma-
tion with parameter Λ(x): δφ = Λφ, δgµν =
−2Λ gµν. This Lagrangian is gauge equivalent
to the Einstein-Hilbert Lagrangian. To see this
one either rewrites it in terms of a scale invariant
metric φ2gµν , or one simply imposes the gauge
condition and sets φ equal to a constant (which
will then be related to Newton’s constant). Again
the Einstein-Hilbert Lagrangian and (34) contain
the same number of off-shell degrees of freedom,
but the latter field configuration is reducible: gµν
describes only 5 degrees of freedom (in view of
the local scale invariance) and the sixth one can
be assigned to the scalar field φ.
Fields such as Φ and φ are called compensat-
ing fields, because they can be used to convert any
quantity that transforms under the gauge symme-
try into a gauge invariant one. Often the gauge
equivalent formulation, based on the introduction
of compensator fields and gauge symmetries at
the same time, is exploited for reasons of renor-
malizability as one can use the gauge freedom to
choose a different gauge that leads to better short-
distance behaviour. This is not the issue here but
the crucial point is that the compensating degrees
of freedom must be contained in full supermulti-
plets. By keeping the gauge invariance manifest
one realizes a higher degree of symmetry which
facilitates the construction of Lagrangians and
clarifies the geometrical features of the resulting
supergravity theories. In this way, pureN = 2 su-
pergravity is, for instance, constructed from two
compensating supermultiplets, one of which is a
vector multiplet and the other one can be a ten-
sor multiplet or a hypermultipet. Of these two
supermultiplet only the vector field will describe
physical degrees of freedom (namely, those corre-
sponding to the graviphoton). The other compo-
nents play the role of either compensating fields
(associated with local scale, R-symmetry and spe-
cial conformal supersymmetry transformations),
or are constrained by the field equations, either
by Lagrange multipliers, or because they are aux-
iliary.
6. BPS ATTRACTORS
As we already discussed in section 4 the BPS
attractor equations follow from the requirement
of full N = 2 supersymmetry at the horizon. In
the context of an off-shell representation of su-
perymmetry, the corresponding equations can be
derived in a way that is rather independent of
the action. As explained in the previous section,
N = 2 vector multiplets contain complex physical
scalar fields which we denoted by XI . In the con-
text of the superconformal framework these fields
are defined projectively, in view of the invariance
BPS Black Holes 13
under local scale and U(1) transformations. The
action for vector multiplets with additional inter-
actions involving the square of the Weyl tensor
is encoded in a holomorphic function F (X,A),
which is homogenous of second degree (c.f. (28)).
Here A is quadratic in the anti-selfdual field Tab
as shown in (27). The normalizations of the La-
grangian and of the charges adopted below differ
from the normalizations used in section 4.
Another issue that we should explain concerns
electric/magnetic duality, which in principle per-
tains to the gauge fields. Straightforward appli-
cation of this duality to an N = 2 supersymmet-
ric Lagrangian with vector multiplets, leads to a
new Lagrangian that no longer takes the canoni-
cal form in terms of a function F (X,A). In order
to bring it into that form one must simultane-
ously apply a field redefinition to the scalar and
spinor fields. On the scalar fields, this redefinition
follows from the observation that (XI , FI(X,A))
transforms as a sympletic vector analogous to the
tensors (Fµν
I , GµνI) discussed previously. Hence
the (2n + 2)-dimensional vector (XI , FI(X,A))
is rotated into a new vector (X̃I , F̃I). When
this rotation belongs to Sp(2n + 2;R) then F̃I
can be written as the derivative of a new func-
tion F̃ (X̃, A) with respect to X̃I . The new func-
tion then encodes the dual action. The need for
this additional field redefinition follows from the
fact that the gauge fields, and thus their field
strengths Fµν
I , and the fields XI have a well-
defined relation encoded in the supersymmetry
transformation rules. Therefore, the transforma-
tions of (XI , FI(X,A)) must be taken into ac-
count when considering electric/magnetic dual-
ity. We refer to [24,28] for further details and a
convenient list of formulae.
To determine the BPS attractor equations one
classifies all possible N = 2 supersymmetric so-
lutions. This is done by studying the supersym-
metry variations of the fermions in an arbitrary
bosonic background. Requiring that these varia-
tions vanish then imposes strong restrictions on
this background. The analysis was performed in
[15] for N = 2 supergravity with an arbitrary
number of vector multiplets and hypermultiplets,
including higher-order derivative couplings pro-
portional to the square of the Weyl tensor. Also
the interpolating BPS solutions were studied in
considerable detail. It was found that N = 2 su-
persymmetric solutions are unique and depend on
a harmonic function with a single center. Hence
the horizon geometry and the values of the rel-
evant fields are fully determined in terms of the
charges. The hypermultiplet scalar fields are co-
variantly constant but otherwise arbitrary. How-
ever, the horizon and the entropy do not depend
on these fields, so that they can be ignored. In the
absence of charges one is left with flat Minkowski
space-time with arbitrary constant moduli and
ij = 0.
As it turns out the attractor equations have
a universal form. Before commenting further
on their derivation, let us present the equa-
tions, which are manifestly covariant under elec-
tric/magnetic duality,
PI = 0 , QI = 0 , Υ = −64 , (35)
where
PI ≡ pI + i(Y I − Ȳ I) ,
QI ≡ qI + i(FI − F̄I) .
Here the Y I and Υ are related to the XI and A,
respectively, by a uniform rescaling and FI and
FΥ will denote the derivatives of F (Y,Υ) with
respect to Y I and Υ. To explain the details of
the rescaling, we introduce the complex quantity
Z, sometimes refered to as the ’holomorphic BPS
mass’, which equals the central charge associated
with the vector supermultiplets. In terms of the
original variables XI it is defined as
Z = exp[K/2] (pIFI(X,A)− qIXI) , (37)
where
e−K = i (X̄IFI(X,A)− F̄I(X̄, Ā)XI) . (38)
At the horizon the variables Y I and Υ are de-
fined by
Y I = exp[K/2] Z̄ XI ,
Υ = exp[K] Z̄2 A .
Note that Y I and Υ are invariant under arbitrary
complex rescalings of the underlying variables XI
14 B. de Wit
and A. The reader may easily verify that for fields
satisfying the attractor equations (35), one estab-
lishes that
|Z|2 ≡ pIFI − qIY I , (40)
which is obviously real and positive, is equal to
i(Ȳ IFI − Y I F̄I).
Finally we wish to draw attention to just one
aspect of the derivation of the attractor equations
(35). Consider the spinor fields belonging to the
vector supermultiplets, and concentrate on their
supersymmetry variation in terms of a two-rank
tensor,
I = 1
I − 1
εklTab
kl X̄I
. (41)
This particular linear combination of the field
strength F−ab
I and the field X̄I arises because the
symmetry transformations are evaluated in a su-
perconformal background. Full supersymmetry
therefore implies that the right-hand side of (41)
vanishes, so that
I = 1
εklTab
kl X̄I . (42)
An extension of this argument gives a similar re-
sult for the conjugate field strengths,
G−abI =
εklTab
kl F̄I . (43)
Note that these two equations are consistent with
respect to electric/magnetic duality. Given the
fact that the field strengths Fab
I and GabI satisfy
the Maxwell equations and therefore become pro-
portional to the magnetic and electric charges, pI
and qI , it is not surprising that one finds the at-
tractor equations (35). For further details of the
analysis we refer to [15].
6.1. The BPS entropy function
The BPS attractor equations follow also from
a variational principle based on the entropy func-
tion [17,18],
Σ(Y, Ȳ , p, q) = F(Y, Ȳ ,Υ, Ῡ)
− qI(Y I + Ȳ I) + pI(FI + F̄I) ,
where pI and qI couple to the correspond-
ing magneto- and electrostatic potentials at the
horizon (c.f. [15]) in a way that is consistent
with electric/magnetic duality. The quantity
F(Y, Ȳ ,Υ, Ῡ), which will be regarded as a ‘free
energy’ in what follows, is defined by
F(Y, Ȳ ,Υ, Ῡ) = −i
Ȳ IFI − Y I F̄I
ΥFΥ − ῩF̄Υ
where FΥ = ∂F/∂Υ. Just as the entropy function
discussed in section 4, the entropy function (44)
transforms as a function under electric/magnetic
duality [28]. Varying this entropy function with
respect to the Y I , while keeping the charges and
Υ fixed, yields the result,
δΣ = PI δ(FI + F̄I)−QI δ(Y I + Ȳ I) . (46)
Here we made use of the homogeneity of the func-
tion F (Y,Υ). Under the mild assumption that
the matrix
NIJ = i(F̄IJ − FIJ), (47)
is non-degenerate, it thus follows that station-
ary points of Σ satisfy the attractor equations.
The macroscopic entropy is equal to the entropy
function taken at the attractor point. This im-
plies that the macroscopic entropy is the Legen-
dre transform of the free energy (45). An explicit
calculation yields the entropy formula [29,30,31],
Smacro(p, q) = πΣ
attractor
|Z|2 − 256 ImFΥ
Υ=−64
Here the first term represents a quarter of the
horizon area (in Planck units) so that the second
term defines the deviation from the Bekenstein-
Hawking area law. The entropy coincides pre-
cisely with the Wald entropy [16] as given by
the right-hand side of (26). In fact, the original
derivation of (48) was not based on an entropy
function and made direct use of the expression
(26).
In the absence of Υ-dependent terms, the ho-
mogeneity of the function F (Y ) implies that the
area scales quadratically with the charges, as was
discussed already at the beginning of section 2.
However, in view of the fact that Υ takes a fixed
BPS Black Holes 15
value, the second term will be subleading in the
limit of large charges.2 Note, however, that also
the area will contain subleading terms, as it de-
pends on Υ.
The entropy equation (48) has been confronted
with the result of microstate counting, for in-
stance, in the situation described in section 3.
In that case the effective supergravity action is
known and based on the function
F (Y,Υ) = −1
CABC Y
AY BY C
− c2A
24 · 64
Substituting this result into (48) and imposing
the attractor equations (35) with p0 = 0, one in-
deed derives the result (13) for the macroscopic
entropy [29]. The entropy formula (48) has also
been put to a test in other cases. Some of them
will be discussed in due course.
The relation between (44) and the entropy
function introduced in section 4 was discussed
in [25], where it was established that both en-
tropy functions lead to identical results for BPS
black holes. When the black holes are not BPS
(i.e. have no residual supersymmetry), then the
entropy function (44) is simply not applicable.
In this connection the question arises whether
other, independent, higher-derivative interactions
associated with matter multiplets will not con-
tribute to the entropy either. For instance, La-
grangians for tensor supermultiplets that con-
tain interactions of fourth order in space-time
derivatives, lead to terms quadratic in the Ricci
scalar that will in principle contribute to the
Wald entropy [32]. Indeed, for non-BPS black
holes these terms yield finite contributions to
the entropy, but for BPS black holes these cor-
rections vanish. A comprehensive treatment of
higher-derivative interactions is yet to be given
for N = 2 supergravity, but it seems that this
result is generic. At any rate, this observation
seems in line with more recent findings [33,34]
based on heterotic string α′-corrections encoded
in a higher-derivative effective action in higher
2 Here one usually assumes that F (Y,Υ) can be expanded
in positive powers of Υ.
dimensions. In four dimensions this action leads
to additional matter-coupled higher-derivative in-
teractions. When these are taken into account,
the matching of the macroscopic entropy with the
microscopic result is established [34].
Another modification concerns possible non-
holomorphic corrections to the function F (X,A).
This holomorphic function leads to a supersym-
metric action that corresponds to the so-called
effective Wilsonian action, based on integrating
out the massive degrees of freedom. The Wilso-
nian action describes the correct physics for en-
ergies between appropriately chosen infrared and
ultraviolet cutoffs. However, this action does not
reflect all the physical symmetries. To preserve
those symmetries non-holomorphic contributions
should be included associated with integrating
out massless degrees of freedom. In the special
case of heterotic black holes in N = 4 supersym-
metric compactifications, the requirement of ex-
plicit S-duality invariance of the entropy and the
attractor equations allows one to determine the
contribution from these non-holomorphic correc-
tions, as was first demonstrated in [31] for BPS
black holes. In [35,18] it was established that
non-holomorphic corrections to the BPS entropy
function (44) can be encoded into a real function
Ω(Y, Ȳ ,Υ, Ῡ) which is homogeneous of second
degree. The modifications to the entropy func-
tion are then effected by substituting F (Y,Υ) →
F (Y,Υ) + 2iΩ(Y, Ȳ ,Υ, Ῡ). There are good rea-
sons to expect that this same substitution should
be applied to the more general entropy function
based on (23) [25]. Note that when Ω is harmonic,
i.e., when it satisfies ∂2Ω/∂Y I ∂Ȳ J = 0, it can
simply be absorbed into the original holomorphic
funtion F (Y,Υ).
Finally we point out the existence of a formu-
lation in terms of real, rather than the complex
fields (Y I ,Υ), that we used before. This for-
mulation is manifestly covariant with respect to
electric/magnetic duality. We first decompose Y I
and FI into their real and imaginary parts,
Y I = xI + iuI , FI = yI + ivI , (50)
where FI = FI(Y,Υ). The real parametriza-
tion is obtained by taking (xI , yI ,Υ, Ῡ) instead
of (Y I , Ȳ I ,Υ, Ῡ) as the independent variables.
16 B. de Wit
This reparametrization is only well defined pro-
vided det(NIJ) 6= 0. Subsequently one defines the
Hesse potential, the real analogue of the Kähler
potential, which equals twice the Legendre trans-
form of the imaginary part of the prepotential
with respect to uI = ImY I ,
H(x, y,Υ, Ῡ) = 2 ImF (x+ iu,Υ, Ῡ)− 2 yI uI ,
Owing to the homogeneity of the function
F (Y,Υ) one can show that the free energy (45)
equals twice the Hesse potential. The entropy
function (44) is now replaced by
Σ(x, y, p, q) = 2H(x, y,Υ, Ῡ)− 2 qIxI + 2 pIyI ,
and it is straightforward to show that the ex-
tremization equations are just the attractor equa-
tions (35), expressed in terms of the new variables
(xI , yI). The value of Σ(x, y, p, q) at the attrator
point coincides again with the macroscopic en-
tropy.
6.2. Partial Legendre transforms
It is, of course, possible to define the macro-
scopic entropy as a Legendre transform with re-
spect to only a subset of the fields, by substitut-
ing part of the attractor equations such that the
variational principle remains valid. These par-
tial Legendre transforms constitute a hierarchy of
Legendre transforms for the black hole entropy.
Here we discuss two relevant examples, namely,
the one proposed in [36] where all the magnetic
attractor equations are imposed, and the dila-
tonic one for heterotic black holes, where only two
real potentials are left which together constitute
the complex dilaton field [31]. A possible dis-
advantage of considering partial Legendre trans-
forms is that certain invariances may no longer
be manifest. As it turns out, the dilatonic for-
mulation does not suffer from this shortcoming.
Apart from that, there is no reason to prefer one
version over the other. This will change in sec-
tion 7 when we discuss corresponding partition
functions and inverse Laplace transforms for the
microscopic degeneracies in semiclassical approx-
imation.
Let us first impose the magnetic attractor equa-
tions so that only the real parts of the Y I will re-
main as independent variables. Hence one makes
the substitution,
Y I = 1
(φI + ipI) . (53)
The entropy function (44) then takes the form,
Σ(φ, p, q) = FE(p, φ,Υ, Ῡ)− qI φI , (54)
where the corresponding free energy FE(p, φ)
equals
FE(p, φ) =
ImF (Y,Υ) + Ω(Y, Ȳ ,Υ, Ῡ)
Y I=(φI+ipI)/2
To derive this result one makes use of the
homogeneity of the functions F (Y,Υ) and
Ω(Y, Ȳ ,Υ, Ῡ). The latter function may contain
possible non-holomorphic terms. When extrem-
izing (55) with respect to φI we obtain the at-
tractor equations qI = ∂FE/∂φI . This shows
that the macroscopic entropy is a Legendre trans-
form of FE(p, φ) subject to Υ = −64, as was first
noted in [36] in the absence of Ω. In the latter
case this Legendre transform led to the conjecture
that there is a relation with topological strings, in
view of the fact that exp[FE] equals the modulus
square of the topological string partition function
[37]. We return to this in subsection 7.1.
Along the same line one can now proceed and
eliminate some of the φI as well. A specific exam-
ple of this is the dilatonic formulation heterotic
black holes, where we eliminate all the φI with
the exception of two of them which parametrize
the complex dilaton field. This leads to an en-
tropy function that depends only on the charges
and on the dilaton field [31,35]. Here it is conve-
nient to include all the Υ-dependent terms into
Ω, which also contains the non-holomorphic cor-
rections. The heterotic classical function F (Y ) is
given by
F (Y ) = −Y
1 Y aηabY
, a = 2, . . . , n, (56)
with real constants ηab. In the application that we
will be considering the function Ω depends only
BPS Black Holes 17
linearly on Υ and Ῡ, as well as on the dilaton
field S = −iY 1/Y 0 and its complex conjugate
S̄. The result for the BPS entropy function then
takes the form,
Σ(S, S̄, p, q) = −q
2 − ip · q (S − S̄) + p2 |S|2
S + S̄
+ 4Ω(S, S̄,Υ, Ῡ) ,
where q2, p2 and p · q are T-duality invariant bi-
linears of the various charges, defined by
q2 = 2q0p
1 − 1
abqb ,
p2 = −2p0q1 − 2paηabpb ,
q · p = q0p0 − q1p1 + qapa .
Note that these bilinears are not positive defi-
nite. Furthermore, Ω captures the Υ-dependent
corrections to the classical result (56). Its form
was derived for N = 4 heterotic string compact-
ifications by requiring S-duality of the attractor
equations and of the entropy [31,35],
Ω(Y, Ȳ ,Υ, Ῡ) =
Υ log η24(S) + Ῡ log η24(S̄)
(Υ + Ῡ) log(S + S̄)12
where η(S) denotes the Dedekind function. Note
the presence of the last term which is non-
holomorphic. This term is in accord with the
result for the effective action obtained from
five-brane instantons [38]. The attractor equa-
tions associated with the dilaton take the form
∂SΣ(S, S̄, p, q) = 0.
It is interesting to consider the consequences
of (57) in the classical case (Ω = 0). Then the
attractor equations yield the following values for
the real part of the dilaton field and the macro-
scopic entropy,
S + S̄ = − 2
p2q2 − (p · q)2 ,
Smacro = π
p2q2 − (p · q)2 .
Obviously there are two types of black holes, de-
pending on whether p2q2 − (p · q)2 is positive or
zero. In the context of N = 4 heterotic string
compactifications these correspond to 1/4- and
1/2-BPS black holes, respectively. The 1/4-BPS
states are dyonic, so that they necessarily carry
both electric and magnetic charges. The 1/2-BPS
states can be purely electric. We derived these
results from the BPS entropy function, but they
have also been obtained directly from the full su-
pergravity solutions [39,40,41].
Clearly the 1/4-BPS black holes are large black
holes as their area (entropy) is nonzero and scales
quadratically with the charges. Note that, de-
pending on the choice of the charges, the com-
plex dilaton field can remain finite in the limit
of large charges. This is relevant when studying
the asymptotic growth of the dyonic degeneracy
of 1/4-BPS dyons in heterotic string theory com-
pactified on a six-torus and in the related class of
heterotic CHL models [42]. These degeneracies
are encoded in automorphic forms Φk(ρ, σ, υ) of
weight k under Sp(2,Z), or an appropriate sub-
group thereof [43,44]. The torus compactifica-
tion corresponds to k = 10 and the CHL mod-
els to k = 1, 2, 4, 6. The three modular param-
eters, ρ, σ, υ, parametrize the period matrix of
an auxiliary genus-two Riemann surface, which
takes the form of a complex, symmetric, two-by-
two matrix. The microscopic degeneracy of 1/4-
BPS dyons is expressed as an integral over an
appropriate 3-cycle,
dk(p, q) =
dρ dσ dυ
eiπ[ρ p
2+σ q2+(2υ−1) p·q]
Φk(ρ, σ, υ)
Since Φk has zeros in the interior of the Siegel
half-space in addition to the zeros at the cusps,
the value of the integral (61) depends sensitively
on the choice of the integration 3-cycles. The
charges are in general integer, with the exception
of q1 which equals a multiple of N , and p
1 which
is fractional and quantized in units of 1/N . Here
N and k are related by (k+2)(N +1) = 24. The
inverse of the modular form Φk takes the form
of a Fourier sum with integer powers of exp[2πiρ]
and exp[2πiυ] and fractional powers of exp[2πiσ]
which are multiples of 1/N . The 3-cycle is then
defined by choosing integration contours where
the real parts of ρ and υ take values in the interval
(0, 1) and the real part of σ takes values in the
18 B. de Wit
interval (0, N). The formula (61) is manifestly
invariant under T-duality (the integrand depends
on the three T-duality invariant bilinears (58)),
as well as under S-duality, which is a subgroup of
the full modular group.
The integral (61) can be evaluated in saddle-
point approximation which yields the leading and
subleading contribtions to dk(p, q) [35,44]. As it
turns out these contributions are precisely en-
coded in (57) and (59), including the Dedekind
eta-functions and the non-holomorphic terms.
The presence of the non-holomorphic terms is
not surprising in view of the S-duality invariance.
Note that the expression (59) refers to k = 10 and
that there exist similar formulae for k = 1, 2, 4, 6.
The 1/2-BPS black holes are small black holes
as their area scales linearly with the charges. Ac-
cording to (60) their entropy (and horizon area)
vanishes while the dilaton field diverges at the
horizon, because we have p2 = p · q = 0. To de-
scribe the situation more accurately one retains
the leading term of (59). In that case one ob-
tains the following result (we restrict ourselves to
k = 10) [31],
S + S̄ ≈
|q2|/2 ,
Smacro ≈ 4 π
|q2|/2− 6 log |q2| ,
where the logarithmic term is due to the non-
holomorphic contribution. Because the dilaton
is large in this case, all the exponentials in the
Dedekind eta-function are suppressed and we are
at weak string coupling gs ∝ (S + S̄)−1/2.
We already stressed that small black holes have
a size of the order of the string scale and, in-
deed, these states are precisely generated by per-
turbative heterotic string states arising in N = 4
supersymmetric compactifications to four space-
time dimensions. In the supersymmetric right-
moving sector these states carry only momen-
tum and winding and contain no oscillations,
whereas in the left-moving sector oscillations are
allowed that satisfy the string matching condi-
tion. The oscillator number is then linearly re-
lated to q2. These perturbative states received
quite some attention in the past [45]. Because
the higher-mass string states are expected to be
within their Schwarzschild radius, it was conjec-
tured that they should have an interpretation as
black holes.3 Their calculable level density, pro-
portional to the exponent of 4π
|q2|/2, implies a
nonzero microscopic entropy for these black holes
[46].
This result was confronted with explicit black
hole solutions [47,48,49] based on standard super-
gravity Lagrangians that are at most quadratic
in derivatives, which have a vanishing horizon
area. Based on the area law one thus obtains
a vanishing macroscopic entropy. The fact that
(48), which takes into account higher-derivative
interactions, can nicely account for the discrep-
ancies encountered in the classical description of
the 1/2-BPS black holes, was first emphasized in
[50,51]. Note also that, since the electric states
correspond to perturbative heterotic string states,
their degeneracy is known from string theory and
given by
d(q) =
eiπσq
η24(σ)
≈ exp
|q2|/2− 27
log |q2|
where the integration contour encircles the point
exp(2πiσ) = 0. This large-|q2| approximation is
based on a standard saddle-point approximation.
Obviously the leading term of (63) is in agreement
with (62, but beyond that there is a disagreement
as the logarithmic corrections carry different co-
efficients. This discrepancy may be regarded as a
first indication that small black holes are not well
understood (for a disucssion, see, for instance,
[18]). Therefore we will mainly concentrate on
large black holes in the next chapter.
7. PARTITION FUNCTIONS AND IN-
VERSE LAPLACE TRANSFORMS
To again make the connection with microstate
degeneracies, we conjecture, in the spirit of [36],
that the Legendre transforms of the entropy are
indicative of a thermodynamic origin of the vari-
ous entropy functions. It is then natural to as-
sume that the corresponding free energies are
3 The idea that elementary particles, or string states, are
behaving like black holes, has been around for quite some
time.
BPS Black Holes 19
related to black hole partition functions corre-
sponding to suitable ensembles of black hole mi-
crostates. Following [18], we define
Z(φ, χ) =
{p,q}
d(p, q) eπ[qIφ
I−pIχI ] , (64)
where d(p, q) denotes the microscopic degenera-
cies of the black hole microstates with black hole
charges pI and qI . This is the partition sum
over a canonical ensemble, which is invariant un-
der the various duality symmetries, provided that
the electro- and magnetostatic potentials (φI , χI)
transform as a symplectic vector. Identifying a
free energy with the logarithm of Z(φ, χ), it is
clear that it should, perhaps in an appropriate
limit, be related to the macroscopic free energy
introduced earlier. On the other hand, viewing
Z(φ, χ) as an analytic function in φI and χI , the
degeneracies d(p, q) can be retrieved by an inverse
Laplace transform,
d(p, q) ∝
dχI dφ
I Z(φ, χ) eπ[−qIφ
I+pIχI ] ,
where the integration contours run, for instance,
over the intervals (φ − i, φ + i) and (χ − i, χ + i)
(we are assuming an integer-valued charge lat-
tice). Obviously, this makes sense as Z(φ, χ) is
formally periodic under shifts of φ and χ by mul-
tiples of 2i.
These arguments suggest to identify Z(φ, χ)
with the Hesse potential (51),
{p,q}
d(p, q) eπ[qIφ
I−pIχI ] ∼
shifts
e2πH(φ/2,χ/2,Υ,Ῡ) ,
where Υ is equal to its attractor value and where
we suppressed possible non-holomorphic contri-
butions for simplicity. However, the Hesse po-
tential is a macroscopic quantity which does not
in general exhibit the periodicity that is charac-
teristic for the partition function. Therefore, the
right-hand side of (66) requires an explicit peri-
odicity sum over discrete imaginary shifts of the φ
and χ.4 When substituting 2πH into the inverse
Laplace transform, we expect that the periodic-
ity sum can be incorporated into the integration
contour.
It is in general difficult to find an explicit repre-
sentation for the Hesse potential, as the relation
(50) between the complex variables Y I and the
real variables xI and yI is complicated. Therefore
we rewrite (66) in terms of the original variables
Y I and Ȳ I , where explicit results are known,
{p,q}
d(p, q) eπ[qI (Y+Ȳ )
I−pI(F̂+ ˆ̄F )I ] ∼
shifts
eπF(Y,Ȳ ,Υ,Ῡ) .
Here F equals the free energy (45) suitably mod-
ified with possible non-holomorphic corrections.
The latter requires that FI is changed into F̂I =
FI + 2iΩI , as was demonstrated in [18]. It is im-
portant to note that both sides of (67) (as well as
of (66)) are manifestly consistent with duality.
Again, it is possible to formally invert (67) by
means of an inverse Laplace transform,
d(p, q) ∝
d(Y + Ȳ )I d(F̂ + ˆ̄F )I e
πΣ(Y,Ȳ ,p,q)
dY dȲ ∆−(Y, Ȳ ) eπΣ(Y,Ȳ ,p,q) ,
where ∆−(Y, Ȳ ) is an integration measure whose
form depends on F̂I+
ˆ̄F I . The expression for ∆
as well as for a related determinant ∆+, reads as
follows,
∆±(Y, Ȳ ) =
ImFKL + 2Re(ΩKL ± ΩKL̄)
As before, FIJ and FI refer to Y -derivatives
of the holomorphic function F (Y,Υ) whereas
ΩIJ and ΩIJ̄ denote the holomorphic and mixed
holomorphic-antiholomorphic second derivatives
4In case that the Hesse potential exhibits a periodicity
with a different periodicity interval, then the sum over the
imaginary shifts will have to be modded out appropriately
such as to avoid overcounting.
20 B. de Wit
of Ω, respectively. In the absence of non-
holomorphic corrections ∆+ = ∆−.
A priori it is not clear whether the integral (68)
is well-defined and we refer to [18] for a discus-
sion. Note that the periodicity sum in (68) is
defined in terms of the variables φ and χ, which
should have some bearing on the integration con-
tour in (68). Leaving aside these subtle points
one may consider a saddle-point approximation of
the intergral representation (68). In view of the
previous results it is clear that the saddle point
coincides with the attractor point. Subsequently
one evaluates the semiclassical Gaussian integral
that emerges when expanding the exponent in the
integrand to second order in δY I and δȲ I about
the attractor point. When Y I − Ȳ I = ipI , the
resulting determinant factorizes into the square
roots of two subdeterminants,
∆+ and
Here the plus (minus) sign refers to the contribu-
tion of integrating over the real (imaginary) part
of δY I . Consequently, the result of a saddle-point
approximation applied to (68) yields,
d(p, q) =
∆−(Y, Ȳ )
∆+(Y, Ȳ )
attractor
eSmacro(p,q) . (70)
In the absence of non-holomorphic corrections the
ratio of the two determinants is equal to unity
and one recovers precisely the macroscopic en-
tropy. In the presence of non-holomorphic terms,
the deviations from unity are usually suppressed
in the limit of large charges, and one recovers the
leading and subleading corrections to the entropy
[18]. Of course, this is only the case when the
saddle-point approximation is appropriate.
Alternatively one may choose to integrate only
over the imaginary values of the fluctuations δY I
in saddle-point approximation. The saddle point
then occurs in the subspace defined by the mag-
netic attractor equations, so that one obtains a
modified version of the OSV integral [36],
d(p, q) ∝
∆−(p, φ) eπ[FE(p,φ)−qIφ
I ] ,
where FE(p, φ) was defined in (55) and ∆−(p, φ)
is defined in (69) with the Y I given by (53).
Hence this integral contains a non-trivial integra-
tion measure factor
∆− in order to remain con-
sistent with electric/magnetic duality. Without
the integral measure this is the integral conjec-
tured in [36]. In view of the original setting in
terms of the Hesse potential, we expect that the
integration contours in (71) should be taken along
the imaginary axes.
Inverting (71) to a partition sum over a mixed
ensemble, one finds,
Z(p, φ) =
d(p, q) eπ qIφ
shifts
∆−(p, φ) eπFE(p,φ) .
It should be noted that this expression and the
preceding one is less general than (68) because
it involves a saddle-point approximation. More-
over the function FE is not duality invariant and
the invariance is only recaptured when complet-
ing the saddle-point approximation with respect
to the fields φI . Therefore an evaluation of (71)
beyond the saddle-point approximation will most
likely give rise to a violation of (some of) the du-
ality symmetries again.
7.1. The integration measure and the
mixed partition function
As mentioned earlier, the partition function
Z(p, φ) was originally conjectured to be equal to
the modulus square of the partition function of
the topological string [36]. Soon thereafter, how-
ever, it was realized that this relationship must
be more subtle. The arguments in [18], based
on electric/magnetic duality clearly indicate some
of the missing ingredients, resulting in the mea-
sure factor
∆− and in the presence of the non-
holomorphic corrections. In fact, it was already
clear from an early analysis of small heterotic
black holes that T-duality was not conserved
when straightforwardly applying these ideas [52].
Although the presence of the measure factor cor-
rects for the lack of duality invariance, the semi-
classical results for small black holes seem to re-
main inconsistent with the analysis of microstate
counting, a fact we already alluded to at the end
of subsection 6.2.
BPS Black Holes 21
It is possible to test the result (72) in the
context of the 1/4-BPS states of the heterotic
N = 4 supersymmetric string compactifications,
by making use of the degeneracy formula (61).
Such a calculation was first performed in [53] and
it was subsequently generalized in [18] for more
general charge configurations and for CHL mod-
els. Using the degeneracies (61) one calculates
the mixed partition sum on the left-hand side of
(72). As it turns out, the resulting expression is
indeed proportional to the square of the partition
function of the topological string,
|Ztop(p, φ)|2 = e−2πi[F (Y,Υ)−F̄(Ȳ ,Ῡ] , (73)
where F (Y,Υ) is now the holomorphic part of
(56) and (59),
F (Y,Υ) = −Y
1 Y aηabY
log η24(−iY 1/Y 0) .
Here Υ = −64 and the Y I are given by (53).5
However, there is a non-trivial proportionality
factor, which, up to subleading contributions, we
expect to coincide with
∆−(p, φ) exp[4πΩnonholo] . (75)
The expression for Ωnonholo follows from the last
term in (59), and is thus equal to
exp[4π Ωnonholo] = (S + S̄)
−12 , (76)
where
S + S̄ =
2(p1φ0 − p0φ1)
(φ0)2 + (p0)2
. (77)
The factor (S + S̄)−12 cancels against a similar
factor in
∆− and, up to subleading terms, (75)
becomes
∆−(p, φ) exp[4πΩnonholo] ≈
i(Ȳ I F̂I − Y I F̂I)
2 |Y 0|2
e−K(Y,Ȳ ,Υ,Ῡ)
2 |Y 0|2
Indeed this result coincides with the result found
in [53,18]. The expression for e−K was already de-
fined in (38) up to non-holomorphic corrections.
5For convenience, we only refer to the case k = 10.
The latter can be dropped as they are subleading.
Note that e−K has been rescaled by replacing the
XI and A by Y I and Υ, respectively. This is to
be expected in view of the fact that the right-
hand side of (78) must be invariant under such
rescalings.
However, we should discuss a subtlety related
to the fact that we derived the expression for√
∆− in the context of N = 2 supergravity,
whereas the evaluation based on (61) is based on
N = 4 supersymmetric compactifications. This
means that the number of scalar moduli (related
to the number of N = 2 vector multiplets) is not
obviously the same. In the case ofN = 4 the rank
of the gauge group is equal to 28 (for simplicity
we restrict ourselves to the case k = 10). Of the
28 abelian vector gauge fields, 6 will correspond
to the graviphotons of pure N = 4 supergrav-
ity, and 22 will each belong to an N = 4 vector
supermultiplet. In the reduction to N = 2 super-
gravity, one of the graviphotons will be contained
in an additional N = 2 vector supermultiplet and
another one will play the role of the gravipho-
ton of N = 2 supergravity. There are thus 24
abelian vector gauge fields, of which 23 are asso-
ciated with N = 2 vector multiplets and one is
associated with the N = 2 graviphoton. The re-
maining 4 graviphotons are associated with two
N = 2 gravitino supermultipets, and these vector
fields seem to have no place in the N = 2 de-
scription. Therefore it is often assumed that the
effective rank of the gauge group in the N = 2
description should be taken equal to 24, rather
than to 28.
Nevertheless, in the calculation of
∆− leading
to (78) we assumed that the N = 2 description
is based on 28 vector fields, corresponding to 27
vector supermultiplets and one graviphoton field.
Only in that case, the factor (S + S̄)−12 cancels
so that one obtains the proportionality constant
noted in (78) based on the N = 2 expression for√
∆−. This somewhat confusing issue seems en-
tirely due to the fact that the N = 2 description
of an N = 4 theory is not fully understood, as the
corresponding description of N = 4 supergravity
is indeed based on 28 electrostatic potentials φ
which are contained in 28, rather than 24, ana-
logues of the N = 2 quantities Y I .
22 B. de Wit
In a recent series of papers [54,55,56] progress
was made towards a further understanding of the
relation between the mixed black hole partition
function and the partition function of the topo-
logical string. Unfortunately no evidence for the
presence of the integration measure factor in (71)
and (72) was presented. However, a more de-
tailed analysis for compact Calabi-Yau [57] mod-
els subsequently revealed the presence of the mea-
sure factor. Based on an extensive analysis of
the factorization formula for BPS indices, it is
shown that the partition function does not com-
pletely factorize into a holomophic and an anti-
holomorphic sector and the measure that is found
agrees (for p0 = 0) with (78). The power of |Y 0|
depends on whether one is discussing N = 2 or
N = 4 black holes. These results seem to be
in line with what was discussed in this section,
although there are many subtleties. Obviously
more work is needed to fully explore their conse-
quences.
The work described in these lectures is based on
various collaborations with Gabriel Lopes Car-
doso, Jürg Käppeli, Swapna Mahapatra, Thomas
Mohaupt and Frank Saueressig. I thank Gabriel
Lopes Cardoso for discussion and valuable com-
ments on the manuscript.
This work is partly supported by NWO
grant 047017015, EU contracts MRTN-CT-2004-
005104 and MRTN-CT-2004-512194, and INTAS
contract 03-51-6346.
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	INTRODUCTION
	DUAL PERSPECTIVES
	BLACK HOLES IN M/STRING THEORY – AN EXAMPLE
	ATTRACTOR EQUATIONS
	N=2 SUPERGRAVITY
	Supermultiplets
	Supersymmetric actions
	Compensator multiplets
	BPS ATTRACTORS
	The BPS entropy function
	Partial Legendre transforms
	PARTITION FUNCTIONS AND INVERSE LAPLACE TRANSFORMS
	The integration measure and the mixed partition function
