Regular 3-charge 4D black holes and their microscopic description
c The Authors.
Institut de Physique Th eorique, CEA Saclay
CNRS URA 2306
The perturbative corrections to Type-IIA String Theory compactified on a Calabi-Yau three-fold allow the construction of regular three-charge supersymmetric black holes in four dimensions, whose entropy scales with the charges as S construct an M-theory uplift of these quantum black holes and show that they can be interpreted as arising from three stacks of M2 branes on a conical singularity. This in turns allow us relate them via a series of dualities to a system of D3 branes carrying momentum
and thus to give a microscopic interpretation of their entropy.
1 Introduction 2 3 4
Type-IIA quantum black holes
Type-IIA string theory on a Calabi-Yau manifold
The quantum-corrected STU model
The M-theory configuration
The string theory dilaton
M2-branes beyond classical supergravity
The microscopic entropy
B A quantum class of black holes
String theory has proven to be extremely successful in reproducing the entropy of
supersymmetric black holes, such as the three-charge black hole in five dimensions  and the
four-charge black hole in four dimensions . The entropies of these black holes scale
like the square root of the product of their charges (or some some duality-invariant form
thereof ), and in the microscopic counting this square root comes from using Cardys
formula to count the states of a certain 1+1 dimensional system of strings and branes.
However, in certain four-dimensional compactifications of string theory one can
construct three-charge black holes whose entropy scales with these charges like S
p1p2p3 3 . The curvature at the horizon of these black holes is small, precisely as
one would expect from the fact that their entropy grows like the square of the charge.
These black holes cannot be constructed in normal four-dimensional supergravity, where
the horizon curvature of three-charge black hole is Planckian, but exist if one adds to the
tions at tree level in gS. Since these black holes do not have a regular limit when these
correction terms are removed1 they are called quantum black holes.
The purpose of this letter is to try to understand the microscopic entropy of the type
IIA quantum black holes constructed in . The first step in this direction is to uplift
these black hole solutions to eleven dimensions and to propose an M-theory interpretation
in terms of three mutually-orthogonal intersecting stacks of M2-branes at the tip of a
fourcompactifications come from higher-derivative terms in String Theory [11, 12], the uplifted
black hole will be a solution of the equations of motion of eleven-dimensional supergravity
modified by the addition of certain higher-derivative terms.
The second step is to argue that if the Calabi-Yau manifold can be written as a (possibly
singular) elliptic fibration, the branes that make the eleven-dimensional solution can be
dualized to a configuration of two intersecting mutually-supersymmetric D3 branes carrying
momentum along their common direction. By counting the possible way of carrying this
momentum and by remembering that these D3 branes sit on top of a conical singularity that
effectively enhances the central charge of the 1+1 dimensional theory on their worldvolume,
the quantum black holes.
In section 2 we introduce the effective theory corresponding to Type-IIA String Theory
compactified on a Calabi-Yau manifold as well as the corresponding Type-IIA quantum
black hole solutions. In section 3 we argue that the M-theory uplift of quantum black holes
can be interpreted as arising from three stacks of intersecting M2 branes and in section 4
we use this to propose a microscopic description of their entropy. Section 5 contains
conclusions and future directions. In the appendices A and B we review the construction
of Type-IIA quantum black holes as well as the H-FGK formalism  and the structure
used in their construction.
Type-IIA quantum black holes
Quantum black holes  are solutions of the effective theory corresponding to Type-IIA
String Theory compactified on a Calabi-Yau manifold in the presence of perturbative
corrections to the Special Kahler geometry of the vector multiplet sector. Despite having only
three charges, these four-dimensional supersymmetric black holes have a macroscopically
large horizon area. It is not hard to see either from this or from their explicit construction
that these black holes do not have a macroscopic horizon in the classical limit, when the
perturbative corrections are turned off, which justifies calling them quantum black holes.
Type-IIA string theory on a Calabi-Yau manifold
Type-IIA String Theory compactified to four-dimensions on a Calabi-Yau three-fold, with
supergravity coupled to vector- and hyper-multiplets. As explained in appendix A, we are
going to truncate the hyperscalars to a constant value, and therefore we will only be
concerned about the vector-multiplet sector of the theory. The corresponding prepotential
third polylogarithmic function. The constant c is proportional to the Euler characteristic
The first two terms in the prepotential correspond to tree level and fourth-loop
perwhile the third term comes from non-perturbative corrections produced by world-sheet
instantons. In this paper we will focus on large-volume compactifications where these
corrections can be safely ignored, and hence focus on the quantum black holes obtained
from the prepotential (2.2). The most general quantum black hole solutions, that are
governed by the prepotential (2.1) have been constructed in .
The scalars zi are given by
zi = X .
Adding the constant term c to the prepotential modifies the geometry of the scalar
manifold, which is no longer homogeneous, and therefore it is said that the geometry has been
corrected by quantum effects. The scalar geometry defined by (2.3) is hence the so-called
quantum corrected d-SK geometry [26, 27]. The attractor points of (2.3) have been
extensively studied in .
The Type-IIA quantum black holes belong to a particular class of purely magnetic
black hole solutions of the theory defined by (2.3). We review them in section 2 and refer
the reader to [6, 21] for more details.
The quantum-corrected STU model
Type-IIA quantum black holes exist in any Type-IIA Calabi-Yau compactification with
h1,1 > h2,1. To make contact to a description of these black holes in terms of intersecting
branes we have to choose a particular model and the best candidate is the popular STU
model, whose black hole solutions and attractors have been extensively studied. Since the
2Actually, the prepotential obtained in a Type-IIA Calabi-Yau compactification is symplectically
equivalent to the prepotential (2.1).
hypermultiplets are truncated, the specific value of h2,1 is irrelevant as long as it is smaller
and therefore the prepotential (2.3) becomes
The normal four- and five-dimensional black holes and attractors of this
quantumcorrected STU model been previously considered in [710, 19, 28], but here we focus on
the black holes that do not have a classical (c 0) limit. The solution corresponding to
these black holes (discussed in detail in appendix B) has
The space-time metric is therefore
ds42 = 31c 3 H1H2H3 2/3 dt2 + 3c 3 H1H2H3 2/3 d~y 2 ,
where d~y 2 = dy1 2
+ dy3 2 is the Euclidean metric on R3. Since in the H-FGK
formalism the H-variables correspond to the imaginary part of the covariantly holomorphic
symplectic section appropriately weighted to be Kahler neutral, supersymmetry require
them to be harmonic functions on the transverse space R3. A single-center black hole
Hi = ai +
i = 1, 2, 3 ,
where r2 = y
3 2, the pi are the three charges of the black hole (A.11) and
the ai are arbitrary constants that can be written in terms of the asymptotic value of the
scalars at spatial infinity zi as
where spi is the sign of the charge pi. The entropy of this black hole is
h1,1 = 1 ,
F (X ) = X X 0X
zi = ic 3
ai = spi 3c
and its mass is the sum of the three charges:
It is easy to see that each term that contributes to the mass is positive definite since
Sign ai = Sign pi and a a a
1 2 3 > 0. In the next section we will try to describe this
threecharge four-dimensional black hole in terms of intersecting branes in M-theory.
Notice that the black hole solution presented in this section is a solution of a standard
Supergravity theory without higher-order curvature terms, and which incorporates all the
perturbative String Theory corrections at the two-derivative level. Therefore, it must be
understood to be valid when the curvature remains small compared to the string length
and is in that regime where the perturbative correction that we are considering corrects
the geometry of the black hole and in particular its entropy, making a singular charge
configuration into a non-singular one.
The M-theory configuration
In order to see whether Type-IIA quantum black holes have an interpretation via
intersecting branes it is desirable to have the precise ten-dimensional configuration corresponding
to the four-dimensional solution. For tree-level Type-IIA Calabi-Yau compactifications
the map between the ten-dimensional and the four-dimensional fields is known ,
but since we are considering four-dimensional solutions to the prepotential that includes
an R4-like term in ten dimensions, no explicit map is known. However, as we will show
below, we will still be able to control the behavior of the dilaton and the Calabi-Yau
volume, which will allow us to obtain the higher-dimensional configuration corresponding to
Type-IIA quantum black holes.
The string theory dilaton
As explained in appendix A, for black hole solutions of ungauged four-dimensional
supergravity, the hyperscalars are truncated to a constant value. In principle, the dilaton
belongs to the universal hypermultiplet, and therefore one may naively conclude that it
should be constant for every black hole solution. However, in the process of obtaining the
ings and redefinitons are performed on the original ten-dimensional fields. In addition,
since we are considering all the perturbative corrections to the Special Kahler sector, the
theory of Type-IIA String Theory at tree level, so we should expect more intrincate
redefinitions. Our purpose is to show now that Type-IIA quantum black holes have a constant
ten-dimensional dilaton and a constant Calabi-Yau manifold volume, and we will do this
q as follows [33, 34]
J J J ,
is the volume of the Calabi-Yau manifold, J being the corresponding Kahler form. The
compactification Calabi-Yau manifold by [33, 34]
e 2 Vol6 = const.
Perturbative corrections. It is also easy to see that this tree level result is not changed
when including perturbative corrections. Indeed, [11, 12] have shown that the loop
corrections in ten dimensions that give rise to the perturbative corrections of the prepotential
from the four-dimensional point of view, only mix the dilaton and the volume among
themselves. Therefore, since they were constant at the tree level, they continue to be
constant after the loop corrections have been taken into account. We thus conclude that for
Type-IIA quantum black holes the dilaton and the volume of the Calabi-Yau manifold are
torus [35, 36]
In order to argue that the M-theory uplift of quantum black holes can be interpreted as
coming from a superposition of M2 branes on a conical singularity it is useful to recall
the usual supersymmetric solution corresponding to three stacks of M2 branes on a
six+ H21 dx52 + dx62 + gmndymdyni ,
directions. Supersymmetry requires the transverse metric, gmn, to
where the Hyper-Kahler space is a Gibbons-Hawking (Taub-NUT) space:
ds2GH = V (y)1 d + Ai(y)dyi 2
i = 1, 2, 3 ,
The eleven-dimensional metric is therefore given by
+ H21 dx52 + dx62 + ds2GHi ,
where now the Hi are harmonic in R3.
In order to interpret Type-IIA quantum black-holes as composed of three stacks of
orthogonally intersecting M2-branes, one should uplift their solution to M-theory. Since the
four-dimensional theory where these black holes are constructed includes all the
perturbathe standard eleven-dimensional supergravity to which one has added the higher-derivative
There are two features of the quantum black hole metric (2.9) that will guide us to
obtain this solution. The first is that the volume of the six-dimensional
torus/CalabiYau manifold is constant to all levels in the corrections and hence, rescaling the time to
p3c 3 t, the eleven-dimensional solution can be put into an M2-brane form:
ds2 = (H1H2H3) 3 3c 3 H1H2H3
1 dt2 + H11 dx12 + dx22 + H31 dx32 + dx42
+ H21 dx52 + dx62 + ds2 .
The four-dimensional base metric ds2 is no longer Gibbons-Hawking but becomes:
i = 1, 2, 3 .
This metric is not Ricci flat and it does not even have constant curvature. This is to
be expected, since the solution (3.9) solves the equations of motion of eleven-dimensional
supergravity modified by appropriate higher curvature terms, which modify in turn the
Gibbons-Hawking character of the base. Indeed, if one tries to compare this metric to a
Gibbons-Hawking one, one finds that on one hand the gauge field Ai(y), corresponding to
D6 brane charge in ten dimensions, is zero, but that on the other hand the corresponding
warp factor is not constant by rather has the form:
which has the same behavior at infinity and near the black hole as the warp factor of a
Taub-NUT space with a nontrivial charge:
the function V becomes harmonic throughout the space
despite the absence of a Gibbons-Hawking (D6) charge. It important to notice that in the
dilaton of the quantum black hole solution being constant. This is an nontrivial check that
the eleven-dimensional brane configuration we propose gives the fundamental constituents
of Type-IIA quantum black holes.
Let us remind the reader that the eleven-dimensional solution that we have presented
here was obtained from the conjectured uplift of a four-dimensional quantum black hole,
and it has not been verified to satisfy the involved equations of motion of eleven-dimensional
est solution and we have presented further evidence that supports the chosen uplift: for
example we have obtained that the dilaton must be constant from two different calculations,
one depending on the uplift and another one independent of it.
The microscopic entropy
Having obtained an eleven-dimensional metric that resembles that of three stacks of
coincident M2 branes, we can easily compactify it along one of the torus directions to obtain a
D2-D2-F1 metric, which upon a further T-duality along the F1 direction becomes a
D3-D3P metric, where the momentum P runs along the direction common to the two D3 branes.
This duality chain transforms the quantum eleven-dimensional black hole into a type IIB
D3-D3-P black hole, whose microscopic entropy can be reproduced straightforwardly by
Strominger-Vafa-type arguments. Indeed, if the numbers of the two types of D3 branes are
N1 and N2 and if NP units of momentum are running along the common directions of these
branes, the most efficient way to carry this momentum when N1 and N2 are co-prime is
to use the strings stretched between the two stacks of D3 branes. These bi-fundamental
strings have a mass gap equal to N1N2R , where R is the radius of the common direction
of the branes, and hence the entropy of the system comes from partitioning the NP units
of momentum between modes that carry integer multiples of N11N2 , or otherwise from the
number of integer partitions of N1N2NP . By taking into account the fact that there are
four bosonic species of bi-fundamentals as well as their fermionic partners, this gives the
hole with entropy 2N1N2NP N [2, 3].
The argument above reproduces the entropy of a D3-D3-P black hole in five dimensions,
which is sourced by a stack of branes in R4. One can write this R4 as a Gibbons-Hawking
stack of D3-D3-P branes in a Taub-NUT space with Kaluza-Klein monopole charge N , the
We would like to use a similar argument to explain microscopically the entropy of
our quantum black holes. However, at first glance there are two problems with this: the
3c 3 (H1H2H3)1/3. Nevertheless, near the branes the warp factor behaves like that of a
Gibbons-Hawking space. Hence, even if the branes do not sit on top of an AN singularity,
they sit on some other conical singularity whose effect on the central charge of the D3-D3
CFT one can calculate.
The second problem is that a generic eleven-dimensional uplift of a quantum black
hole will not be a six-torus but a more complicated CY manifold. The key ingredient
needed to relate the quantum black holes to the D3-D3-P system is the presence of two
U(1) isometries, one of which is used for reducing to a ten-dimensional Type IIA black hole,
and the other for T-dualizing.3 This can be easily done for any CY manifold that has a T 2
fiber. Nevertheless, in order for our construction of quantum black holes to yield regular
solution, this elliptic fibration must be singular: CY manifolds with regular fibration have
zero Euler characteristic and hence c vanishes which makes the black hole horizon singular.
in gS , but the resulting solutions involve the Lambert W function and are much harder
The way out is to focus on singular elliptic fibrations, which give CY manifolds with
nonzero Euler number. The places where the fibration degenerates become seven-branes
upon dualization to the type IIB duality frame. The presence of these seven-branes does
not affect the entropy counting, because this entropy comes from D3-D3 strings carrying
momentum, which do not see the seven-branes.
There are two ways to take into account the effect of the conical singularity on the
entropy. The first is to compare this singularity with a Gibbons-Hawking solution, and
determine its effective Gibbons-Hawking charge. The second is to focus on the near-horizon
geometry of the black hole and to compute the corresponding Brown-Henneaux central
charge , which determines how the central charge of the D3-D3 CFT increases when
the D3 branes are placed on top of the conical singularity. As we will explain below, the
two calculations are equivalent, but since the first is more intuitive we will present it here.
Near the tip of a Gibbons-Hawking metric
ds2GH = V (y)1 d + Ai(y)dyi 2
i = 1, 2, 3 ,
the metric becomes that of R4/ZN
V = 1 +
with = 2 r and d (23) the standard metric on S3/ZN . When the D3-D3 system is placed
at the tip of this space its central charge increases by a factor of N . This is a well-known
phenomenon for the D1-D5-P black hole in Taub-NUT , and our system is just its
T-dual. Now, given a conical metric of the type (4.3), there is a way to extract directly
where VS3 is the volume of the three-sphere S3 and VS3/ZN is the volume of the S3/ZN ,
at the same radius. In the Brown-Henneaux formalism this ratio of the volumes also gives
the decrease of the effective three-dimensional Newtons constant, and hence the increase
of the central charge of the corresponding CFT.
3If these two isometries are not present our microscopic description does not work, but neither does the
microscopic description of normal M2-M2-M2 black holes.
For the quantum black hole metric we discussed in section 3.3, the base metric
ds2 = V (r)1d2 + V (r) dr2 + r2d(22) ,
has a conical singularity in the near-tip region:
i = 1, 2, 3
Hence, the microscopic entropy of the quantum black holes will be given by
where N1 and N2 are the numbers of D3 branes and NP is the number of momentum
quanta. Since the supergravity charges are proportional to these numbers, it is clear
that this microscopic entropy count reproduces the correct charge growth of the quantum
3 c p1p2p3 1/3
Hence, the singularity will increase the central charge of the D3-D3-P CFT by a factor
given by the ratio of the volume of S3 and the volume of the transverse space, which is
nothing but NE .
Upon defining (with hindsight)
which is already an important confirmation that our strategy is correct. Of course, the
ideal would be to find exactly the coefficients that relate the supergravity charges to the
quantized ones, and therefore establish that the macroscopic and the microscopic entropies
are identical. However, since we know neither the eleven-dimensional uplifts of the Maxwell
fields of the four-dimensional quantum black hole, nor the volume of the Calabi-Yau
threefold and its submanifolds, we cannot determine these coefficients from first principles.
However, what we can do is to use the symmetry of the STU model in order to argue
that in the M2-M2-M2 duality frame the supergravity charges are related to the quantized
where we have defined pi
equation (4.11) becomes:
pi to ease the presentation. We can now ask what is the value
1 c 6 .
3 c p1p2p3 1/3 .
NE = (4 N1N2NP )1/3
Hence one may naively infer that NE can never be a natural number. However, it turns
There is a nontrivial check that this value satisfies: the number NE , which gives the increase
of the CFT central charge is expected to be a natural number, at least for some values of
N1, N2 and NP . However, NE is defined in terms of c, and therefore contains both a factor
and therefore NE can easily be an integer for a suitable choice of N1, N2 and NP . The
fact that the transcendental number in c drops out of this relation is a nontrivial check
of our proposed microscopic description. Indeed, in  it was argued that a black hole
entropy expression that contains such a transcendental number can never be reproduced
from microscopic calculations, and our microscopic proposal evades this by absorbing this
transcendental number into the central charge increase of the underlying CFT.
The puzzling aspect of equation (4.16) is that it makes the classical limit very hard
to see. Indeed, as we explained in section 2 if one turns off the quantum corrections
(c 0) while keeping the supergravity charges of the black hole constant, the horizon
becomes singular. This is reflected in our construction by the fact that our black hole has
three charges and, when c 0, the warping of the base space becomes trivial and the
corresponding entropy becomes that of the three-charge system in four dimensions, which
does not give rise to a macroscopic horizon.
However, one can also ask what happens if one takes the c 0 limit while keeping the
quantized black hole charges, N1, N2 and NP constant. This does not affect NE, so it looks
like in this limit the black hole entropy remains macroscopic. This is consistent with the
fact that in this limit the four-dimensional supergravity charges blow up (4.14), but the
factors of c cancel from the expression of the near-horizon limit of the metric (2.9), which
remains a regular AdS2 S2. On the other hand, the expressions of the four-dimensional
moduli diverge, and therefore it appears that in this limit the dictionary between ten- and
four-dimensional solutions breaks down. It would clearly be interesting to try to derive
equation (4.14) from first principles to see precisely how this breakdown occurs.
We have constructed an eleven-dimensional metric (3.9) that upon dimensional reduction
gives the Type-IIA quantum black hole of the STU quantum-corrected model. Because
the four-dimensional metric has constant dilaton and CY volume, this eleven-dimensional
uplift can be interpretation as arising from three stacks of orthogonal M2 branes that sit
at the apex of a cone in a four-dimensional transverse space. Because of the presence of
correction terms in the Lagrangian, this space is not Gibbons-Hawking, although it has
exactly the same kind of warping as a Gibbons-Hawking space. The strength of the conical
singularity is proportional to the cubic root of the product of the three M2 charges
When the dix-dimensional internal space of the compactification has a T 2 fiber we can
dualize this solution to an asymptotically four-dimensional three-charge D3-D3-P solution
in type IIB string theory. In flat space the microscopic entropy of this system is not enough
to give rise to a regular horizon, but we have shown that the conical singularity enhances
the central charge of this system, and the resulting microscopic entropy reproduces the
which we could not determine. We have however been able to show that if this coefficient
is such that the entropies match, a certain dependence of the entropy on transcendental
numbers drops out, which we believe is a nontrivial check of our proposal.
Since we are working in a supergravity theory in the presence of quantum corrections
to the geometry of the scalar manifold, our proposed microscopic description is not at
the same level of rigor as the usual three- and four-charge black hole entropy counting.
However, the fact that we have found a brane interpretation that reproduces the
highlyunusual charge dependence of the entropy of quantum black holes makes us confident that
we have identified the correct microscopic framework for understanding the entropy of all
Type-IIA quantum black holes, which remains as an important open problem in String
We would like to thank T. Ortn for useful discussions and comments. The work of IB was
supported in part by the ERC Starting Independent Researcher Grant 240210,
String-QCDBH, by the John Templeton Foundation Grant 48222: String Theory and the Anthropic
Universe and by a grant from the Foundational Questions Institute (FQXi) Fund, a donor
advised fund of the Silicon Valley Community Foundation on the basis of proposal
FQXiRFP3-1321 to the Foundational Questions Institute. The work of CS was supported by
the ERC Starting Independent Researcher Grant 259133, ObservableString.
Type-IIA quantum black holes are black hole solutions of Type-IIA String Theory
compactified down to four dimensions on a Calabi-Yau three-fold, which is described, up to
venient to review the basic formulation of the theory and its vector multiplet sector, since
the hypermultiplets and the fermions can be always truncated for black hole solutions. The
can be written as follows [39, 40]
S =Z d4xp|g| nR + Gij(z, z)zizj + 2I(z, z)F F
of the symplectic complex period matrix N . Hence, the period matrix determines the
equations of motion following from the action (A.1) are given by
G + 2Gij zi zj 2 g zizj + 8Im NF +F = 0 ,
(Gijzj) 2 iGjkzj zk +
where we have defined
g F = Re NF + Im N F .
identities can be written in terms of differential forms as follows
be arranged into a symplectic vector AM =
. Supersymmetry constrains the
couplings of all the fields of the theory in a very precise way which, for the vector-multiplet
sector, is elegantly encoded in the language of Special Kahler Geometry [39, 41]. In fact, the
Gij = ijlog i XF X F
Since the metric is spherically symmetric, we will assume that all the fields of the theory
be together arranged into a symplectic vector Q
. In the background given
by (A.9) Maxwells equations can be integrated explicitly, in such a way that the complete
T be a symplectic vector made from the time components of
where MMN is a symplectic and symmetric matrix constructed from the couplings of the
scalars and the vector fields as
Therefore, choosing a second-degree homogeneous function F (X ) automatically determines
which has the appropriate matter content for constructing black hole solutions. The most
general static and spherically symmetric metric that solves the equations of motion (A.1)
is given by 
MMN (N )
We choose to express all Maxwell field strengths in terms of the time components of the
electric and the magnetic connection one-forms. For the electric field strengths this gives:
equation (A.14) using equation (A.3). Since the connection one-forms can be explicitly
supergravity action coupled to vector multiplets can then be shown to be completely
equivalent, assuming the space-time background given (A.9) and radial dependence for all the
SFGK [U, z] =
d nU 2 + Gijzizj e2U Vbh(z, z, Q)o ,
together with the Hamiltonian constraint,
Here Vbh is the so-called black hole potential, which is given by 
U 2 + Gijzizj + e2U Vbh(z, z, Q) = r02 .
Vbh(z, z, Q) 2 MMN (N )QM
We are now ready to introduce the H-FGK formalism. The H-FGK formalism [14, 16, 17,
20, 44] consists of a particular change of variables from the (2nv +1)-real U, zi to a new set
the U-duality group of the theory, and become harmonic functions in Euclidean R3 in the
2 P MN log W
+ P M log W H M
HP = 0 (A.18)
together with the Hamiltonian constraint
2 MN log W
r02 = 0 (A.19)
e2U = W(H) HM (H)HM ,
H M + iHM = V
M being the covariantly holomorphic symplectic section that determines the
vectorH M (I) H M (H) stands for the real part of V
M /X Kahler invariant.
The symplectic vector
M written as a function of the imaginary
part, HM ; this can always be done by solving the stabilization equations. The function
W(H) is usually known in the literature as the Hesse potential.
The effective theory is now expressed in terms of 2 (nv + 1) variables HM . The solution
depends on 2 (nv + 1) + 1 parameters, namely the 2 (nv + 1) charges Q
and the non-extremality parameter r0, from which it is always possible to reconstruct
the complete solution in terms of the four-dimensional fields of the theory. The H-FGK
formalism introduces an extra real degree of freedom. Hence the H-FGK action enjoys
an extra gauge symmetry which, by gauge fixing, allows to get rid of the extra degree of
A quantum class of black holes
In this appendix we present the solution of the equations (A.18) and (A.19) that correspond
to the quantum black holes of Type-IIA String Theory. Type-IIA quantum black holes are
based on the following truncation of the H-variables and the charges
Using now equation (B.1) together with equations (A.21) and (2.3) we find
H0 = H0 = Hi = 0 ,
p0 = q0 = qi = 0 .
e2U = W(H) =
metric, c must be positive, that is, h1,1 > h2,1 is a necessary condition in order to obtain
an admissible solution. There are plenty of Calabi-Yau manifolds that satisfy this condition,
so we will not worry any more about it. The scalar fields, purely imaginary, are
zi = i (3!) 31 c 31
It is easy to see that the solution is not consistent in the classical limit c 0, and also that
solving the equations of motion. Hence, we conclude that the corresponding solutions
are genuinely quantum solutions, i.e., they only exist when the perturbative quantum
corrections are incorporated into the action, and thus they are called Type-IIA quantum
automatically know that 
r0 = 0 ,
solutions we have to take r0 0 and therefore the general metric (A.9) simplifies to
ds42 = e2 U()dt2 + e2 U()mndxmdxn ,
The entropy of the Type-IIA quantum black holes is given by
using equation (A.12). For Type-IIA quantum black holes we obtain
R00 = 0 ,
I0i = Ii0 = 0 ,
Rij = 0 ,
which in turn implies that the following components of MMN are zero
M0i = Mi0 = 0 ,
M00 = M00 = Mij = Mij = 0 ,
0i = M
i0 = 0 .
From equations (A.12) and (B.1) we obtain
1 Z e2U
Ai t =
1 Z e2U
A0t = 2
1 Z e2U
This implies that the connection one-forms Ai have only magnetic components, which give
rise to the magnetic charges pi of the black hole solution (A.11). Notice however that the
time component of graviphoton A0 is non-zero, although the corresponding charges are
precise cancellation in the corresponding formula for q0:
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