AdS3: the NHEK generation
HJE
AdS3: the
Iosif Bena 0 1 3 6 7 8
Lucien Heurtier 0 1 3 4 5 7 8
Andrea Puhm 0 1 2 3 7 8
0 Boulevard du Triomphe , CP225, 1050 Brussels , Belgium
1 F91128 Palaiseau , France
2 Department of Physics, University of California
3 F91191 Gif sur Yvette , France
4 Centre de Physique Theorique, Ecole Polytechnique, CNRS, Universite ParisSaclay
5 Service de Physique Theorique, Universite Libre de Bruxelles
6 Institut de Physique Theorique, CEA Saclay
7 vedimensional nearhorizon extremal Kerr
8 Santa Barbara , CA 93106 , U.S.A
It was argued in [1] that the (NHEK) geometry can be embedded in String Theory as the infrared region of an innite family of nonsupersymmetric geometries that have D1, D5, momentum and KK monopole charges. We show that there exists a method to embed these geometries into asymptoticallyAdS3 S3=ZN solutions, and hence to obtain in nite families of ows whose infrared is NHEK. This indicates that the CFT dual to the NHEK geometry is the IR
AdSCFT Correspondence; Black Holes in String Theory; Dbranes

xed
point of a Renormalization Group ow from a known local UV CFT and opens the door
to its explicit construction.
2.1
2.2
2.3
3.1
3.2
3.3
1 Introduction 2 In nite families of NHEK embeddings in String Theory
AlmostBPS D1D5PKK solutions
NonBPS D1D5PKK solutions with a NHEK infrared
Towards the full explicit form of the RR elds
3
An AdS3 throat with a NHEK
Solutions with NHEK infrared
Hunting for AdS asymptotics
Constraints for AdS asymptotics
3.3.1
3.3.2
The KerrCFT conjecture [2] relates the nearhorizon geometry of an extremal Kerr black
hole [3] to a putative 1+1 dimensional conformal eld theory whose central charges are
given by the angular momenta of this black hole. This connection allows one to count
microscopically the entropy of extremal Kerr black holes, and its discovery is an important
step in extending the powerful machinery of string theory to analyze nonsupersymmetric
black holes.
Unfortunately, besides the central charge, very little is known about the CFT dual to
the NearHorizon Extremal Kerr (NHEK) geometry. In particular, it is not known whether
this CFT can be realized as the infrared xed point of a Renormalization Group (RG) ow
from a known UV theory, or as a lowenergy theory on a system of strings and branes.
In order to achieve such a construction it is important to embed the NHEK geometry in
String Theory, and to look for a system of branes that, when gravity is turned on, gives rise
to a NHEK geometry. The rst attempt at such an embedding was made by the authors
of [4, 5], who constructed a solution with D1, D5 and TaubNUT charges, that has a
vedimensional NHEK geometry in the infrared. However, this solution has xed moduli, and
hence does not allow one to search for a microscopic theory that ows in the infrared to
the CFT dual to NHEK.
{ 1 {
In [
1
], it was shown that one can embed the vedimensional NHEK geometry in a very
large family of supergravity solutions, parameterized by several continuous parameters, and
that moreover one can obtain multicenter solutions where the geometry near one of the
centers is NHEK. All these solutions belong to a class of extremal nonsupersymmetric
solutions that can be obtained by performing a duality sequence known as generalized
spectral ow [6] on the wellknown almostBPS solutions [7, 8].1 The existence of this very
large family of solutions that have a NHEK region in the infrared raises the hope that one
may be able to nd a ow from a UV that is AdS to the NHEK geometry in the infrared.
This would imply that the UV CFT is a \nice" local CFT, with wellunderstood operators,
etc. One could then go ahead and investigate this UV CFT and
nd which operators
procedures to embed the in nite families of multicenter solutions with NHEK regions [
1
]
into asymptoticallyAdS3 solutions. The rst procedure is to use the explicit form of the
solutions [
1
] and to investigate various limits of the parameters that control the UV of these
solutions in order to produce an AdS factor in the metric. The second is to make clever
use of the fact that the only di erence between the asymptotics of BPS and almostBPS
solutions is the sign of one of the electric elds [7], of the fact that BPS solutions transform
under generalized spectral ows into other BPS solutions [6], and of the fact that BPS
solutions develop an AdS3 UV region when certain of their moduli are put to zero, in order
to perform a systematic search for solutions with a NHEK infrared and an AdS3 UV.
At rst glance, both these procedures should be automatically successful. The
asymptotics of the solutions is controlled by 17 parameters, and if one sets to zero the constant,
1 and cos2 terms in g
and certain divergent components of the elds one should obtain
a 10parameter family of solutions that have the leading radial component of the metric of
the form d 22 , which is the hallmark of an asymptoticallyAdS solution. However, things are
not so simple. Most of the solutions obtained in this class have closed timelike curves, and
if one tries to naively impose all the conditions that eliminate the closed timelike curves
none of the asymptoticallyAdS solution seem to survive. Similarly, if one tries to obtain
these solutions by relating BPS and almostBPS solutions (as we will explain in detail
in appendix A) at the end one has to solve 7 equations for 17 variables, which however
appear at rst glance to have no solution. Thus, despite the presence of a large number of
available constants, neither of the two hunting methods we use seems to be very willing to
yield asymptoticallyAdS solutions.
Fortunately, a careful analysis reveals that things are not so bleak. Indeed, we nd
that among the many ways to solve the constraints associated to the absence of closed
timelike curves, there is one that produces nontrivial solutions that are asymptotically
AdS3
S3=ZN . Furthermore, the second method produces exactly the same solution, which
we take as a remarkable con rmation that we have really identi ed the way to construct
solutions that have an AdS3 factor in the UV and NHEK in the IR. The existence of such
a class of ows has several important implications.
1Other approaches to understanding the microstates of Kerr black holes include e.g. [9].
2See [10{12] for approaches along similar lines.
{ 2 {
The rst is for the debate whether the theory dual to the NHEK geometry can be
described as the infrared limit of a local CFT2 or only of a nonlocal one. Since the NHEK
geometry can be obtained by a certain identi cation of an (uncompacti ed) warped AdS3
geometry, it was argued in [13] that the theory dual to NHEK is the DLCQ of a
nonlocal theory which is dual to warped AdS3 (oftentimes known as a \dipole" quantum
eld theory).
Another proposal is that the dual to NHEK is given by the identi
cation of a more exotic type of conformal eld theory, called warpedCFT (wCFT) [14{16],
which would be a local theory [17]. Our construction shows that the theory dual to the
NHEK geometry could be equally well obtained as the IR
xed point of many RG
ows
of \vanilla CFT's" and hence it can be understood without resorting to wCFT's or dipole
needs to turn on to ow to a NHEK infrared,3 and in particular whether this deformation
is similar to the one that triggers the RG
ow of the asymptotically AdS3
the nearhorizon AdS3 of a BPS black ring [20], or whether it is rather a deformation of the
Lagrangian. Thus, the existence of this family of ows opens a new route for determining
S3 solution to
what the CFT dual to 5D NHEK is.
In addition to embedding the NHEKcontaining solutions of [
1
] in AdS3, we also
construct the full form of their RR threeform
eld strength. These solutions can be obtained
by dualizing the twicespectrally owed almost BPS solutions that were constructed in the
M2M2M2 duality frame in [21] to the D1D5P duality frame. In [
1
] this duality
transformation was performed for the metric and the dilaton, which was enough to ascertain
the existence of these solutions and to perform some basic regularity checks. However,
to perform all the regularity checks and to be able to understand all the properties of
these solutions one must also construct this threeform explicitly. As we will see in
appendix B, even if this construction involves several duality transformations that act rather
nontrivially on the RR
elds, the
nal implicit form of the expression that give
threeform
eld strength is quite simple. However, its explicit form is much more complicated
than for BPS and almostBPS solutions, even after making several simplifying assumptions
(equation (2.21)).
Besides its importance for the programme of embedding the NHEK geometry in String
Theory, the calculation of the RR threeform also lls an important gap in our knowledge of
almostBPS solutions and generalized spectral ows thereof. Indeed, the full solution that
comes from applying three generalized spectral ows on an almostBPS solution has so far
only been constructed in the M2M2M2 duality frame [21]. Writing some of these solutions
in the D1D5P duality frame allows one to embed these solutions into sixdimensional
3And conversely what is the deformation of the NHEK geometry that allows it to UV ow to a nice
local CFT in the UV [13].
{ 3 {
ungauged supergravity and explore whether these solutions belong to a larger class of
wiggly solutions, as it happens when supersymmetry is preserved [22, 23].
This paper is organized as follows. In section 2 we review the family of supergravity
solutions that contain NHEK regions in the infrared and that can be obtained by a
sequence of generalized spectral ow transformations from almostBPS solutions with D1,
D5, momentum and KK monopole charges. We then
nd the explicit RR
elds for these
solutions whose derivation is given in appendix B. In section 3 and appendix A we
develop two di erent systematic procedures to search for solutions with an AdS UV, and
identify a subclass of the large family of supergravity backgrounds with a NHEK infrared
constructed in [
1
] that have an AdS3
S3=ZN asymptotic region.
2
In nite families of NHEK embeddings in String Theory
In [
1
] it was shown that an in nite family of IIB supergravity solutions with a NHEK
infrared can be obtained by performing generalized spectral ow transformations [6, 21]
on a class of nonsupersymmetric, \almostBPS", multicenter supergravity solutions [7, 8]
whose charges correspond to D1 and D5 branes, momentum and KK monopoles. However,
in [
1
] only the metric and dilaton have been constructed. While this was enough to ascertain
the existence of such solutions, in order to perform all regularity checks, calculate their
asymptotic charges or use holography to read o the features of the UV CFT, the explicit
expressions for the RR
elds are needed. It is the purpose of this section to complete
the construction of the supergravity solutions that contain a NHEK infrared by explicitly
computing these RR elds.
2.1
AlmostBPS D1D5PKK solutions
The vedimensional Hodge star is with respect to the time bration over the
fourdimensional base space
ds42 = V 1(d
+ A)2 + V d 2 + 2(d 2 + sin2 d 2)
with
?3 dA =
dV;
(2.3)
4We work in units where 0 = gs = 1 and use the conventions of [21]. This solution can be derived from
a sequence of duality transformation acting on an M2M2M2 solution where the three M2 branes wrap
that the relative sign in the metric (2.1) between dz and A3 di ers from the one in [
25
]. We explain the
reason for this in appendix B where we explicitly carry out the sequence of duality transformations on the
spectrally owed M2M2M2 solution to get the spectrally owed D1D5P solution reviewed in section 2.2.
{ 4 {
The metric of the extremal BPS and almostBPS D1D5PKK solutions is [
20, 24, 25
]4
ds2 =
1
Z3pZ1Z2
(dt + k)2 + pZ1Z2ds42 + p
(A3
in (2.3) speci es the orientation of the TaubNUT base and distinguishes
between BPS and almostBPS solutions [7]. We will consider almostBPS solutions
corresponding to the minus sign in (2.3) for which A =
qKK cos d . The oneforms AI consist
of an \electric" and a \magnetic" part:
AI =
dt + k
ZI
+ aI ;
with the warp factors ZI with I = 1; 2; 3 encoding respectively the asymptotic electric
D1, D5 and momentum charges and the vector potentials aI encoding the local magnetic
dipole charges. In the base space (2.3) the magnetic oneform potentials aI and the angular
momentum oneform k can be decomposed as
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
(2.9)
(2.10)
(2.11)
(2.12)
HJEP05(216)
aI = KI (d
+ A) + wI ;
k = (d
+ A) + ! ;
where KI and
are functions and wI and ! are oneforms in the threedimensional at
space. Solution speci ed by ZI , aI , k are obtained by solving the almostBPS equations
2
d ?3 dZI =
CIJK V d ?3 d(KJ KK ) ;
?3dwI = KI dV
V dKI ;
?3d! = d( V )
V ZI dKI ;
d ?3 d(V ) = d(V ZI ) ?3 dKI :
where CIJK = j IJK j. Acting with d?3 on (2.9) yields an equation for :
and
obtained by solving5
The solutions ZI and
denote by LI and M .
to (2.7) and (2.10) contain harmonic functions which we will
An almostBPS solution is then determined by the functions
(V; fKI g; fLI g; M ). In anticipation of section 2.2 we also de ne [21] the oneforms vI
?3dvI = dZI
2
CIJK [V d(KJ KK )
KJ KK dV ] ;
?3d = ZI dKI
KI dZI + V d(K1K2K3)
K1K2K3dV:
2.2
NonBPS D1D5PKK solutions with a NHEK infrared
Applying a sequence of supergravity transformations known as generalized spectral ows [6]
to the solution (2.1){(2.2) yields a large class of solutions that contains the NHEK
geometry as an infrared limit. We refer to [21] for details about the generalized spectral ow
5Note that we renamed the oneform v0 of [21] to
since we already use v0 as the constant in the
TaubNUT potential.
{ 5 {
transformations and summarize here the solution for the NSNS elds [
1
]. We then go on
to compute the RR
elds of the spectrally owed solution.
The spectrally owed extremal D1D5PKK metric is [21]:
ds2 =
e
1
q
Ze3
dt + ek + eaI ;
decomposed as
and the magnetic oneform potentials eaI and the angular momentum oneform ek can be
e
aI = PeI (d
+ Ae) + weI ;
ek = e(d
+ Ae) + !e ;
with !e = !. The oneforms weI and Ae in the threedimensional at space are given by
Ae = A
I wI
CIJK
2
J K vI + 3 ;
e
wI = wI + CIJK J vK
CIJK
2
J K ;
(2.17)
and functions ZeI , WfI , PeI , e
and Ve are given by
(2.13)
(2.14)
(2.15)
(2.16)
(2.18)
ZeI =
WfI =
PeI =
e
=
Ve =
NI
V
e
1
1
NI
V 2
e
;
2
NI
CIJK
2
CIJK
2
T 3V + CIJK J K TI ZI
2
CIJK I J TK ZK
V ZI TI KI +
I ZJ ZK
(2T1
1)V
CIJK J2 K2 TI2ZI2
2
CIJK I J K TJ ZJ TK ZK
T 3V (CIJK J K TI ZI ) + T 6V 2 + 8 3T 3V
;
CIJK
2
+ e ;
WfI
1=2
;
3 Z3 +
J K ZI TI V
I V TJ ZJ TK ZK + T 3V 2
;
where T = (T1T2T3)1=3 and
= ( 1 2 3)1=3 and we de ned
TI = 1 + I KI ;
NI =
2
CIJK I2ZJ ZK + V TI2ZI
2 I V TI :
(2.19)
In order to complete the solution of [
1
] we need to construct the RR threeform
ux
elds and are summarized appendix B. The implicit
expression for this ux has a similar form as that of almostBPS solutions:
where the vedimensional Hodge star is with respect to the time bration over the
fourdimensional base space (2.14).
HJEP05(216)
2.3
Towards the full explicit form of the RR elds
Despite its apparent simplicity, the explicit expression of Fe(3) in terms of the harmonic
functions determining the solutions is a very complicated nested expression. In particular
the action of the vedimensional Hodge star involves repeated use of several of the tilded
forms and functions and the application of the almostBPS equations (2.7){(2.12).
The purpose of this subsection is to give the complete explicit form of this eld for a
certain subclass of solutions. We will restrict ourselves to solutions with
= 0 (yet keeping
! 6= 0), that only have constant terms in the KI harmonic functions (corresponding to
Wilson lines along the TaubNUT direction in
ve dimensions and to axion vev's in four
dimensions) but no poles. The class of solutions whose explicit threeform
eld we
nd
does not include the asymptoticallyAdS3 solutions that are the main focus of this paper.
However, we hope that the (rather complicated) expression we nd will be an important
stepping stone for nding the threeform of that more complicated class of solutions.
To facilitate the calculation we note that while the rst two generalized spectral ows
with parameters 1
; 2 act nontrivially on the solution the third spectral ow with
parameter 3 corresponds to a coordinate transformation. Hence, without loss of generality
we can set 3 = 0 (which implies that T3 = 1). To obtain an explicit expression for the
threeform
ux from (2.20) we express all vedimensional Hodge stars in terms of
threedimensional ones and make successive use of the almostBPS equations (2.7){(2.12). We
refer to appendix B for the details.
The explicit expression for the threeform
ux is:
We now explore the asymptotics of the supergravity solutions of section 2 and, in particular,
whether it is possible to construct a geometry with an AdS3 ultraviolet and a NHEK
infrared. We will pursue two strategies. The rst is to investigate various limits of the
parameters controlling the UV of these solutions in order to produce an AdS factor in
the metric. The asymptotics of the solutions is controlled by 17 parameters. Setting to
zero the constant, 1 , cos
and cos2 terms in g
and certain divergent components of the
elds should yield a 10parameter family of solutions whose leading radial component of
the metric is of the form d 2
2 characteristic of AdS solutions. The second strategy, which
we will present in appendix A, makes clever use of the relations between the asymptotics
of BPS and almostBPS solutions and turns out to give exactly the same class of solutions
as the rst strategy.
3.1
Solutions with NHEK infrared
The supergravity solutions of section 2.2 can have multiple centers where the geometry near
one of the centers is NHEK. Our starting point to look for solutions with AdS asymptotics
is a twocenter solution where in addition to the nonBPS D1D5PKK black hole that
becomes the NHEK geometry in the infrared we add another smooth center corresponding
to a supertube. The asymptotics of this solution are prototypical for all solutions in the
class constructed in [
1
] and, as we will see, the conditions that ensure that the nal solution
is asymptoticallyAdS3 do not depend on the particular distribution of centers and charges
in the infrared.
One can ask whether there is any reason behind our strategy to try to embed the
NHEK solution in an asymptoticallyAdS3 solution that has two or more centers, and
hence topologicalnontrivial threecycles. Our original inspiration came from threecharge
BPS black ring solutions [24, 26, 27] embedded in an asymptotically AdS3
S3 solution [20]:
these rings have another AdS3 region in the vicinity of the black ring center, with smaller
AdS radius; thus the black ring solution can be thought of as an RG
ow from a CFT in
the UV to a CFT with lower central charge in the IR. The full solution has a
topologicallynontrivial threesphere at whose North Pole this smaller AdS3 sits.
One can also see by direct calculation that a multicenter solution is necessary to get
a NHEK region in the infrared. One could try to start from a singlecenter solution and
{ 8 {
play with the constants in the harmonic functions to obtain a cohomogeneityone solution
with an AdS3 UV, but one will nd that the infrared of this solution is always AdS3. This
is essentially because in a singlecenter solution the KI harmonic functions are constant,6
and hence their value is the same at in nity and at the black hole. The only way to get
something other than AdS3 in the infrared is to makes those values di erent by introducing
additional sources for KI , and thus additional centers.
Hence, we begin by considering a twocenter solution with a TaubNUT base containing
a nonBPS black hole and a supertube at distance R in the R3 base of the TaubNUT space.
The solution is easiest to describe using two sets of spherical coordinates, one centered at
the black hole ( ; ) and another centered at the supertube ( ;
), related by
= p 2 + R2
2 R cos ;
cos
=
cos
R
;
cos 0 =
R cos
ux of the solution are given by (2.13) and (2.20) of section 2.2
and are speci ed by the functions V; KI ; LI ; M;
and the oneforms wI ; !; vI ; .7 The
functions V; KI ; LI ; M are
qKK
dI
;
;
M = m0 +
LI = `I0 +
J
+
QI +
j
eI
+ c
:
cos
2 ;
The supertube charges, dipole charges and angular momentum are denoted by feI g =
fe1; e2; 0g, fdI g = f0; 0; d3g and j = e21de32 , and the black hole has charges fQI g = fQ1; Q2;
Q3g and angular momentum J . The KK gauge potential is A =
qKK cos d . The warp
factors are given by ZI = LI . The angular momentum oneform k =
(d
+ A) + ! is
given by8
=
! =
M
V
+ `03d3 +
2V
Q3d3v0 +
+ J cos + j cos
2
2
`03d3v0 cos
(qKK`30 + v0Q3)d3 cos 0 d :
Q3d3qKK cos
V
c
R
sin2
6In an almostBPS solutions the KI functions cannot have poles where the TaubNUT harmonic function
V has poles, as these give rise to singularities or diverging moduli [28].
7In section 6 of [8] the solution for a nonBPS black ring and a black hole in TaubNUT was found in
the type IIA duality frame. We adapt this solution for our purpose, by turning o a charge and two dipole
charges of the black ring to transform it into a supertube.
8Note that in the expression for
given in [8] there is a typo: the factor of cos in the last term in (3.3)
is missing.
{ 9 {
qKK
R
qKK
R
;
Regularity requires the absence of Dirac strings at the poles
= 0; of the twosphere
which implies that ! must vanish there:
0 = ! =0 =
0 = ! =
=
+ J + s j
s
2
qKK
R
2R
+
Q3d3v0
2R
:
between the centers in the R
in the vedimensional solution.
The second equation is interpreted as the bubble equation that determines the distance R
3 base of the TaubNUT space, or the radius of the supertube
3.2
Hunting for AdS asymptotics
We now explore the UV structure of the metric (2.13) speci ed by the functions in
section 3.1. To simplify the analysis we make use of the fact that the third spectral ow
corresponds to a coordinate transformation and so we can set 3 = 0 without loss of
generality. On the other hand, for the IR to correspond to NHEK, 1 and 2 must be
nonvanishing. For the UV to be AdS we need the radial part of the metric to behave as
g
d 2= 2 at large . From (2.13) and (2.14) we read o the radial part of the metric:
and the oneforms ; vI in (2.17) are
and
v3 = (Q3 + k10k20qKK) cos d :
(3.6)
(3.7)
where we made use of (2.18). The large expansion of the NI gives
q
g
= Ve
Ze1Ze2 = pN1N2 ;
NI = nI + I = + I = 2 + O(1= 3) ;
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
with
similarly for 2), and imposing
In general this expression contains terms proportional to cos
that would give rise to
warped AdS asymptotics, and since our purpose is to obtain \normal" AdS we require
these terms to vanish. To leading order this can be achieved by writing I =
I0 + I cos ,
1 corresponding respectively to the rst and second line of (3.16) (and
The leading radial part of the metric is then simply given by
and similar expressions for n2; 2
; 2 (obtained by exchanging 1 $ 2), where tI = 1 + kI0 I
is de ned by expanding the function TI = tI + O(1= ). To have the constant and 1= terms
in (3.12) vanish in the large
expansion requires
After imposing these constraints the leadingorder term in the radial part of the metric
becomes
g
g
p
(3.14)
(3.15)
(3.16)
cos ;
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
subject to relations between the parameters of the solution following from the
constraints (3.17) and (3.19). Regularity imposes further constraints: for absence of closed
timelike curves (CTCs) we need
as well as
Z1Z2Z3V
!2 > 0 ;
2V 2
+
(CTC)0 + (CTC)1= + (CTC)2= 2 +
> 0 ;
where in the second line we have expanded the noCTC condition in the UV. Since the
constants `I0 are nonnegative and the charges QI , e1, e2 in ZI are strictly positive, to avoid
CTCs we also need the constant v0 to be nonnegative and the KK monopole charge qKK
to be strictly positive. Altogether we have:
`I0; v0 > 0 ;
QI ; e1; e2; qKK > 0 :
The condition (3.22) is equivalent to the existence of a global time function [29].
In the following we will solve the constraints (3.17) and (3.19) of section 3.2. It is useful
to consider separately the solutions where `03 = 0 and `03 6= 0.
3.3.1
One can then solve n1 = 0 to obtain
The second constraint n2 = 0 is solved by
negative leadingorder contribution (CTC)0 =
have to be imposed:
[n1 = n2 = (CTC)0 = 0]
,
Further solving 1 = 0 for d3 gives
In principle we have to consider both solutions. However, it turns out that canceling the
m20 + `01`02`03v0 in (3.23) both conditions
v0 = `30 t11t22
; `02 =
With this, the constraint 2 = 0 is automatically satis ed. For the metric to correspond
to unwarped AdS at large
we have to impose (3.19) which can be achieved by setting
c =
( R
2
+
`
0
For the solution to be free of CTCs we need to impose (3.22) subject to the
constraints (3.28), (3.29) and (3.30). The leading nonvanishing term is (CTC)2= 2 which,
evaluated at
= 0 or
= , is:
0 6 (CTC)2 =
1
4
(`30)2
t2
2 (Q1 + e1)
+ (`10)2 t1
1
qKK +
2
t2
t1
Q3
1 (Q2 + e2)
2
2
+ 2(`10)2 t
2
1
12 qK2K +
2
(3.25)
(3.26)
(3.27)
(3.28)
(3.29)
however, kills the sought after d 2
= 2 term in the metric. Hence, the solutions in section 3.1
with `03 6= 0 do not have an asymptotic AdS metric.
When `03 vanishes, the nonnegativity of the leading part of the noCTC condition (3.22)
imposes:
,
2 = 0 one can see that one must also have `01 = `
The constraints n1 = n2 = 0 are then automatically satis ed. To satisfy the noCTC
condition (3.22) at order 1= 2 implies
(CTC)2
c + Q3d3(qKK=R + v0=2) 2 sin4
> 0
,
c =
d3Q3(Rv0 + 2qKK) :
2R
Finally, the constraints 1 = 2 = 0 are satis ed by imposing9
tI = 0
,
kI0 =
1= I
for I = 1; 2 :
Notice that this implies that T1 = T2 = 0.10
With these constraints the bubble
equation (3.11) becomes
(3.33)
(3.34)
(3.35)
(3.36)
(3.37)
where we used the usual relation between the supertube charges, dipole charges and
angular momentum. The distance between the centers, R, is determined once the constants
controlling the UV are xed. With these constraints the leading radial metric component
is d 2
= 2 and the ultraviolet is free of CTCs. To ensure that there are no CTCs in the
infrared we expand (3.22) for small :
0 6 Q1Q2Q3qKK
c2 cos2 ;
which can always be satis ed by taking Q1; Q2 su ciently large. The noCTC
conditions (3.21) are automatically satis ed near the black hole. It is also trivial to check that
the determinant of the UV metric at leading order is constant and negative:
Det(gUV)
12(Q2 + e2)
22(Q1 + e1)
3
Q32 sin2
< 0 :
9One can also satisfy 1 = 2 = 0 by setting v0 = 0. This would, however, set j = 0 which is inconsistent
since the supertube has nonvanishing angular momentum.
10This is a consequence of having only one nonvanishing dipole charge (d3 in K3). Replacing the
supertube by a black ring turns on extra dipole charges, d1 and d2, in respectively K1 and K2 and in such
a solution T1 6= 0, T2 6= 0.
3.4
We now summarize the features of the solutions that contain a NHEK region [
1
] and have
an AdS3 UV. The complete bulk metric is given by (2.13) of section 2.2 and is speci ed
by the harmonic functions (3.2) of section 3.1. For the UV of the metric (2.13) to have
a leading radial component d 2
= 2 we have to constrain the constants that appear in the
harmonic functions that determine the solution:
Absence of CTCs requires positivity of the charges (3.24) and
c =
This equation enforces the cancelation of the sin2
term in the asymptotics of !,
and is necessary for avoiding CTCs only when the asymptotics is AdS. This cancelation
is equivalent to the vanishing of the total fourdimensional angular momentum (JR) of
the solution. In the particular solution we consider, this angular momentum has both a
contribution from the black hole as well as a contribution coming from the interaction
between the magnetic dipole charge of the supertube and the electric charge of the black
hole, and equation (3.39) forces these contributions to cancel each other.
For a more general multicenter solution containing one or more black holes (whose
nearhorizon regions are transformed by the generalized spectral ows into a NHEK region),
there will be more contributions to JR (see for example [30]), and we expect that the only
requirements to have asymptoticallyAdS3 solutions are that the constants entering in the
harmonic functions satisfy (3.38) and that the total JR vanishes.
Of course, the more general multicenter solutions will also have to satisfy the
corresponding bubble equations, which for the twocenter solution we focus on are quite simple:
but in general will be much more complicated [8].
In terms of a more standard radial coordinate r2 = 4qKK the UV metric at leading
order is
ds2UV
where
(3.38)
(3.39)
+ 2
1
lUV
r
2
4lU2V ( dT 2 +dY 2)+lU2V (d 2 +sin2 d 2)+
d
N
( dT + dY )2
N (dT + 3dY )
sgn( 1 1) cos d
d
N
sgn( 1 1) cos d
cos ( dT 2 + dY 2) + UV ds2T 4 ;
T =
s Q3v0 1 2 t ;
Y =
s Q3v0
1 2
v0
t + z
k
0
3
+
d3
R
N sgn( 1 2)
;
(3.42)
N = j 1 2jQ3 ;
lU2V = N p(Q1 + e1)(Q2 + e2) ;
= d3pQ3qKKv0 ;
= qKK
e1
Q1 +e1
+
e2
Q2 +e2 2
R
;
UV =
The term in (3.41) proportional to
corresponds to a spectral ow and so we can remove
it by a coordinate transformation:
=
N 2
2lU4V (T + 3Y ) ;
and we de ned
where
(3.43)
1
2
s Q2 +e2
Q1 +e1
:
(3.44)
2
(3.45)
(3.46)
lUV
;
S3=ZN
lUV
T 4 with subleading
perturThe leading part of the UV metric (3.45) is AdS3
bations that trigger the RG ow to NHEK.
This establishes that all the geometries containing NHEK regions in the infrared [
1
]
can be embedded into asymptoticallyAdS3 solutions, and hence that the theories dual to
NHEK can arise as xed points of Renormalization Group ow from the D1D5 CFT dual
to AdS3. Di erent multicenter solutions will correspond to di erent RG ows, and we leave
the detailed study of these ows for future work.
r
2
4lU2V ( dT 2 +dY 2)+lU2V (d 2 +sin2 d 2)+
d
N
+ 0( dT + dY )2
2 0dY (dT + dY )
0 cos ( dT 2 + dY 2) + UV ds2T 4 ;
sgn( 1 2) cos d
Acknowledgments
We would like to thank Monica Guica for valuable discussions and collaboration in the early
state of this project and Geo rey Compere and Diego Hofman for interesting discussions
and comments on the manuscript. The work of I.B. was supported in part by the ERC
Starting Grant 240210 StringQCDBH, by the John Templeton Foundation Grant 48222
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 (this grant was administered by Theiss Research). The work of A.P. is supported by
National Science Foundation Grant No. PHY1205500. I.B. and A.P. are grateful to the
Centro de Ciencias de Benasque Pedro Pascual and to the Aspen Center for Physics for
hospitality and support via the National Science Foundation Grant No. PHYS1066293.
Throats with a NHEK  an alternative approach
In section 3 we carried out a systematic analysis for embedding the multicenter solutions
containing NHEK regions in an asymptotic AdS geometry and identi ed a set of constraints
that have to be satis ed. We now describe a second approach that yields the same set of
constraints.
The leading and subleading asymptotics of an almostBPS solution is controlled by 17
parameters, and the same is true for its BPS equivalent. Generalized spectral ows shu e
these parameters in a certain way, and at the end one has to
x 7 of these parameters in
order to obtain an AdS3 UV. Since the nearhorizon of any almostBPS black hole can be
transformed into a NHEK geometry by choosing the appropriate generalized spectral ow
parameters [
1
], this implies that there should be at least a 10parameter family of solutions
with NHEK IR and an AdS3 UV. However, as emphasized in the Introduction, most of the
solutions have closed timelike curves, which have to be eliminated (as we also did in the
approach described in section 3). At the end we will nd that this approach only produces
one way to get a solution that is asymptotically AdS3 and free of CTCs and this is exactly
the same solution as the one obtained in section 3!
The method. The procedure which we will now develop makes clever use of the fact that
the only di erence between the asymptotics of BPS and almostBPS solutions is the sign
of one of the electric elds [7], of the fact that BPS solutions transform under generalized
spectral ows into other BPS solutions [6], and of the fact that BPS solutions develop an
AdS3 UV region when certain of their moduli are put to zero.11 The procedure we follow
is summarized in this illustration:
BPS: AdS3 UV
# 2 BPS gSFs
BPS+
nonBPS: NHEK IR and AdS3 UV?
UV
" 2 nonBPS gSFs
almostBPS
We start with a BPS solution that has an asymptotic AdS3 geometry. We then perform
two BPS generalized spectral ow transformations (gSF) that yields another BPS solution
whose asymptotics we identify with those of the almostBPS solution of section 3.1 (to
distinguish the BPS and almostBPS solutions whose asymptotics we are matching we use,
respectively, + and
sub/super scripts). This xes some of the moduli of the almostBPS
solution. Applying two nonBPS generalized spectral ow transformations to this
almostBPS solution yields a nonBPS solution which contains a NHEK geometry in the IR and
whose UV we show to correspond to an AdS3 geometry.
P
Hunting for AdS3. In a BPS solution that is determined by 8 harmonic functions [27,
31, 32] one can read o the asymptotics and charges from H = h + P
i ii , where h is the
vector of constants and tot = lim !1
i ii is the total charge vector as determined by
11As in section 3, without loss of generality, we set to zero the third spectral ow.
the harmonic functions H = (V; fKI g; fLI g; M ):
tot = vtot; fkItotg; f`tIotg; mtot :
A BPS solution has AdS3 asymptotics if the vector of constants takes the form [33]:
with l03 6= 0. The requirement that12
h tot; hAdSi = 0 xes m0 =
kt3ot=vtot with vtot 6= 0.
Performing two BPS spectral ow transformations with parameters 1 and 2, corresponds
to the supergravity transformations [6]:13
M ! M+ = M ;
LI ! LI+ = LI
KI ! KI+ = KI
2 I M ;
CIJK J LK + CIJK J K M ;
V ! V+ = V + I KI
1
1
2 CIJK I J LK +
3 CIJK I J K M :
This yields another BPS solution with the new asymptotics:
h2gSF AdS =
For (A.4) to correspond to the asymptotics of the almostBPS solution of section 3.1 we
have to identify up to order 1= :
V+
V ;
Z+
I
ZI ;
(aI+
k+)
(aI
k ) ;
k+
k :
(A.5)
The gauge potentials, warp factors and angular momentum of the almostBPS solution are
given in section 3.1. Those of the BPS solution are given by
a
I+ =
K+
V+
I (d
ZI+ = LI+ +
+ = M+ +
+ A+) + wI+ ;
CIJK KJ+KK+ ;
2
V+
2
V+
1 KI+LI+ +
CIJK KI+KJ+KK+ ;
6
V+2
with the constants in the harmonic functions H+ = fV+; fKI+g; fLI+g; M+g speci ed
by (A.4). The identi cations (A.5) then become [34]
;
K+
I
V+
+
Z+
I
K
I
Z
I
:
(A.9)
12The symplectic product is given by h tot; hi = 2(vtotm0
mtotv0) + (kItot`0
I
`totkI0).
I
13Note that work in the conventions of [21] which have an extra factor of 2 multiplying M+ when compared
to the conventions of [6].
(A.1)
(A.2)
(A.3)
(A.6)
(A.7)
(A.8)
This yields the following relations between the BPS and almostBPS solutions (up to
order 1= ):
from which we can read o h+ = (v0;+; fkI0;+g; f`I0;+g; m0;+):
V+
K+
I
L+
I
M+
CIJK KI+KJ+KK+ ;
;
+ v02; m0; CIJK kJ0; `0J; kK0; `0;
K
4v0; m0; `10; `02; `0;
3
Note, that as in section 3.3 we can use the bubble equation and the supertube relation
between charges, dipole charge and angular momentum to obtain the relation:
From h+ = h2gSF AdS with (A.4) we can solve for the constants of the almostBPS solution:
v0; =
`
0;
I
= 0 ;
=
=
I
CIJK (`0J; kJ0: v0;
2m0; )(`0K; kK0: v0;
2m0; )
;
v0; `0J; `0;
K
8m03;
We can use this to solve for l03 in (A.19) and plug the result into the expression for k30; .
Since m0 is a free parameter, k30; is in fact unconstrained. With this remark and dropping
the \ " in the constraints (A.19) we
nd that the constants entering in the harmonic
functions determining the solution are:
(A.10)
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
(A.17)
!
:
(A.18)
(A.19)
(A.20)
which agrees precisely with the results (3.38) of our analysis in section 3.3. The extra
condition on c in (3.39) follows from the requirement that there be no closed timelike
curves  a condition we also have to impose here. Hence, the method described here
yields exactly the same solution as the one discussed in section 3.3 and we take this as a
remarkable con rmation that we have really identi ed the ow from AdS3 to NHEK.
B
In [
1
] it was shown that a large class of supergravity solutions with a NHEK infrared
can be obtained by dualizing the twicespectrally owed almostBPS solutions that were
constructed in the M2M2M2 duality frame in [21] to the D1D5P duality frame. In [
1
]
this duality transformation has been performed on the metric and dilaton, but not the RR
elds. We now complete this class of supergravity solutions by computing the RR elds
in the D1D5P duality frame.
From M2M2M2 to D1D5P.
The metric and threeform gauge potential of the
twicespectrally owed M2M2M2 solution obtained in [21] are given by
where dTI denote the volume forms of the three twotori TI2 and ds21 = dx24 + dx25, ds22 =
dx26 + dx27, ds23 = dx28 + dx29, denote the metrics on the latter. We use the shortcut notation
(Z~1Z~2Z~3)1=3 and all tilded quantities are de ned in (2.17){(2.18). To bring this
solution to the D1D5P duality frame we have to perform a KaluzaKlein reduction on
one of the torus legs followed by a sequence of three Tduality transformations. Note that
this parallels the chain of dualities derived in [
25
] for dualizing BPS solutions.
• KK reduction on x9. The KaluzaKlein reduction of the solution (B.1) along the x9
direction (renaming the remaining leg of the torus x10
z) yields14
(B.1)
(B.2)
F (4) = dC(3) + dB(2)
^ C(1) = dA~1 ^ dT1 + dA~2 ^ dT2 :
14Note that the sign in the expression for B(2) in (B.2) depends on which torus leg we compactify; if we
chose to reduce along x10 rather than x9 the minus sign in (B.2) would turn into a plus sign. Anticipating
that B(2) will become part of the metric and threeform
ux after the
nal Tduality along z (B.7) this
explains the relative sign between dz and, respectively, A3 in (2.1) and Ae3 in (2.13).
C(1) = 0 ;
1
Z~3pZ~1Z~2
1
4
=
log
Z~1Z~2
~
Z3
;
(dt + k~)2 +
qZ~1Z~2ds42 +
s ~
Z1
Z~2 ds12 +
s ~
Z2
Z~1 ds22 +
pZ~1Z~2 dz2;
~
Z3
B(2) =
~
A3 ^ dz ;
C(3) = A~1 ^ dT1 + A~2 ^ dT2 ;
? (dA
10
e2 ^
dx
5 ^
dx
7 ^
dx ) =
8
? dA
5 e2 ^
^
dz dx ;
6
(B.4)
Z
~5
2
Z Z
~2 ~3
where the vedimensional Hodge star is with respect to the vedimensional metric
• Tduality along x
ds
2
10
F
log
Z Z
~ ~ ~
1 2 3
Z
Z
~3
1
Z Z
~ ~2
• Tduality along z
ds
2
10
F
(3)
=
=
=
1
2
~
Z
3
log
1
p
Z
~5
2
Z Z
~2 ~3
3 1
~ ~
Z Z
1 2
~
Z
~
Z
1
2
;
1=4
q
(dt + k) +
~ 2
~ ~
Z Z
B
F
(2)
(4)
=
=
~
A
3 ^
dz ;
Z
~5
2
Z Z
~2 ~3
3 1
p
~ ~
Z Z
Z
3
1=4
s
~
Z
~
Z
1
2
(dt + k) +
~ 2
~ ~
2
Z Z ds +
1 2 4
p
~
Z
3
~ ~
Z Z
Z
~
Z
1
2
(dz A ) +
3
~ 2
2 2
(ds + ds ) ;
1 2
B
(2)
= 0 ;
? dA
5 e2
dA
e1 ^
(dz
~
A ) :
3
• Tduality along x
C
F
log
~ ~ ~
Z Z Z
Note that the tendimensional Hodge star can be expressed as
(dt+k) +
Finally the RR threeform ux in the D1D5P duality frame is given by
F
e
(3)
=
5
Z
e2
Note that this expression has the same form as the RR threeform ux (2.2) of the
almostBPS solution. However, (B.8) is considerably more complicated. In particular, the action
(B.5)
(B.7)
(B.8)
of the vedimensional Hodge star on this expression involves repeated use of expressions of
the tilded forms and functions, as well the simpli cation of the result using the almostBPS
equations (2.7){(2.12).
To give the explicit form we make some simplifying assumptions. We will restrict
ourselves to solutions with
= 0 (yet keeping ! 6= 0), and allow the KI to be constants
(dKI = 0). Computing the explicit form of the threeform
ux for more general solutions
is much more complicated and we will not address it here.
As mentioned before, the third spectral ow corresponds to a coordinate
transformation in the D1D5P duality frame, so without loss of generality we only focus on the
solution obtained after two spectral ows and set 3 = 0. This leads to considerable
simpli cations. Since T3 = 1 + 3K3 and K3 = const < 1 we have T3 = 1 and the tilded
functions (2.17){(2.18) relevant here simplify to
Ve = jT1T2V
1 2Z3j ;
Wf1 =
N1
T1T2V
1 2Z3
;
e
=
Wf2 =
and we write
where
Pe1 =
N1
G1 + e ;
Wf1
( 1T2Z2 + 2T1Z1)Z3V
;
T1T2V
Pe2 =
N2
V 2
e
1 2Z3
N2
G2 + e ;
Wf2
;
Wf3 =
N3
T1T2V + 1 2Z3
Pe3 =
N3
G3 + e ;
Wf3
G1 = T1K1Z1V + 1Z2Z3 ;
G2 = T2K2Z2V + 2Z1Z3 ;
G3 = K3Z3V:
(B.12)
We recall the form of the gauge potentials (2.15)
(B.9)
;
(B.10)
(B.11)
(B.13)
(B.14)
(B.15)
(B.16)
dek !
Wf2
;
AeI = weI + PeI (d
+ Ae)
To give the explicit form of Fe(3) we trade all vedimensional Hodge stars for
threedimensional Hodge stars and subsequently use the almostBPS equations (2.7){(2.12) to
replace as many of the threedimensional Hodge stars as possible.
First term:
?5 dAe2.
We want to express
1=4
Ze25
e
1
q
Ze3
?5 dAe2 = ?5 dw2 + dPe2 ^ (d
+ Ae) + Pe2dAe +
d
^ (dt + ek)
whose vedimensional Hodge stars are with respect to the metric
q
ds52 =
in term of threedimensional Hodge stars with respect to the at threedimensional base
spanned by yi with i = 1; 2; 3. We introduce the vielbeine
e
e0 = f0(dt + ek) ;
e
e1 = f1(d
eei = fidyi ;
where
With and thus 1
W
f2
? d
3
W
f2
1=4
f = (Z Z )
0
e1 e2
1=4
Z
e3
1=2
;
f = (Z Z )
1
e1 e2
+1=4
V
1=2
;
f = (Z Z )
i
e1 e2
+1=4
+1=2
V
e
:
(B.17)
? (e
5
? (e
5
? (e
5
i
i
1
e
e
^
^
^
e
e
^
1
1
e
^
i
e ) = f f
0 i ijk
^ ijk
ijk j
2 e
j
e
e
^
k
e
e
^
k
e ;
^
0
e ;
k
e ;
e
? (dy
5
i
^
j 1
dy ) = f f f (dt + k) (d + A)
0 1 i
e ^
e ^ ijk
k
dy ;
? dy
5
i
^
1
(dt + k) = f f f (d + A)
0 1 i
e ^
? (d + A) dy
5
e ^
i
^
ijk
2
dy
j
(dt + k) ;
^
e
k
dy ;
1
= f f f (dt + dk)
0 1 i
e ^
1
f f f (dt + dk) (d +A) e
0 1 i
e ^
e
With this, the Hodge star combinations appearing in (B.14) are
? dA =
3 e
T T dV
1 2
3 f2
? dW = (V ? dZ
3
3
the terms in (B.14) can be expressed as
d
^
e
1
(dt+k) = f f f (d + A)
0 1 i
e ^
1
1
1
? dw = f f f
5 2
0 1 i
e ^
(dt + k) (d + A) ? dw ;
3 2
e
e
? dP
5
e2 ^
(d + A) = f f
? dW
3 f2
W
f2
2
;
? d
3
W
f2
e
V Z
3
T T V
1 2
Finally we get for (B.14):
? dA = (dt + !) (d + A)
^
e ^
(V ? dZ
3
3
Z ? dV ) +
3 3
( T ? dZ +
1 2 3 2
dz). The second term in equation (B.8) is given by
dAe1 ^ (Ae3
dz) = d
(dt + ek) + Pe1(d + Ae) + we1
^
(dt + ek) + Pe3(d + Ae) + we3
dz :
(B.25)
Using (B.9){(B.12) we can write (B.25) as
dAe1 ^ (Ae3
dz) = (dt + !) ^ (we3
dz) ^ d
+ (d + Ae) ^ (we3
dz) ^ d
(dt + ek) +
1
Wf3
G1
N1
N3
G3 + e
Wf3
G3
N3
d
1
Wf1
(d +Ae) ^
where, using the almostBPS equations (2.7){(2.12) one can write
dAe =
T1T2 ?3 dV
1 2 ?3 dZ3 ;
d!1 = K1T2 ?3 dV + 2 ?3 dZ3 :
(B.27)
Putting (B.24) and (B.26) together we arrive at the following expression for the RR eld
strength:
+ (w3 dz)
e
(B.28)
(B.29)
e
where
e
and we recall from section 2.2:
As a check, a tedious but straightforward exercise shows that the expression (B.28) is
closed.
Open Access.
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any medium, provided the original author(s) and source are credited.
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