Regular 3charge 4D black holes and their microscopic description
Iosif Bena
0
C.S. Shahbazi
0
GifsurYvette
0
F
0
France
0
Open Access
0
c The Authors.
0
0
Institut de Physique Th eorique, CEA Saclay
,
CNRS URA 2306
The perturbative corrections to TypeIIA String Theory compactified on a CalabiYau threefold allow the construction of regular threecharge supersymmetric black holes in four dimensions, whose entropy scales with the charges as S construct an Mtheory 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
TypeIIA quantum black holes
TypeIIA string theory on a CalabiYau manifold
The quantumcorrected STU model
The Mtheory configuration
The string theory dilaton
M2branes 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 threecharge black hole in five dimensions [1] and the
fourcharge black hole in four dimensions [24]. The entropies of these black holes scale
like the square root of the product of their charges (or some some dualityinvariant form
thereof [5]), 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 fourdimensional compactifications of string theory one can
construct threecharge black holes whose entropy scales with these charges like S
2
p1p2p3 3 [6]. 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 fourdimensional supergravity, where
the horizon curvature of threecharge 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 [6]. The first step in this direction is to uplift
these black hole solutions to eleven dimensions and to propose an Mtheory interpretation
in terms of three mutuallyorthogonal intersecting stacks of M2branes at the tip of a
fourcompactifications come from higherderivative terms in String Theory [11, 12], the uplifted
black hole will be a solution of the equations of motion of elevendimensional supergravity
modified by the addition of certain higherderivative terms.
The second step is to argue that if the CalabiYau manifold can be written as a (possibly
singular) elliptic fibration, the branes that make the elevendimensional solution can be
dualized to a configuration of two intersecting mutuallysupersymmetric 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 TypeIIA String Theory
compactified on a CalabiYau manifold as well as the corresponding TypeIIA quantum
black hole solutions. In section 3 we argue that the Mtheory 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 TypeIIA quantum black holes as well as the HFGK formalism [1321] and the structure
used in their construction.
TypeIIA quantum black holes
Quantum black holes [6] are solutions of the effective theory corresponding to TypeIIA
String Theory compactified on a CalabiYau manifold in the presence of perturbative
corrections to the Special Kahler geometry of the vector multiplet sector. Despite having only
three charges, these fourdimensional 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.
TypeIIA string theory on a CalabiYau manifold
TypeIIA String Theory compactified to fourdimensions on a CalabiYau threefold, with
supergravity coupled to vector and hypermultiplets. 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 vectormultiplet 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 fourthloop
perwhile the third term comes from nonperturbative corrections produced by worldsheet
instantons. In this paper we will focus on largevolume 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 [21].
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 socalled
quantum corrected dSK geometry [26, 27]. The attractor points of (2.3) have been
extensively studied in [28].
The TypeIIA 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 quantumcorrected STU model
TypeIIA quantum black holes exist in any TypeIIA CalabiYau 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 TypeIIA CalabiYau 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 fivedimensional 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 spacetime metric is therefore
1 1
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 HFGK
formalism the Hvariables 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 singlecenter 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
,
S =
and its mass is the sum of the three charges:
M =
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 fourdimensional black hole in terms of intersecting branes in Mtheory.
Notice that the black hole solution presented in this section is a solution of a standard
Supergravity theory without higherorder curvature terms, and which incorporates all the
perturbative String Theory corrections at the twoderivative 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 nonsingular one.
The Mtheory configuration
In order to see whether TypeIIA quantum black holes have an interpretation via
intersecting branes it is desirable to have the precise tendimensional configuration corresponding
to the fourdimensional solution. For treelevel TypeIIA CalabiYau compactifications
the map between the tendimensional and the fourdimensional fields is known [2932],
but since we are considering fourdimensional solutions to the prepotential that includes
an R4like 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 CalabiYau
volume, which will allow us to obtain the higherdimensional configuration corresponding to
TypeIIA quantum black holes.
The string theory dilaton
As explained in appendix A, for black hole solutions of ungauged fourdimensional
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 tendimensional fields. In addition,
since we are considering all the perturbative corrections to the Special Kahler sector, the
theory of TypeIIA String Theory at tree level, so we should expect more intrincate
redefinitions. Our purpose is to show now that TypeIIA quantum black holes have a constant
tendimensional dilaton and a constant CalabiYau manifold volume, and we will do this
q as follows [33, 34]
Vol6 =
J J J ,
is the volume of the CalabiYau manifold, J being the corresponding Kahler form. The
compactification CalabiYau manifold by [33, 34]
eK =
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 fourdimensional 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
TypeIIA quantum black holes the dilaton and the volume of the CalabiYau manifold are
torus [35, 36]
In order to argue that the Mtheory 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 HyperKahler space is a GibbonsHawking (TaubNUT) space:
ds2GH = V (y)1 d + Ai(y)dyi 2
i = 1, 2, 3 ,
The elevendimensional metric is therefore given by
+ H21 dx52 + dx62 + ds2GHi ,
where now the Hi are harmonic in R3.
In order to interpret TypeIIA quantum blackholes as composed of three stacks of
orthogonally intersecting M2branes, one should uplift their solution to Mtheory. Since the
fourdimensional theory where these black holes are constructed includes all the
perturbathe standard elevendimensional supergravity to which one has added the higherderivative
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 sixdimensional
torus/CalabiYau manifold is constant to all levels in the corrections and hence, rescaling the time to
1
p3c 3 t, the elevendimensional solution can be put into an M2brane form:
ds2 = (H1H2H3) 3 3c 3 H1H2H3
1 dt2 + H11 dx12 + dx22 + H31 dx32 + dx42
+ H21 dx52 + dx62 + ds2 .
The fourdimensional base metric ds2 is no longer GibbonsHawking 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 elevendimensional
supergravity modified by appropriate higher curvature terms, which modify in turn the
GibbonsHawking character of the base. Indeed, if one tries to compare this metric to a
GibbonsHawking 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
TaubNUT space with a nontrivial charge:
the function V becomes harmonic throughout the space
despite the absence of a GibbonsHawking (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 elevendimensional brane configuration we propose gives the fundamental constituents
of TypeIIA quantum black holes.
Let us remind the reader that the elevendimensional solution that we have presented
here was obtained from the conjectured uplift of a fourdimensional quantum black hole,
and it has not been verified to satisfy the involved equations of motion of elevendimensional
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 elevendimensional 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
D2D2F1 metric, which upon a further Tduality along the F1 direction becomes a
D3D3P metric, where the momentum P runs along the direction common to the two D3 branes.
This duality chain transforms the quantum elevendimensional black hole into a type IIB
D3D3P black hole, whose microscopic entropy can be reproduced straightforwardly by
StromingerVafatype 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 coprime is
to use the strings stretched between the two stacks of D3 branes. These bifundamental
1
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 bifundamentals as well as their fermionic partners, this gives the
hole with entropy 2N1N2NP N [2, 3].
The argument above reproduces the entropy of a D3D3P black hole in five dimensions,
which is sourced by a stack of branes in R4. One can write this R4 as a GibbonsHawking
stack of D3D3P branes in a TaubNUT space with KaluzaKlein 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
GibbonsHawking 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 D3D3
CFT one can calculate.
The second problem is that a generic elevendimensional uplift of a quantum black
hole will not be a sixtorus but a more complicated CY manifold. The key ingredient
needed to relate the quantum black holes to the D3D3P system is the presence of two
U(1) isometries, one of which is used for reducing to a tendimensional Type IIA black hole,
and the other for Tdualizing.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 [21], 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 sevenbranes
upon dualization to the type IIB duality frame. The presence of these sevenbranes does
not affect the entropy counting, because this entropy comes from D3D3 strings carrying
momentum, which do not see the sevenbranes.
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 GibbonsHawking solution, and
determine its effective GibbonsHawking charge. The second is to focus on the nearhorizon
geometry of the black hole and to compute the corresponding BrownHenneaux central
charge [37], which determines how the central charge of the D3D3 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 GibbonsHawking 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 D3D3 system is placed
at the tip of this space its central charge increases by a factor of N . This is a wellknown
phenomenon for the D1D5P black hole in TaubNUT [38], and our system is just its
Tdual. Now, given a conical metric of the type (4.3), there is a way to extract directly
N =
where VS3 is the volume of the threesphere S3 and VS3/ZN is the volume of the S3/ZN ,
at the same radius. In the BrownHenneaux formalism this ratio of the volumes also gives
the decrease of the effective threedimensional 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 M2M2M2 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 neartip region:
i = 1, 2, 3
S =
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
2r
ds2
2r
NE
3 c p1p2p3 1/3
Hence, the singularity will increase the central charge of the D3D3P 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 elevendimensional uplifts of the Maxwell
fields of the fourdimensional quantum black hole, nor the volume of the CalabiYau
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 M2M2M2 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 .
NE =
3 c p1p2p3 1/3 .
NE = (4 N1N2NP )1/3
the
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 [7] 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 threecharge 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 fourdimensional supergravity charges blow up (4.14), but the
factors of c cancel from the expression of the nearhorizon limit of the metric (2.9), which
remains a regular AdS2 S2. On the other hand, the expressions of the fourdimensional
moduli diverge, and therefore it appears that in this limit the dictionary between ten and
fourdimensional 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 elevendimensional metric (3.9) that upon dimensional reduction
gives the TypeIIA quantum black hole of the STU quantumcorrected model. Because
the fourdimensional metric has constant dilaton and CY volume, this elevendimensional
uplift can be interpretation as arising from three stacks of orthogonal M2 branes that sit
at the apex of a cone in a fourdimensional transverse space. Because of the presence of
correction terms in the Lagrangian, this space is not GibbonsHawking, although it has
exactly the same kind of warping as a GibbonsHawking space. The strength of the conical
singularity is proportional to the cubic root of the product of the three M2 charges
When the dixdimensional internal space of the compactification has a T 2 fiber we can
dualize this solution to an asymptotically fourdimensional threecharge D3D3P 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 fourcharge 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
TypeIIA 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,
StringQCDBH, 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
FQXiRFP31321 to the Foundational Questions Institute. The work of CS was supported by
the ERC Starting Independent Researcher Grant 259133, ObservableString.
TypeIIA quantum black holes are black hole solutions of TypeIIA String Theory
compactified down to four dimensions on a CalabiYau threefold, 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 d4xpg 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 vectormultiplet
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
zi
Therefore, choosing a seconddegree 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 [4244]
MMN (N )
RI1
We choose to express all Maxwell field strengths in terms of the time components of the
electric and the magnetic connection oneforms. For the electric field strengths this gives:
equation (A.14) using equation (A.3). Since the connection oneforms can be explicitly
supergravity action coupled to vector multiplets can then be shown to be completely
equivalent, assuming the spacetime 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 socalled black hole potential, which is given by [43]
U 2 + Gijzizj + e2U Vbh(z, z, Q) = r02 .
Vbh(z, z, Q) 2 MMN (N )QM
We are now ready to introduce the HFGK formalism. The HFGK 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 Uduality group of the theory, and become harmonic functions in Euclidean R3 in the
2 P MN log W
2 Q
+ P M log W H M
H P
HP = 0 (A.18)
together with the Hamiltonian constraint
2 MN log W
2 Q
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 nonextremality parameter r0, from which it is always possible to reconstruct
the complete solution in terms of the fourdimensional fields of the theory. The HFGK
M =
formalism introduces an extra real degree of freedom. Hence the HFGK 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 TypeIIA String Theory. TypeIIA quantum black holes are
based on the following truncation of the Hvariables 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 CalabiYau 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 TypeIIA quantum
automatically know that [4547]
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 TypeIIA quantum black holes is given by
S =
using equation (A.12). For TypeIIA 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 oneforms 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 nonzero, although the corresponding charges are
precise cancellation in the corresponding formula for q0:
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
q0 =
iG0 .
JHEP 12 (1997) 002 [hepth/9711053] [INSPIRE].
Phys. Rev. D 53 (1996) 5344 [hepth/9602014] [INSPIRE].
JHEP 01 (2013) 089 [arXiv:1210.2817] [INSPIRE].
the ST model, Nucl. Phys. B 506 (1997) 267 [hepth/9704095] [INSPIRE].
heterotic string vacua, Nucl. Phys. B 514 (1998) 227 [hepth/9705150] [INSPIRE].
vacua, Phys. Lett. B 429 (1998) 297 [hepth/9802140] [INSPIRE].
theories on CalabiYau spaces, Nucl. Phys. B 507 (1997) 571 [hepth/9707013] [INSPIRE].
universal hypermultiplet, Class. Quant. Grav. 20 (2003) 5079 [hepth/0307268] [INSPIRE].
Equations, Class. Quant. Grav. 27 (2010) 235008 [arXiv:1006.3439] [INSPIRE].
JHEP 07 (2012) 163 [arXiv:1112.2876] [INSPIRE].
Phys. Lett. B 707 (2012) 178 [arXiv:1107.5454] [INSPIRE].
supergravity, JHEP 07 (2011) 041 [arXiv:1105.3311] [INSPIRE].
supergravity with a quantum correction, in the HFGK formalism, JHEP 04 (2013) 157
[arXiv:1212.0303] [INSPIRE].
[26] B. de Wit and A. Van Proeyen, Special geometry, cubic polynomials and homogeneous
quaternionic spaces, Commun. Math. Phys. 149 (1992) 307 [hepth/9112027] [INSPIRE].
Nucl. Phys. B 400 (1993) 463 [hepth/9210068] [INSPIRE].
Class. Quant. Grav. 8 (1991) 789 [INSPIRE].
dimensions, hepth/0408044 [INSPIRE].
Fortsch. Phys. 53 (2005) 1179 [hepth/0507153] [INSPIRE].
scalar manifolds: Symplectic covariance, gaugings and the momentum map,
arXiv:1307.3064 [INSPIRE].
Nucl. Phys. B 500 (1997) 75 [hepth/9702103] [INSPIRE].