Gauge/Gravity Correspondence and Black Hole Attractors in Various Dimensions A dissertation presented by Wei Li to The Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Physics Harvard University Cambridge, Massachusetts June 2008
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Gauge/Gravity Correspondence and Black HoleAttractors in Various Dimensions
6.3 N = 2 SLFT and its Boundary States . . . . . . . . . . . . . . . . . . 1706.3.1 N = 2 SLFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 1706.3.2 N = 2 Ishibashi States and Cardy States . . . . . . . . . . . . 1736.3.3 Falling Euclidean D0-brane in N = 2 SLFT . . . . . . . . . . 1756.3.4 Falling D0-brane in N = 2 SLFT . . . . . . . . . . . . . . . . 177
6.4 Falling D0-brane in N = 1, 2D Superstring Theory . . . . . . . . . . 1786.4.1 Using N = 2 SLFT to study boundary states in 2D superstring 1786.4.2 Number of D0-branes after GSO projection . . . . . . . . . . . 179
The results in Chapter 2 were obtained in collaboration with M. Guica, L. Huangand A. Strominger. The text has appeared in the following published work:
“R2 Corrections for 5D Black Holes and Rings”, M. Guica, L. Huang, W.Li and A. Strominger. JHEP 10, 036 (2006), hep-th/0505188;
I would like to thank M. Cyrier, D. Gaiotto and X. Yin for helpful discussions relatedto the material in this chapter.
The results in Chapter 3 were obtained in collaboration with A. Strominger. Theyhas been published in:
“Supersymmetric probes in a rotating 5D attractor”, W. Li and A. Stro-minger. Phys. Lett. B659, 407-415 (2007), hep-th/0605139;
I would also like to acknowledge M. Ernebjerg, D. Gaiotto, L. Huang, J. Lapan andX. Yin for valuable discussions throughout this project.
The results in Chapter 4 were obtained in collaboration with D. Gaiotto and M.Padi. The text has been published in:
“Non-Supersymmetric Attractor Flow in Symmetric Spaces”, D. Gaiotto,W. Li and M. Padi. JHEP 12, 093 (2007), 0710.1638 [hep-th];
I am grateful to A. Neitzke, J. Lapan and A. Strominger for many discussions relatedto this work.
The results in Chapter 5 were obtained in collaboration with W. Song and A.Strominger. They have previously appeared in:
“Chiral Gravity in Three Dimensions”, W. Li, W. Song and A. Strominger.JHEP 04, 082 (2008), arXiv:0801.4566 [hep-th];
I would also like to thank E. Witten and X. Yin for stimulating discussions regardingthis work.
Finally, the results in Chapter 6 were obtained in collaboration with J. Lapan.They have previously appeared in:
“Falling D0-Branes in 2D Superstring Theory”, J. Lapan and W. Li.arXiv:hep-th/0501054;
I would also like to thank A. Strominger and T. Takayanagi for many enlighteningdiscussions regarding this work.
Electronic preprints (shown in typewriter font) are available on the Internet atthe following URL:
http://arXiv.org
viii
Acknowledgments
I would like to express my deepest gratitude to my advisor Andrew Strominger
for his guidance, support and encouragement, to which this thesis owns its existence.
Besides all the physics he taught me, I am immensely grateful for his sharing with
me his passion for physics, his devotion to his students, and his constant reminding
me with his own example that difficulties are irrelevant in the quest for truth.
I am also deeply indebted to my committee members Frederik Denef, Gary Feld-
man and Lubos Motl for their tireless guidance over the years. I must further extend
my profound gratitude to all other members of high energy theory group at Harvard.
In particular, Nima Arkani-Hamed, Shiraz Minwalla, and Cumrun Vafa are always
there to answer questions, offer advice, or simply sprinkle insights. I would also like
to thank post-doctoral fellows Tadashi Takayanagi, Davide Gaiotto, Chris Beasley,
Alessandro Tomasiello, and Mboyo Esole for many stimulating discussions.
It has been truly a great honor and pleasure to work with my collaborators Josh
Lapan, Monica Guica, Lisa Huang, Andrew Strominger, Davide Gaiotto, Megha Padi,
Wei Song and Dionysios Anninos. All the excitement, joy, and even occasional frus-
trations we shared throughout our projects will always remain precious memories.
I also wish to thank my fellow graduate students Matt Baumgart, Michelle Cyrier,
Liam Fitzpatrick, Tom Hartman, Jon Heckman, Lisa Huang, Dan Jafferis, Subhaneil
Lahiri, Joe Marsano, Suvrat Raju, Jihye Seo, and Xi Yin, for all the physics discus-
sions, for their company during those late working nights, and for sharing parts of
their lives with me.
I want to thank Adriana, Dayle, Jean, Nancy, and Sheila for making the depart-
ment feel like home. And I thank my friends outside physics department for all the
ix
Acknowledgments x
wonderful moments we shared during these years. I would mention in particular
The existence of multi-centered BPS bound states is crucial in understanding the
microscopic entropy counting of BPS black holes and the exact formulation of OSV
formula [48]. One expects that a similarly important role should be played by multi-
centered non-BPS solutions in understanding non-BPS black holes microscopically.
However, the analytical non-BPS multi-centered black hole solutions with generic
background dependence have been elusive — in contrast to the case of BPS attractor
flows, the difficulty of solving the second-order non-BPS attractor equations makes the
construction of generic non-BPS multi-centered attractor flows a highly non-trivial
problem. In fact, even their existence has been in question.
In the BPS case, the construction of multi-centered attractor solutions is a simple
generalization of the full attractor flows of single-centered black holes: one needs
simply to replace the single-centered harmonic functions in a single-centered BPS
flow with multi-centered harmonic functions. However, the full attractor flow of a
generic single-centered non-BPS black hole has not been constructed analytically,
owing again to the difficulty of solving second-order equations of motion. Ceresole et
al. obtained an equivalent first-order equation for non-BPS attractors in terms of a
“fake superpotential,” but the fake superpotential can only be explicitly constructed
for special charges and asymptotic moduli [34, 98]. Similarly, the harmonic function
procedure was only shown to apply to a special subclass of non-BPS black holes, but
has not been proven for generic cases [91].
Aiming towards the construction of generic black hole attractors with arbitrary
charges and asymptotic moduli, Chapter 4 develops a new framework to encompass
Chapter 1: Introduction and Summary 26
generic black hole attractor solutions, both BPS and non-BPS, single-centered as well
as multi-centered, in all models for which the 3D moduli spaces obtained via c∗-map
are symmetric coset spaces. All attractor solutions in such a 3D moduli space can
be constructed algebraically in a unified way. Then the 3D attractor solutions are
mapped back into four dimensions to give 4D extremal black holes.
The non-BPS configurations are found to be drastically different from their BPS
counterparts. For example, in the particular model that we focused on — N = 2
supergravity coupled to one vector multiplet, the non-BPS single-centered attractor
constrains all attractor flows with different asymptotic moduli to flow toward the
attractor point along a common tangent direction. And in great contrast to the
BPS counterpart, the non-BPS multi-centered attractors in these systems are found
to enjoy complete freedom in the placement of attractor centers but suffer severe
constraints on the allowed D-brane charges. The constraint on the charges is expected
to be released by allowing a coupling between the moduli fields and 3D gravity, thus
generating 4D bound state solutions with non-zero angular momentum.
1.5 Chiral gravity in three dimensions
Chapter 5 concerns three dimensional topologically massive gravity (TMG), namely,
three dimensional Einstein gravity modified by the addition of a gravitational Chern-
Simons action.
Pure Einstein gravity in three dimensions is trivial classically. It has no local
propagating degrees of freedom, as can be seen from a degrees-of-freedom counting.
The degrees-of-freedom counting of Einstein gravity in D-dimensions receives contri-
Chapter 1: Introduction and Summary 27
butions from the spatial part of the metric and the corresponding momenta, minus
the number of constraints from the diffeomorphism symmetry and Bianchi identities:
Spatial metric + Momenta−Diff− Bianchi
=D(D − 1)
2+
D(D − 1)
2−D −D
= D(D − 3) ,
which vanishes at D = 3. Therefore there is no gravitational waves traveling in the
bulk of 3D spacetime in Einstein gravity.
That the 3D pure Einstein gravity is classically trivial also manifests itself in the
fact that 3D Riemann tensor is completely determined by the Ricci tensor:
Rµνρσ = gµρRνσ + gνσRµρ − gνρRµσ − gµσRνρ
−1
2(gµρgνσ − gµσRνρ)R . (1.34)
They both have six degrees of freedom. This means that all solutions of 3D pure
Einstein gravity have constant sectional curvature.
One way to render the 3D pure Einstein gravity non-trivial is to quantize the
theory. In fact, the existence of BTZ black holes [16] in 3D pure Einstein gravity
already signifies that the theory is non-trivial quantum mechanically. In the presence
of a negative cosmological constant Λ, there exist asymptotically AdS3 black hole
solutions [16] as well as massless gravitons which can be viewed as propagating on
the boundary. These BTZ black holes obey the laws of black hole thermodynamics
and have an entropy given by the area law. The microscopic origin of the black hole
entropy in the classically trivial theory can only be understood in the full quantum
version of the theory.
Chapter 1: Introduction and Summary 28
Three dimensional pure gravity can also be rendered non-trivial by adding a gravi-
tational Chern-Simons term. In fact, the Chern-Simons term appears naturally during
renormalization of quantum field theory in a three-dimensional gravitational back-
ground. It also arises in the compactification of string theory down to three dimen-
sions. The resulting theory has one single massive, propagating graviton degree of
freedom at generic Chern-Simons coupling, hence the name “topologically massive
gravity” (TMG).7
All solutions of Einstein gravity are automatically solutions of TMG. Moreover,
with an extra degree of freedom, TMG allows more solutions than its Einstein gravity
counterpart. For example, when Λ = 0, pure Einstein gravity allows only Minkowski
spacetime and no black hole solution, whereas TMG has ACL black holes even at
Λ = 0 [109].8 At Λ < 0, all solutions of Einstein gravity are locally AdS3; they are
AdS3 vacuum and BTZ black holes. In contrast, in addition to AdS3 vacuum and
BTZ black holes, TMG with Λ < 0 also allows Squashed AdS3 solution and “NG/BC”
black holes [116, 82, 24].
In Chapter 5, we will focus on theories with negative cosmological constant, and
consider only asymptotically AdS3 spacetimes. This will allow us to employ the con-
jectured AdS3/CFT2 correspondence to study properties of the bulk theory. Theories
with Λ = 0 is far less interesting than the one with Λ < 0, and theories with Λ > 0 has
dS3 space as its vacuum, which is metastable and does not have a globally conserved
energy, and the conjectured dS3/CFT2 correspondence is not well understood enough
7Note that the naive degrees-of-freedom counting no longer works for 3D TMG since there arenow third-time-derivatives in the action.
8ACL black hole is a type of 3D non-Einstein black hole solution living in TMG with zerocosmological constant.
Chapter 1: Introduction and Summary 29
to be of much use in understanding the bulk theory.
It is conjectured that all asymptotically AdS3 spacetimes are also locally AdS3,
namely, they are AdS3 vacuum solution and BTZ black holes, which belong to Ein-
stein solutions of TMG. Then taking both BTZ black holes and massive gravitons
propagating in the AdS3 vacuum into account, we will show that TMG with Λ < 0
is unstable/inconsistent for generic Chern-Simons coupling:9 either the BTZ black
hole or the massive gravitons would have negative energies. We then showed that
the theory is only unitary and stable when the parameters obey: µ` = 1.10 At this
special point, the theory has several interesting features:
1. The central charges of the dual boundary CFT2 become
cL = 0 , cR =3`
G. (1.35)
2. The conformal weights as well as the wave function of the massive graviton de-
generate with those of the left-moving weight (2, 0) massless boundary graviton.
They are both pure gauge in the bulk, and the gauge transformation parameter
does not vanish at infinity.
3. The mass of the massive graviton vanishes.
4. Both the massive graviton and the left-moving massless graviton have zero
energy.
5. Both BTZ black holes and the right-moving massless graviton have non-negative
energies.
9The Chern-Simons coupling is 1/µ
10Here ` is the radius of the AdS3 vacuum solution.
Chapter 1: Introduction and Summary 30
6. BTZ black holes become right moving, namely, their mass and angular momen-
tum obey
J = `M . (1.36)
This suggests the existence of a stable, consistent quantum gravity theory at
µ` = 1 which is dual to a holomorphic boundary CFT (i.e. containing only right-
moving degrees of freedom) with cR = 3`/G. We conjecture that for a suitable choice
of boundary conditions, the zero-energy left-moving graviton excitations at µ` = 1
can be discarded as pure gauge. We will refer to this theory as 3D chiral gravity. If
such a dual CFT does exist and is unitary, an application of Cardy formula can then
provide a microscopic derivation of the BTZ black hole entropy [139, 95, 96].
1.6 Time-dependent D-brane Solution in 2D Su-
perstring.
The final chaper — chapter 6 — of this thesis discusses the time-dependent D-
brane solutions in two-dimensional superstring theory.
The duality between matrix quantum mechanics and two-dimensional critical
string theory in the linear dilaton background is one of many different manifestations
of Gauge/Graivity correpondence. In this conjecture, the matrix quantum mechanics
is dual to the Liouville field theory (LFT) coupled to the c = 1 matter field, i.e. the
one-dimensional noncritical string theory in flat background. The latter is in turn
equivalent to the two-dimensional critical string theory in the linear dilaton back-
ground once we identify the Liouville mode of LFT with the spatial dimension of the
Chapter 1: Introduction and Summary 31
critical string, and the worldsheet cosmological constant of the former theory with
the amplitude of the tachyon field in the latter one.
In particular, the bosonic Liouville field theory coupled to c = 1 matter is dual to
the matrix model with the inverse harmonic oscillator potential with matrix eigenval-
ues filled at only one side of the potential. In the supersymmetrized version of this
duality, the N = 1 supersymmetric Liouville field theory (SLFT) coupled to c = 1
matter field, which is considered as the c = 1 noncritical Type 0 string theory in the
flat 2D target space, is dual to the matrix model with the same inverse harmonic
oscillator potential but with matrix eigenvalues filled on both sides of the potential.
The purpose of Chapter 6 is to construct and study the time-dependent D0-brane
solutions living in this two-dimensional N = 1, c = 1 noncritical Type 0 string theory.
Below, we will first give a lightening review on the description of D-branes in terms
of boundary states in the closed string sector.
The properties of a D-brane can be described by the boundary condition of the
open string attached to it. According to the string Open/Close duality, the boundary
condition of the open string corresponds to a certain state (boundary state) in the
closed string Hilbert space. This boundary state then serves as a description of the
corresponding D-brane in the closed string sector.
For a state in the closed string Hilbert space to be a boundary state, it must
satisfy two conditions. The most basic constraint that the boundary states must
satisfy comes from the requirement that the corresponding boundary vertex operator
preserve the symmetries of the original theory. In the case of the bosonic Liouville
field theory, this amounts to requiring conformal invariance which, in the language of
Chapter 1: Introduction and Summary 32
boundary states, translates to the constraints
(Lm − L−m)|α〉 = 0 , (1.37)
where |α〉 labels a boundary state, corresponding to a given boundary condition of
open string. Similarly, in the N = 1 cm = 1 supersymmetric Liouville field theory, a
closed string boundary state must satisfy [3],
(Lm − L−m)|α; η, σ〉 = 0 ,
(Gr − iηG−r)|α; η, σ〉 = 0 , (1.38)
where the two additional indices σ = NS − NS, R − R and η = ± label the sector
and spin structure of the boundary states.
More importantly, a boundary state must satisfy constraints coming from
Open/Closed duality of a cylinder diagram. In the open string sector, a cylinder
diagram is interpreted as the open string partition function11
Zαβ(τo) ≡ TrHαβ[qHo
o ] , (1.39)
where qo = e2πiτo and τo denotes the open string modulus. Ho is the open string
Hamiltonian; and the trace is taken over the open string spectrum Hαβ that satisfies
the specified boundary conditions α and β at the two ends of the open string .
In the string Open/Close duality, the same cylinder diagram can also be inter-
preted as a two point function in the closed string sector:
Zαβ(τc) = 〈α|(q1/2c )Hc |β〉 , (1.40)
11Here we will use bosonic string to illustrate the main points. The computations in Chapter 6concern superstring, which is slightly more complicated, with two more indices σ and η to consider.
Chapter 1: Introduction and Summary 33
where qc = e2πiτc with τc denoting the closed string modulus; and Hc is the closed
string Hamiltonian. It computes the evolution of the initial closed string state |β〉
(which corresponds to the right boundary condition β) to the final closed string state
〈α| (which corresponds to the left boundary condition α). The open and closed string
moduli are related through worldsheet duality by a modular transformation:
τc = − 1
τo
. (1.41)
The string Open/Close duality is realized through the modular invariance of the
open string partition function:
Zαβ(τo) = Zαβ(τc) =⇒ TrHαβ[qHo
o ] = 〈α|(q1/2c )Hc|β〉 . (1.42)
The above equation can be considered as the definition of boundary states |α〉 and
|β〉 — the boundary states are the closed string description of the corresponding
D-branes.
Given a set of boundary conditions α and β at the two ends of an open string, we
can use the modular bootstrap to construct the corresponding boundary states |α〉
and |β〉. Now we will review the procedure of modular bootstrap construction.12
First we expand the boundary states with a set of orthonormal states called
Ishibashi states |i〉〉 [89]. They are defined to satisfy the additional constraints
〈〈i|q12Hc
c |j〉〉 = δijTrHi[qHo
c ] = δijχi(τc) , (1.43)
where i, j now label closed string conformal family. Hi is spanned by the conformal
family corresponding to representation i of the constraint algebra. χi(τc) is the char-
12Again, we will focus on bosonic string here. The computation forN = 1, cm = 1 supersymmetricLiouville field theory can be found in Chapter 6.
Chapter 1: Introduction and Summary 34
acter of representation i of Virasoro algebra with closed string modulus τc. Thus a
boundary state |α〉 can be expanded by Ishibashi states:
|α〉 =∑
i
Ψα(i)|i〉〉 , (1.44)
and the problem of finding |α〉 is translated into finding the wave function Ψα(i).
Expand the closed string two point function (1.40) with a set of Ishibashi states
|i〉〉:
Zαβ(τc) = 〈α|(q1/2c )Hc |β〉 =
∑i
Ψ†α(i)Ψβ(i)χi(τc) . (1.45)
On the other hand, it was shown by Cardy in [33] that the trace in the open string
partition function (1.39) may be further rewritten as
Zαβ(τo) = TrHαβ[qHo
o ] =∑
i
niαβχi(τo) , (1.46)
where χi(τo) is the character of representation i of Virasoro algebra with open string
modulus τo. The non-negative integer niαβ counts multiplicty of Hi in Hαβ, and it
depends on the boundary conditions α, β at the two ends of the open string. That it,
the dependence of open string partition function Zαβ(τo) on the boundary conditions
α, β of the open string is only through niαβ.
Expressions (1.45) and (1.46) are equal due to Open/Close duality. Therefore,
using the modular transformation to relate χ(τc) and χ(τo), we can determine the
“wave functions” Ψα(i), after fixing the additional freedom by noting that these
wave functions are one-point functions on the disk and have specific transformation
properties under reflection [65]. This is what is known as the modular bootstrap
construction. As will be shown in Chapter 6 using the supersymmetrized version
of the modular boostrap construction, one can derive boundary state solutions that
Chapter 1: Introduction and Summary 35
correspond to falling D-branes N = 1, 2D superstring theory in the linear dilaton
background.
In the bosonic 2D string in the linear dilaton background with Euclidean time,
Lukyanov, Vitchev, and Zamolodchikov showed the existence of a time-dependent
boundary state, the so-called paperclip brane. This paperclip brane breaks into two
hairpin-shaped branes in the UV region [100]. Under the Wick-rotation from Eu-
clidean time into Minkowski time, the hairpin brane is reinterpreted as the falling
D0-brane.
We will show that in N = 1, 2D superstring theory with a linear dilaton back-
ground — which we will use interchangeably with cm = 1 N = 1 SLFT — there exists
a similar, time-dependent boundary state corresponding to the falling D0-brane. The
naive argument for the existence of the falling D0-brane is as follows. Since the mass
of the D0-brane is inversely related to the string coupling as
m = e−φ , (1.47)
the mass of the D0-brane decreases as it runs along the Liouville direction from the
weak coupling region (φ → −∞) to the strong coupling region (φ → +∞). Thus,
if we set a D0-brane free at the weak coupling region, it will roll along the Liouville
direction towards the strong coupling region until it is reflected back by the boundary
Liouville potential. This is the falling D0-brane solution which can be described by
a time-dependent closed string boundary state of the N = 1, 2D superstring.
In the bosonic case, the hairpin brane satisfies symmetries in addition to those
of the action (conformal symmetry). The additional symmetry is known as the W-
symmetry and is generated by higher spin currents [100]. The hairpin brane is then
Chapter 1: Introduction and Summary 36
constructed from the integral equations that are defined by the W-symmetry. In the
N = 1, 2D superstring, it should be possible to use the supersymmeterized version
of the W-symmetry to go through a similar construction and find a falling D0-brane.
However, we will argue that it can also be obtained by adapting the falling D0-brane
solution in N = 2 SLFT [112, 58], to the N = 1, 2D superstring.
We will also show that there exist four types of stable, falling D0-branes (two
branes and two anti-branes) in the Type 0A projection and two unstable ones in the
Type 0B projection. Type 0, N = 1, 2D superstring theory has a dual description in
the language of matrix models. An interesting question then would be to understand
these falling D0-branes in the context of the dual matrix model.
Chapter 2
R2 Corrections for 5D Black Holes
and Rings
2.1 Introduction
Recently a surprisingly powerful and precise relationship has emerged between
higher dimension F-terms in the 4D effective action for N = 2 string theory (as
captured by the topological string [23]) and the (indexed) BPS black hole degeneracies
[99, 119]. Even more recently [70] a precise relationship has been conjectured between
the 4D and 5D BPS black hole degeneracies. This suggests that there should be a
direct relationship between higher dimension terms in the 5D effective action and 5D
degeneracies which does not employ four dimensions as an intermediate step. Five
dimensions is in many ways simpler than four so such a relation would be of great
interest. It is the purpose of this work to investigate this issue.
The 4D story benefitted from a well understood superspace formulation [108, 45].
37
Chapter 2: R2 Corrections for 5D Black Holes and Rings 38
The relevant supersymmetry-protected terms are integrals of chiral superfields over
half of superspace and can be classified. In 5D the situation is quite different (see e.g.
[22]). There is no superfield formulation and we do not have a general understanding
of the possible supersymmetry-protected terms. In general, the uplift to 5D of most
of the 4D F-terms vanishes. However, the area law cannot be the exact answer for
the black hole entropy (for one thing it doesn’t give integer numbers of microstates!)
so there must be some kind of perturbative supergravity corrections.
As a first step towards a more general understanding, in this work we will study
the leading order entropy correction arising from R2 terms, which are proportional to
the 4D Euler density. Such terms give the one loop corrections in 4D, and — unlike
the higher order terms — do not vanish upon uplift to 5D. They are also of special
interest as descendants of the interesting 11D R4 terms [77, 93]. These terms correct
the entropy of the both the 5D black ring [59] and the 5D BMPV spinning black
hole [26]. We find that the macroscopic black ring correction matches, including the
numerical coefficient, a correction expected from the microscopic analysis of [41]. For
the BMPV black hole, we find the correction matches, to leading order, one expected
from the 4D-5D relation conjectured in [70].
The next section derives the R2 corrections to the 5D entropy as horizon integrals
of curvature components using Wald’s formula. Section 2.3 evaluates this formula for
the black ring, while section 2.4 evaluates it for BMPV. Section 2.5 contains a brief
summary.
Chapter 2: R2 Corrections for 5D Black Holes and Rings 39
2.2 Wald’s Formula in 5D
In this section we will use Wald’s formula to derive an expression for R2 corrections
to the 5D entropy.
The Einstein-frame low energy effective action for the compactification of M-
theory on a Calabi-Yau threefold CY3 down to five dimensions contains the terms
[12]
I0 + ∆I = − 1
32π2
∫d5x
√|g5|R(5)
− 1
29 · 3π2
∫d5x
√|g5|c2AY A(RαβµνR
αβµν − 4RαβRαβ + R2) (2.1)
in units in which G5 = 2π (for compactification on a circle of unit radius, this choice
leads to G4 = 1 and hence facilitates 4D/5D comparisons). Here Y A, A = 1, . . . , nV
are scalar components of vector multiplets. They are proportional to the Kahler
moduli of CY3, normalized so that
DABCY AY BY C = 1 . (2.2)
c2A are the components of the second Chern class of CY3 and DABC the corresponding
intersection numbers. The R2 term in ∆I arises from dimensional reduction of the
much studied R4 term [77, 93] in eleven dimensions. It is also the uplift from four
dimensions of an F term whose coefficient is computed by the N = 2 topological
string on CY3 at one loop order [23].
When we add R2 corrections to the action the entropy is no longer given by the
area law; instead, we need to use the more general formula found by Wald [146]
SBH = 2π
∫
Hor
d3x√
h∂L
∂Rµνρσ
εµνερσ (2.3)
Chapter 2: R2 Corrections for 5D Black Holes and Rings 40
where, εαβ is the binormal to the horizon, defined as the exterior product of two null
vectors normal to the horizon and normalized so that εαβεαβ = −2. We can then
identify two types of first-order corrections implied by this formula:
• modifications to the area law due to the additional terms in the action — these
terms are evaluated using the zeroth order solutions for the metric and the other
fields.
• modification of the area due to the change of the metric on the horizon, which
follows from the fact that adding extra terms to the action may change the
equations of motion.
In 4D, the second type of modification is absent at leading order for this particular
R2 form of ∆I obtained by reduction of (2.1) [104]. This and the 4D-5D agreement
we find to leading order suggest that this may be the case in 5D as well. In order
to understand all R2 corrections to the entropy this should be ascertained by direct
calculation. In the following we consider only the first type of modification.
The corresponding correction to the entropy is then (see also [110])
∆S = −4πc2 · Y29 · 3π2
∫
Hor
d3x√
h ( Rµνρσεµνερσ − 4 Rµρgνσε
µνερσ + Rεµνεµν) , (2.4)
where h is the induced metric on the horizon and the moduli are fixed at their attractor
values. In the following we will evaluate this correction for the spinning black hole
and black ring solutions.
Chapter 2: R2 Corrections for 5D Black Holes and Rings 41
2.3 Black Ring
The black ring solution was discovered in [59] and its entropy understood from a
microscopic perspective in [41]. It represents a supersymmetric solution to 5D super-
gravity coupled to a number of abelian vector (and hyper)multiplets that describes
a charged, rotating black ring. It is characterized by electric charges qA, magnetic
dipole charges pA, and the angular momentum around the ring, Jψ. The macroscopic
entropy formula for the black ring can be written in the suggestive form
SBR = 2π
√cLq0
6(2.5)
where, in terms of the macroscopic charges,
cL = 6D = 6DABCpApBpC , (2.6)
with DABC being (one sixth) the intersection numbers of the Calabi-Yau, and
q0 = −Jψ +1
12DABqAqB +
cL
24, (2.7)
where DAB is the inverse of DAB ≡ DABCpC . The microscopic origin of the entropy
is from the quantum degeneracy of a 2D CFT with central charge cL and left-moving
momentum q0 available for distribution among the oscillators. The last term in (2.7)
is ascribed to the left moving zero point energy.
2.3.1 Macroscopic entropy correction
Now we evaluate the correction to the black ring entropy induced by ∆I. Due to
the 5D attractor mechanism [90] the moduli take the horizon values
Y A =pA
D13
. (2.8)
Chapter 2: R2 Corrections for 5D Black Holes and Rings 42
Next, all we need to do is to find the binormal to the horizon for the black ring metric,
evaluate the relevant curvature terms at the horizon, and integrate. We obtain1
∆SBR =π
6c2 · p
√q0
D. (2.9)
2.3.2 Microscopic entropy correction
The microscopic entropy comes from M5 branes wrapping 4-cycles associated to
pA in CY3. As shown in [104], these are described by a CFT with the left-moving
central charge
cL = 6D + c2 · p . (2.10)
In [41] the leading entropy at large charges was microscopically computed using the
leading approximation (2.6) to cL at large charges. Subleading modifications should
arise from using the exact formula (2.10) in (2.5). This leads to
∆SBR =π
6c2 · p
√q0
D+
π
24c2 · p
√D
q0
+ . . . (2.11)
The first term comes from correcting cL in (2.5), while the second comes from correct-
ing the zero point shift in (2.7). We see that the macroscopic R2 correction matches
1Note that in the following we have employed the following relationships between various quan-tities used in this work, in [59] and in [26]:
Qeemr = (16πG)23 µbmpv =
(4G
π
) 23
q ,
qeemr =(
4G
π
) 13
p ,
J = Jeemr = 16πJbmpv = 4π2µω .
The value of Newton’s constant used in [26] is G5 = (16π)−1, so we needed to rescale their metricby (16πG)
23 in order to get ours. Also recall we are setting G = 2π in the text.
Chapter 2: R2 Corrections for 5D Black Holes and Rings 43
precisely the first term. We do not understand the matching of the second term, but
note that it is subleading in the regime q0 À D where Cardy’s formula is valid.
2.4 BMPV Black Hole
Let us now turn now to BMPV — the charged rotating black hole in 5D character-
ized by electric charges qA and angular momenta J in SU(2)L. Its leading macroscopic
entropy is given by
SBMPV = 2π√
Q3 − J2 (2.12)
where
Q32 = DABCyAyByC (2.13)
where the yA’s are determined from
qA = 3DABCyByC . (2.14)
We find the correction to this entropy following from the application of Wald’s formula
to (2.1) to be
∆SBMPV = − π
24
√Q3 − J2 c2 · Y
(− 3
Q− J2
Q4
)=
π
6A c2 · Y
(1
Q− A2
4Q4
), (2.15)
where we defined A =√
Q3 − J2 and the moduli fields take the horizon values2
Y A =yA
Q12
. (2.16)
In general, the microscopic origin of the entropy for the 5D spinning black holes in
M-theory on CY3 (unlike for black rings) is not known,3 so we will not try herein
2From now on we will take ‘Y A’ to mean the horizon value of the modulus field Y A.
3It is of course known for N = 4 compactifications [140, 26], so it would be interesting to interpretthe macroscopic correction for that case.
Chapter 2: R2 Corrections for 5D Black Holes and Rings 44
to understand the microscopic origin of ∆SBMPV . We will however compare it to
corresponding corrections in 4D and the topological string partition function. As
argued in [70], the exact 5D BMPV entropy is equal to the entropy of the D6-D2-D0
system in 4D, with the same 2-brane charges qA, D6-brane charge p0 = 1, and D0-
brane charge q0 = 2J . In the same paper, the following relationship for the partition
functions of 5D black holes, 4D black holes and consequently of the topological string
— see [119] — was conjectured
Z5D(φA, µ) = Z4D(φA, φ0 =µ
2+ iπ) =
∣∣∣∣Ztop
(gtop =
8π2
µ, tA =
2φA
µ
)∣∣∣∣2
, (2.17)
where φA are the electric potentials conjugate to qA, while φ0 is conjugate to q0 in 4D,
and µ to J in 5D. The absolute value in the last expression is defined by keeping φ0
real. With this in mind, we can start from Ftop — the topological string amplitude —
and compute the entropy of the BMPV (including first order corrections) as follows.
Up to one-loop order Ftop is
Ftop =i(2π)3
g2top
DABCtAtBtC − iπ
12c2AtA (2.18)
=i
π
DABCφAφBφC
µ− iπ
6
c2AφA
µ.
The entropy of the black hole is given by the Legendre transform of
F(φA, Reµ) = ln ZBH = Ftop + Ftop . (2.19)
To first order we have
F = − 1
π2
DABCφAφBφC − π2
6c2AφA
(Reµ2π
)2 + 1, (2.20)
Chapter 2: R2 Corrections for 5D Black Holes and Rings 45
which gives
qA =1
π2
3DABCφBφC − π2
6c2A
(Reµ2π
)2 + 1(2.21)
J = −Reµ
2π4
DABCφAφBφC − π2
6c2AφA
((Reµ
2π)2 + 1
)2
and therefore
S = 2π√
Q3 − J2(1 +1
12
c2AY A
Q+ . . .) (2.22)
where the . . . stand for higher order corrections in |gtop|2 = 16π2A2/Q3.
We see that to the 5D R2 corrections (2.15) to the entropy do not exactly match
the 4D corrections (2.22). This is possible of course because dimensional reduction
of the 5D R2 gives the 4D R2 term plus more terms involving 4D field strengths.
However we also see that the mismatch is subleading in the expansion in gtop, and
we can therefore conclude that the 5D R2 term captures the subleading correction to
the area law.
2.5 Summary
We have shown that higher dimension corrections to the 5D effective action do
give corrections to the black hole/black ring entropy just as in 4D, but that the 5D
situation is currently under much less control than the 4D one. Some leading order
computations were performed and found to give a partial match between macroscopic
and microscopic results. We hope these computations will provide useful data for
finishing the 5D macro/micro story.
Chapter 3
Supersymmetric Probes in a
Rotating 5D Attractor
3.1 Introduction
The near-horizon attractor geometry of a BPS black hole has twice as many
supersymmetries as the full asymptotically flat solution. In four dimensions, such
geometries admit BPS probe configurations which preserve half of the enhanced su-
persymmetry of the near-horizon AdS2 × S2 ×CY3 attractor geometry, but break all
of the supersymmetries of the original asymptotically flat solution [137]. The quan-
tization of these classical configurations gives rise to the superconformal quantum
mechanics system which is conjectured to be the holographic dual of the IIA string
theory on AdS2×S2×CY3 [70]. In particular, the supersymmetric black hole ground
states are identified with the chiral primaries of this near-horizon superconformal
quantum mechanics, which form the lowest Landau levels that tile the black hole
46
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 47
horizon [68]. The counting of the degeneracy of the lowest Landau levels reproduces
the Bekenstein-Hawking black hole entropy [69].
Furthermore, a novel feature of these probe brane configurations is that branes
and anti-branes antipodally located on the S2 preserve the same supersymmetries.
In the dilute gas approximation, the black hole partition function is dominated by
the sum over these chiral primary states [71]. An appropriate expansion thus yields
a derivation of the OSV relation [119], with branes and anti-branes contributing to
the holomorphic and anti-holomorphic parts of the partition function.
These interesting 4D phenomena should all have closely related 5D cousins [70].
In five dimensions, the generic supersymmetric black hole is the BMPV rotating
black hole [26]. We are interested in the N = 2 BMPV black hole, which can be
constructed by wrapping M2-branes on the holomorphic two-cycles of the Calabi-Yau
threefold. Unlike the BMPV black hole in N = 4 and N = 8 compactifications,
whose holographic dual has been known for a while, the microscopic description of
the N = 2 BMPV black hole has been eluding our search. For some recent progress
towards this goal, see [79, 87].
The present paper extends the 4D classical BPS probe analysis of [137] to five
dimensions. The 5D problem is considerably enriched by the fact that 5D BMPV BPS
black holes can carry angular momentum J and have a U(1)L × SU(2)R rotational
isometry group [26]. BPS zero-brane probes are constructed by wrapping the M2-
brane on the holomorphic two-cycles of CY3, and are found to orbit the S3 using
a κ-symmetry analysis. Their location in AdS2 depends on the azimuthal angle on
S3, the background rotation J , and the angular momentum of the probe. The BPS
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 48
one-branes are constructed by wrapping M5-branes on the holomorphic four-cycles
of CY3. We find BPS configurations with momentum and winding around a torus
generated by a U(1)L × U(1)R rotational subgroup.1 A one-brane in five dimensions
can carry the magnetic charge dual to the electric charge supporting the BMPV black
hole. Interestingly, we find that this allows for static BPS “black ring” configurations,
where the angular momentum required for saturation of the BPS bound is carried by
the gauge field.
3.2 Review of the BMPV Black Hole
The generic five-dimensional N = 2 supersymmetric rotating black hole arises
from M2-branes wrapping holomorphic two-cycles of a Calabi-Yau threefold X. It is
characterized by electric charges qA, A = 1, 2, . . . , b2(X), and the angular momentum
J in SU(2)left. The metric is [26]
ds2 = −(
1 +Q
r2
)−2 [dt +
J
2r2σ3
]2
+
(1 +
Q
r2
) (dr2 + r2dΩ2
3
), (3.1)
dΩ23 =
1
4
[dθ2 + dφ2 + dψ2 + 2 cos θdψdφ
]=
1
4
3∑i=1
(σi)2, (3.2)
where the ranges of the angular parameters are
θ ∈ [0, π], φ ∈ [0, 2π], ψ ∈ [0, 4π]. (3.3)
1Inclusion of these states in the partition function of [71] could lead to non-factorizing correctionsto the OSV relation.
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 49
σi are the right-invariant one-forms:2
σ1 = − sin ψdθ + cos ψ sin θdφ ,
σ2 = cos ψdθ + sin ψ sin θdφ , (3.4)
σ3 = dψ + cos θdφ ,
and we choose Planck units l5 = (4G5
π)1/3 = 1. The graviphoton charge Q is deter-
mined via the equations
Q32 = DABCyAyByC , (3.5)
qA = 3DABCyByC , (3.6)
with DABC the intersection form on X.
The near-horizon limit (r → 0) of the metric is
ds2 = −[r2
Qdt +
J
2Qσ3
]2
+ Qdr2
r2+ QdΩ2
3 . (3.7)
Rescaling t to absorb Q, defining sin2 B = J2
Q3 and r2 = 1/σ, we obtain the metric in
Poincare coordinates:
ds2 =Q
4
[−(
dt
σ+ sin Bσ3)
2 +dσ2
σ2+ σ2
1 + σ22 + σ2
3
]. (3.8)
The graviphoton field strength in these coordinates is
F[2] = dA[1] , A[1] =
√Q
2[1
σdt + sin Bσ3] . (3.9)
2The SU(2) rotation matrix is parameterized as:
ei σz2 ψei
σy2 θei σz
2 φ =(
cos θ2ei(ψ+φ)/2 sin θ
2ei(ψ−φ)/2
− sin θ2e−i(ψ−φ)/2 cos θ
2e−i(ψ+φ)/2
).
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 50
It was shown in [47] that the black hole entropy is given by the quantum eigenstates
with respect to the global, rather than the Poincare, time of the near horizon AdS2.
Therefore, we will mostly work in the global coordinates (τ, χ, θ, φ, ψ). Using the
coordinate transformation between the global coordinates and Poincare ones:
t =cos B cosh χ sin τ
cosh χ cos τ + sinh χ,
σ =1
cosh χ cos τ + sinh χ,
ψPoincare = ψglobal + 2 tan B tanh−1 (e−χ tanτ
2) ,
we obtain the metric in the global coordinates:
ds2 =Q
4
[− cosh2 χdτ 2 + dχ2 + (sin B sinh χdτ − cos Bσ3)2 + σ2
1 + σ22
], (3.10)
in which
A[1] =
√Q
2[cos B sinh χdτ + sin Bσ3] . (3.11)
The near horizon geometry of the BMPV black hole is a kind of squashed AdS2×
S3. The near-horizon isometry supergroup is SU(1, 1|2)×U(1)left, where the bosonic
subgroup of SU(1, 1|2) is SU(1, 1) × SU(2)right [74]. When J = 0, U(1)left is pro-
moted to SU(2)left and the full SO(4) ∼= SU(2)right×SU(2)left rotational invariance is
restored. The unbroken rotational symmetries for J 6= 0 are generated by the Killing
vectors
ξL3 = ∂ψ , (3.12)
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 51
and
ξR1 = sin φ∂θ + cos φ(cot θ∂φ − csc θ∂ψ) ,
ξR2 = cos φ∂θ − sin φ(cot θ∂φ − csc θ∂ψ) , (3.13)
ξR3 = ∂φ .
The supersymmetries arise from Killing spinors ε which are the solutions of the
equation [d +
1
4ωabΓ
ab +i
8
(eaΓbcΓaFbc − 4eaΓbFab
)]ε = 0 . (3.14)
To solve this in global coordinates we choose the vielbein
e0 =
√Q
2[cosh (sin B cos Bψ) cosh χdτ + sinh (sin B cos Bψ)dχ] ,
e1 =
√Q
2[sinh (sin B cos Bψ) cosh χdτ + cosh (sin B cos Bψ)dχ] ,
e2 =
√Q
2[− sin (cos2 Bψ)dθ + cos (cos2 Bψ) sin θdφ] , (3.15)
e3 =
√Q
2[cos (cos2 Bψ)dθ + sin (cos2 Bψ) sin θdφ] ,
e4 =
√Q
2[− sin B sinh χdτ + cos Bσ3] .
The Killing spinors are then [9, 10]
ε = e[−12(sin B cos BΓ01+cos2 BΓ23)ψ]e[+
12(cos BΓ24+i sin BΓ2)θ]e[−
12(cos BΓ34+i sin BΓ3)φ]
e[+12(sin BΓ04−i cos BΓ0)χ]e[−
12(sin BΓ14−i cos BΓ1)τ]ε0
≡ Sε0 , (3.16)
where ε0 is any spinor with constant components in the frame (3.15).
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 52
As a comparison, for Poincare coordinates we choose the vielbein
e0 =
√Q
2[dt
σ+ sin Bσ3], e1 =
√Q
2
dσ
σ,
e2 =
√Q
2σ1, e3 =
√Q
2σ2, e4 =
√Q
2σ3. (3.17)
The Killing spinors are [74]
ε+ =1√σ
R(θ, φ, ψ)ε+0 , (3.18)
ε− =
[√σ(1− sin BΓ04)− t√
σΓ01
]R(θ, φ, ψ)ε−0 , (3.19)
where
R(θ, φ, ψ) = e−12Γ23ψe
12Γ24θe−
12Γ23φ ,
iΓ0ε±0 = ±ε±0 , (3.20)
for constant ε±0 .
3.3 Supersymmetric Probe Configurations
In this section, we find classical brane trajectories which preserve some super-
symmetries of the rotating attractor (3.7). The worldvolume action has a local κ-
symmetry (parameterized by κ) as well as a spacetime supersymmetry transformation
(parameterized by ε) which acts nonlinearly. A spacetime supersymmetry is preserved
if its action on the worldvolume fermions Θ can be compensated by a κ transformation
[19, 21]:
δεΘ + δκΘ = ε + (1 + Γ)κ(σ) = 0 , (3.21)
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 53
where Γ is given in various cases analyzed below. This gives the condition
(1− Γ)ε = 0, (3.22)
which must be solved for both the Killing spinor and the probe trajectory.
3.3.1 Zero-brane probe
The zero-brane can be obtained by wrapping M2-branes on the holomorphic two-
cycles of the Calabi-Yau threefold X. It carries electric charges vA, A = 1, 2, . . . , b2(X).
For the zero-brane the (bosonic part of the) κ-symmetry projection operator is
Γ =1√h00
Γ0 , (3.23)
where h and Γ0 are the pull-backs of the metric and Dirac matrix onto the worldline
of the zero-brane, respectively:
h00 = ∂0Xµ∂0X
νGµν , (3.24)
Γ0 = ∂0Xµea
µΓa . (3.25)
Global coordinates
First, let’s look at the global coordinates. In the static gauge, where we set the
worldvolume time σ0 equal to the global time τ , the κ-symmetry operator is
Γ =1√h00
dXµ
dτea
µΓa . (3.26)
To solve for the classical trajectory of a supersymmetric zero-brane, we plug the
Killing spinors (3.16) into the κ-symmetry condition (3.22) of the supersymmetric
zero-brane. A zero-brane following a classical trajectory, given by
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 54
(χ(τ), θ(τ), φ(τ), ψ(τ)), is supersymmetric if, in the notation of (3.16),
1√h00
dXµ
dτea
µS−1ΓaSε0 = ε0 , (3.27)
for some constant ε0, where S = S(χ, τ, θ, φ, ψ). The explicit prefactors are
S−1ea0ΓaS =
√Q
2[(cosh χ cos τ cos2 B + sin θ cos φ sin2 B)Γ0
+i cosh χ sin τ cos BΓ01 − i cos θ sin BΓ02 − i sin θ sin φ sin BΓ03
+i(cosh χ cos τ − sin θ cos φ) sin B cos BΓ04] ,
S−1ea1ΓaS = (−1)
√Q
2[sin θ cos φ cos τΓ1
− sin τ sin Be12(cos BΓ34+i sin BΓ3)φe−(cos BΓ24+i sin BΓ2)θ
e12(cos BΓ34+i sin BΓ3)φΓ4
−i cos τ cos θ sin BΓ12 − i cos τ sin θ sin φ sin BΓ13
+ie(sin BΓ14−i cos BΓ1)τ sinh χ cos BΓ01
+i(cosh χ− sin θ cos φ cos τ) cos B(sin BΓ14 − i cos BΓ1)] ,
S−1ea2ΓaS = (−1)
√Q
2[cosh χ cos τ cos φΓ3 − cosh χ cos τ sin φ cos BΓ4
+e(cos BΓ34+i sin BΓ3)φ(+i sinh χ cos BΓ03
−i cosh χ sin τ cos BΓ13 + i cos θ sin BΓ23)
+i(cosh χ cos τ cos φ− sin θ) sin B(cos BΓ34 + i sin BΓ3)] ,
S−1ea3ΓaS = (−1)
√Q
2[(cosh χ cos τ cos2 B + sin θ cos φ sin2 B)Γ2
+i sinh χ cos BΓ02 − i cosh χ sin τ cos BΓ12 − i sin θ sin φ sin BΓ23
+i(cosh χ cos τ − sin θ cos φ) sin B cos BΓ24] ,
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 55
S−1ea4ΓaS = (−1)
√Q
2cos Be+ 1
2(sin BΓ14−i cos BΓ1)τe−(sin BΓ04−i cos BΓ0)χ
e+ 12(sin BΓ14−i cos BΓ1)τe+ 1
2(cos BΓ34+i sin BΓ3)φe−(cos BΓ24+i sin BΓ2)θ
e+ 12(cos BΓ34+i sin BΓ3)φΓ4 . (3.28)
We first see that a probe static in the global time τ cannot be supersymmetric.
For such a probe we have dχdτ
= dθdτ
= dφdτ
= dψdτ
= 0 and the κ-symmetry condition
reduces to
1√−1− cos2 B sinh2 χ
· [(cosh χ cos τ cos2 B + sin θ cos φ sin2 B)Γ0
+i cosh χ sin τ cos BΓ01 − i cos θ sin BΓ02 − i sin θ sin φ sin BΓ03 (3.29)
+i(cosh χ cos τ − sin θ cos φ) sin B cos BΓ04]ε0 = ε0 .
The terms in this equation proportional to cos τ , sin τ and 1 must all vanish separately,
which is clearly impossible. The lack of such configurations is not surprising, because
angular momentum must be nonzero for a nontrivial BPS configuration.
Now we allow the probe to orbit around the S3. Solving the κ-symmetry condition
(3.22) using (3.28) for Killing spinors obeying
Γ02ε0 = ∓ε0 , (3.30)
we find the supersymmetric trajectory at a generic (χ, θ, ψ) to be
dχ
dτ=
dθ
dτ=
dψ
dτ= 0 ,
dφ
dτ= ±1 . (3.31)
This is a probe orbiting along the φ-direction.
The constraint on the Killing spinor (3.30) projects out half of the components
of ε0, i.e. the orbiting zero-brane probe is a half-BPS configuration. We will show
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 56
in the next subsection, using the BPS bound, that this supersymmetric trajectory is
unique up to rotations.
A BPS bound
The worldline action of a zero brane probe, with mass m and the electric charge
q, can be written as
S = −m
∫ √hdσ0 + q
∫A[1] , (3.32)
where A[1] is the 1-form gauge field (3.11). The zero-brane obtained by wrapping M2-
branes on the holomorphic two-cycles of the Calabi-Yau threefold X carries electric
charges vA, A = 1, 2, . . . , b2(X). For the supersymmetric branes we are considering,
m = q = vAyA√Q/2
.
In global coordinates with σ0 = τ , the Lagrangian of the system is
L =
√Q
2−m
√cosh2 χ− χ2 − [sin B sinh χ− cos B(ψ + cos θφ)]2 − θ2 − sin2 θφ2
+m[cos B sinh χ + sin B(ψ + cos θφ)] . (3.33)
The corresponding Hamiltonian is
H = cosh χ
√P 2
χ + P 2θ + (
cos θPφ − Pψ
sin θ)2 + P 2
φ + (sin BPψ −
√Q2
m
cos B)2
+ sinh χ(sin BPψ −
√Q2
m
cos B) , (3.34)
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 57
where the momenta are
Pχ =m√
Q
2√
hχ ,
Pθ =m√
Q
2√
hθ ,
Pφ =m√
Q
2[
1√h
(− cos B cos θ[sin B sinh χ− cos B(ψ + cos θφ)] + sin2 θφ
)
+ sin B cos θ] ,
Pψ =m√
Q
2[
1√h
(− cos B[sin B sinh χ− cos B(ψ + cos θφ)]
)+ sin B] , (3.35)
and
h = cosh2 χ− χ2 − [sin B sinh χ− cos B(ψ + cos θφ)]2 − θ2 − sin2 θφ2 . (3.36)
The unbroken rotational symmetries lead to the conserved charges:
J1right = sin φPθ + cos φ(cot θPφ − csc θPψ) ,
J2right = cos φPθ − sin φ(cot θPφ − csc θPψ) ,
J3right = Pφ ,
J3left = Pψ . (3.37)
It is easy to see that there are no static solutions. They would have to minimize
the potential energy according to
0 =∂H
∂χ=
√Q
2m cos B cosh χ(
cos B sinh χ√cos2 B sinh2 χ + 1
− 1) , (3.38)
which has no solutions for finite χ. Physically, the probe is accelerated to χ = ±∞.
Now we allow the probe to orbit. Solutions of this type can be stabilized by the
angular potential. The supersymmetric configuration turns out to be at constant
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 58
radius in the AdS2, i.e. Pχ = 0. The Hamiltonian is minimized with respect to χ
when
tanh χ = − 1√P 2
θ + (cos θPφ−Pψ
sin θ)2 + P 2
φ + (sin BPψ−
√Q2
m
cos B)2
(sin BPψ −
√Q2
m
cos B) . (3.39)
The value of H at the minimum is
Hmin =
√P 2
θ + (cos θPφ − Pψ
sin θ)2 + P 2
φ = | ~Jright| , (3.40)
where | ~Jright|2 = (J1right)
2 + (J2right)
2 + (J3right)
2. This implies the BPS bound
H ≥ | ~Jright| (3.41)
for generic χ.
Up to spatial rotations, we may always choose static BPS solutions to satisfy
H = J3right = ±Pφ , J1
right = J2right = 0 . (3.42)
This implies
Pθ = 0 , cos θPφ = Pψ . (3.43)
Hence, the azimuthal angle is determined by the ratio of left and right angular mo-
menta:
cos θ =J3
left
J3right
. (3.44)
We can rewrite φ and ψ in terms of Pφ and Pψ. With χ = θ = 0,
φ =cosh χ(
Pφ−cos θPψ
sin2 θ)√
P 2θ + (
cos θPφ−Pψ
sin θ)2 + P 2
φ + (sin BPψ−
√Q2
m
cos B)2
, (3.45)
ψ =cosh χ[tan B(
sin BPψ−√
Q2
m
cos B)− (
cos θPφ−Pψ
sin2 θ)]√
P 2θ + (
cos θPφ−Pψ
sin θ)2 + P 2
φ + (sin BPψ−
√Q2
m
cos B)2
+ tan B sinh χ . (3.46)
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 59
Eliminate χ through (3.39),
φ =1√
P 2θ + (
cos θPφ−Pψ
sin θ)2 + P 2
φ
(Pφ − cos θPψ
sin2 θ) , (3.47)
ψ =1√
P 2θ + (
cos θPφ−Pψ
sin θ)2 + P 2
φ
(Pψ − cos θPφ
sin2 θ) . (3.48)
Plug in (3.43), the solution is
θ = 0 , φ = ±1 , ψ = 0 , (3.49)
for which (Pφ, Pψ) are
Pψ = ±√
Q
2m
cos θ
cos B sinh χ± sin B cos θ, (3.50)
Pφ = ±√
Q
2m
1
cos B sinh χ± sin B cos θ. (3.51)
The energy of the particle following this trajectory is equal to ±Pφ:
H =
√Q
2m
1
cos B sinh χ± sin B cos θ= ±Pφ . (3.52)
We see that the solution with φ = 1 (φ = −1) corresponds to a chiral (anti-chiral)
BPS configuration.
Therefore, we have confirmed that the supersymmetric trajectories (3.31) obtained
by solving the κ-symmetry condition correspond to the BPS states.
Poincare coordinates
In Poincare coordinates and static gauge σ0 = t, the κ-symmetry condition for a
static probe is
1√− 1
σ2
[− 1
σΓ0
]ε = iΓ0ε = ε . (3.53)
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 60
This equation is solved by simply taking ε = ε+ = 1√σR(θ, φ, ψ)ε+
0 . Again, we find
a half-supersymmetric solution, although the broken supersymmetries are different
than in the global case. It can be seen that there are no supersymmetric orbiting
trajectories in Poincare time.
3.3.2 One-brane probe
In this subsection, we find some supersymmetric one-brane configurations. The
one-brane probe is constructed by wrapping the M5-brane on a holomorphic 4-cycle
of the CY3. We consider a specific Ansatz with no worldvolume electromagnetic field
and with the one-brane geometry:
τ = σ0 ,
φ = φσ0 + φ′σ1 ,
ψ = ψσ0 + ψ′σ1 , (3.54)
where (σ0, σ1) are worldvolume coordinates, and φ, ψ, φ′ and ψ′ are all taken to be
constant. Note that since (ψ, φ) are the orbits of (J3L, J3
R), they may be viewed as
one-brane momentum-winding modes on the torus generated by (J3L, J3
R). This torus
degenerates to a circle at the loci θ = 0, π. One-branes of the form (3.54) at these
loci are therefore static (up to reparametrizations).
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 61
With no electromagnetic field the κ-symmetry condition is3
1
2εijΓijε = ε , (3.55)
where h and Γi are the pull-backs of the 5D metric and gamma matrices onto the
one-brane worldsheet. With the Ansatz (3.54), we have explicitly
Γ0 = Γτ + φΓφ + ψΓψ , (3.56)
Γ1 = φ′Γφ + ψ′Γψ , (3.57)
1
2εijΓij =
1
2√
deth[φ′Γτφ + ψ′Γτψ + (φψ′ − ψφ′)Γφψ] , (3.58)
and
h00 =Q
4− cosh2 χ + [sin B sinh χ− cos B(ψ + cos θφ)]2 + sin2 θ φ2 ,
h11 =Q
4cos2 B(ψ′ + cos θφ′)2 + sin2 θ φ′2 , (3.59)
h01 =Q
4[sin B sinh χ− cos B(ψ + cos θφ)](− cos B)(ψ′ + cos θφ′) + sin2 θ φφ′ ,
and hence
deth = (Q
4)2cosh2 χ[cos2 B(ψ′ + cos θφ′)2 + sin2 θφ′2]
− sin2 θ[sin B sinh χφ′ − cos B(−ψ′φ + φ′ψ)]2 . (3.60)
It is simplest to analyze the κ-symmetry condition in the form
S−1 1
2εijΓijSε0 = ε0 . (3.61)
3There is a simple kappa-symmetric action in six dimensions, but not in five. In 5D we expect anextra scalar field along with the transverse coordinates to fill out the supermultiplet. For the case ofthe M5-brane wrapping a Calabi-Yau 4-cycle, the scalar in the effective one-brane arises as a modeof the antisymmetric tensor field. The Ansatz of this section corresponds to taking this extra scalarto be a constant.
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 62
The rotated gamma matrices appearing in this expression are explicitly
S−1ΓτφS (3.62)
= −Q
4[(cosh2 χ cos2 B + sin2 θ sin2 B)Γ02 − i(cosh χ cos τ cos2 B + sin θ cos φ sin2 B
−i cosh χ sin τ cos BΓ1 + i sin θ sin φ sin BΓ3
−i(cosh χ cos τ − sin θ cos φ) sin B cos BΓ4)(cos θ sin BΓ0 + sinh χ cos BΓ2)] ,
S−1ΓτψS (3.63)
=Q
4cos B− cosh2 χ cos θ cos BΓ02
+ cos B sinh χ[i cosh χ sin θ cos τ cos φΓ4 + cosh χ sin θ sin φ sin τΓ13
− cosh χ sin θ cos τ sin φ(sin BΓ34 − i cos BΓ3)
+ cosh χ sin θ cos φ sin τ(cos BΓ14 + i sin BΓ1)]
−(cos2 B cosh2 χ sin θ cos φ + sin2 B cosh χ cos τ)Γ04
− sin B cos B cosh χ sinh χ cos τ cos θΓ24
− cosh χ sin τ sin BΓ01 + cos B cosh χ sinh χ cos θ sin τΓ12
− cosh2 χ sin θ sin φ cos BΓ03
−i cosh χ(cosh χ sin θ cos φ− cos τ) sin B cos BΓ0
+i cos2 B cosh χ sinh χ cos τ cos θΓ2 ,
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 63
S−1ΓφψS (3.64)
=Q
4cos B+ sinh χ sin2 θ sin BΓ02
+ sin B cos θ[i cosh χ sin θ cos φ cos τΓ4 + cosh χ sin θ sin φ sin τΓ13
+ cosh χ sin θ cos φ sin τ(cos BΓ14 + i sin BΓ1)
− cosh χ sin θ sin φ cos τ(sin BΓ34 − i cos BΓ3)]
− sin B cos B sinh χ sin θ cos θ cos φΓ04
+(cosh χ sin2 θ cos τ sin2 B + sin θ cos φ cos2 B)Γ24
− cosh χ sin2 θ sin τ sin BΓ12
− sin B sin θ cos θ sin φ sinh χΓ03 + sin θ sin φ cos BΓ23
−i sin2 B sinh χ sin θ cos θ cos φΓ0
−i sin θ(cosh χ sin θ cos τ − cos φ) cos B sin BΓ2 .
This all simplifies at points obeying
sinh χ = ± tan B cos θ (3.65)
when −ψ′φ + φ′ψ = ±ψ′. Under these conditions
√deth =
Q
4(φ′ + cos θψ′) , (3.66)
and
S−1[φ′Γτφ + ψ′Γτψ + (φψ′ − ψφ′)Γφψ]S
=Q
4[−(φ′ + cos θψ′)Γ02 + (φ′D1 + ψ′D2)(Γ
0 ± Γ2)] , (3.67)
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 64
where
D1 = i cos θ sin B[cosh χ cos τ cos2 B + sin θ cos φ sin2 B
−i cosh χ sin τ cos BΓ1 + i sin θ sin φ sin BΓ3
−i(cosh χ cos τ − sin θ cos φ) sin B cos BΓ4] ,
D2 = − cos B(cos2 B sin θ cos φ + sin2 B cosh χ cos τ)Γ4 + cos B sin B cosh χ sin τΓ1
+ cos2 B sin θ sin φΓ3 − i sin B cos2 B(sin θ cos φ− cosh χ cos τ) . (3.68)
So far we have not chosen which supersymmetries are to be preserved. We take those
generated by spinors obeying Γ02ε0 = ±ε0, or equivalently Γ2ε0 = ∓Γ0ε0. In this
case, the last term in (3.67) can be dropped and the supersymmetry conditions are
satisfied.
To summarize, any configuration satisfying
− ψ′φ + φ′ψ = ±ψ′ , χ = θ = 0 ,
sinh χ = ± tan B cos θ (3.69)
preserves those supersymmetries corresponding to
Γ02ε0 = ±ε0 . (3.70)
Other BPS configurations preserving other sets of supersymmetries can be obtained
by SL(2, R)× SO(4) rotations of these ones.
Note that, as for the zero-branes, there are generic solutions for any θ. These
include θ = 0, π, which correspond to static one-branes because the (ψ, φ) torus
degenerates to a circle along these loci. Static solutions are possible because a one-
brane probe in 5D couples magnetically to the dual of the spacetime gauge field F[2]
of (3.11) hence there is nonzero angular momentum carried by the fields.
Chapter 3: Supersymmetric Probes in a Rotating 5D Attractor 65
3.4 Conclusion
In this chapter, we constructed supersymmetric brane probe solutions in the
squashed AdS2 × S3 near-horizon geometry of the BMPV black hole.
We expect that the quantization of the moduli space of these classical configura-
tions will provide a microscopic description of the five-dimensional N = 2 rotating
black holes. This will be carried out in the future work.
Chapter 4
Non-Supersymmetric Attractor
Flow in Symmetric Spaces
4.1 Introduction
Soon after the attractor mechanism was first discovered in supersymmetric (BPS)
black holes [63], it was reformulated in terms of motion on an effective potential for
the moduli [62]. Ferrara et al demonstrated that the critical points of this potential
correspond to the attractor values of the moduli. More recently, several groups used
the effective potential to show that non-supersymmetric (non-BPS) extremal black
holes can also exhibit the attractor mechanism, thereby creating a new and exciting
field of research [92, 76]. Many connections between non-BPS attractors and other
active areas of string theory soon revealed themselves. Andrianopoli et al found that
both BPS and non-BPS black holes embedded in a supergravity with a symmetric
moduli space can be studied using the same formalism, and they uncovered many
66
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 67
intricate relations between the two [11, 43]. Dabholkar, Sen and Trivedi proposed
a microstate counting for non-BPS black holes (albeit subject to certain constraints
[42]). Saraikin and Vafa suggested that a new extension of topological string theory
generalizes the Ooguri-Strominger-Vafa (OSV) formula such that it is also valid for
non-supersymmetric black holes [133]. Studying non-BPS attractors could also give
insight into non-supersymmetric flux vacua. Given all these possible applications, it
is important to characterize non-BPS black holes as fully as possible.
There has been a great deal of progress in understanding the near-horizon region
of these non-BPS attractors. The second derivative of the effective potential at the
critical point determines whether the black hole is an attractor, and the location of
the critical point yields the values of the moduli at the horizon; in this way, one can
compute the stability and attractor moduli for all models with cubic prepotential
[145, 113]. However, the effective potential has only been formulated for the leading-
order terms in the supergravity lagrangian. If one wants to include higher-derivative
corrections, one can instead use Sen’s entropy formalism, which incorporates Wald’s
formula, to characterize the near-horizon geometry in greater generality [135]. Sen’s
method has led to many new results [132, 13, 131, 36, 8, 136]. The tradeoff is that
this method cannot be used to determine any properties of the solution away from
the horizon.
The BPS attractor flow is constructed from the attractor value z∗BPS = z∗BPS(pI , qI)
by simply replacing the D-brane charges with the corresponding harmonic functions:
We know that k must be an element of G2(2), hence also of SO(4, 3). As in the
pure gravity case, we choose the representation such that
S0kS0 = kT , S0k2S0 = (k2)T . (4.112)
In this base a G2(2) Lie algebra element is given as
k =
Aj1i1 εi1j2kv
k√
2wi1
εi2j1kwk −Ai2j2 −√2vi2
−√2vj1√
2wj2 0
. (4.113)
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 99
Here A is a traceless 3× 3 matrix. S is a symmetric element in G2(2) with signature
1,−1,−1, 1,−1,−1, 1, i.e. S = MS0MT with
S0 =
η1 0 0
0 η1 0
0 0 1
= Diag(1,−1,−1, 1,−1,−1, 1) (4.114)
where η1 is the one for pure gravity.
If the gauge field is turned off, then S is block diagonal
S|F=0 =
Sgr 0 0
0 S−1gr 0
0 0 1
(4.115)
where Sgr is the same as the one for pure 5D gravity. Turning on a non-zero 5D
vector field corresponds to a more general S:
S = ekT1 (S|F=0)e
k1 (4.116)
with k1 a G2(2) Lie algebra matrix with w1 equal to the fifth component of the gauge
field, v2 equal to the time component of the gauge field and w3 equal to the scalar
dual to the three-dimensional part of the gauge field.
In this representation, (x, y) can be extracted from the symmetric matrix S via:
x(τ) = −S35(τ)
S33(τ), y2 =
S33(τ)S55(τ)− S35(τ)2
S33(τ)2. (4.117)
And u via:
u =1√
S33(τ)S55(τ)− S35(τ)2. (4.118)
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 100
The 4D gauge currents sit in J = S−1∇S, where J12(J31) is again the elec-
tric(magnetic) current for the KK photon, J32 the timelike NUT current, and J72(J51)
the electric(magnetic) current for the reduction of the 5D gauge field.
J32 = −2Jσ J12 =√
2JA0 J72 =2
3JA1 J51 = −
√2JB1 J31 =
√2JB0 .
(4.119)
Moreover,
J22 − J33 = 2JU . (4.120)
We use Q to denote the charge matrix, where it relates to the D-brane charge
p0, p1, q1, q0 and the vanishing NUT charge k by
(Q31,Q51,Q72,Q12) = (√
2p0,−√
2p1,2
3q1,√
2q0) , Q32 = −2k = 0 . (4.121)
Since k is nilpotent: k3 = 0,
S = ekτS0 = (1 + kτ +1
2k2τ 2)S0 . (4.122)
The AdS2 × S2 near-horizon geometry of the 4D attractor dictates u = 1VBH |∗ τ
−2 as
τ →∞. Therefore, the flow generator k can be obtained by
k2 = 2VBH |∗(uS|u→0)S0 . (4.123)
Computing k2 using S constructed from the solvable elements Σ(φ) shows that k2 is
of rank two, its Jordan form has two blocks of size 3.4 It can be written as
k2 =∑
a,b=1,2
vavTb cabS0 (4.124)
4In fact, the real G2(2) group has two third-degree nilpotent orbits, and it can be shown al-gebraically that in both orbits, k2 is of rank two and has Jordan form with two blocks of size 3[38].
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 101
with va null and orthogonal to each other: va · vb ≡ vTa S0vb = 0, and cab depends on
the particular choice of k. Thus k can be expressed as:
k =∑a=1,2
(vawTa + wav
Ta )S0 (4.125)
where each wa is orthogonal to both va: wa · vb = 0, and wa satisfy
wa · wb = cab . (4.126)
Parallel to the pure gravity case, the single-centered attractor flow is constructed
as S(τ) = eK(τ)S0, where we choose K(τ) to have the same properties as the generator
k:
K3(τ) = 0 and K2(τ) rank two. (4.127)
This determines K(τ) = kτ + g where
k =∑a=1,2
[vawTa + wav
Ta ]S0 and g =
∑a=1,2
[vamTa + mav
Ta ]S0 (4.128)
where the two 7-vectors ma’s are orthogonal to va and contain the information of
asymptotic moduli. Using [[k, g], g] = 0, the current is reduced to
J =S0(k + 1
2[k, g])S0
r2~r (4.129)
from which we obtain va and wa in terms of the charges and the asymptotic moduli.
4.5 Flow Generators in the G2(2)/(SL(2,R)×SL(2,R))
Model
We now explicitly construct the generators of single-centered attractor flows. We
start with the BPS flow which is associated with a specific combination of the coset
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 102
algebra generators aαA. It can be derived from the condition of preservation of super-
symmetry. We then construct the non-BPS attractor flow generator in analogy with
the BPS one. In Section 4.5.2, we write kBPS and knonBPS in terms of the va and
wa vectors. This form will be especially helpful in generalizing to the multi-centered
case.
4.5.1 Construction of flow generators
Constructing kBPS using supersymmetry
To describe BPS trajectories it is useful to remember that the stabilizer of S
in G2(2) is SO(1, 2) × SO(1, 2), corresponding to the elements of G2(2) which are
antisymmetric after multiplication by S0. Geodesics are exponentials of elements that
are symmetric after multiplication by S0. Such elements sit in a (2,4) representation
of SO(1, 2) × SO(1, 2). A BPS trajectory is highest weight for the first SO(1, 2).
Labelling the symmetric generators as aαA under the two SO(1, 2) groups, a BPS
trajectory is generated by
kBPS = aαACAzα . (4.130)
The twistor z and the coefficients CA are fixed in terms of the charges of the extremal
BPS black hole and the condition of zero time-like NUT charge.
To see why this is true, expand the coset element kBPS that generates the BPS
attractor flow using aαA:
kBPS = aαACαA (4.131)
where CαA are conserved along the flow. On the other hand, the conserved currents in
the homogeneous space are constructed by projecting the one-form valued Lie algebra
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 103
g−1 · dg onto k, which gives the vielbein in the symmetric space:
g−1dg|k = aαAV αA (4.132)
where V αA is conserved:
d
dτ
(V αA
a φa)
= 0 . (4.133)
Therefore, the expansion coefficients of kBPS are
CαA = V αAa φa . (4.134)
In terms of the vielbein, the supersymmetry condition that gives the BPS geodesics
are written as [127]:
V αAzα = 0 . (4.135)
That is:
V αAa φazα = 0 =⇒ CαAzα = 0 . (4.136)
Define zα = εαβzβ,
CαA = CAzα . (4.137)
Therefore, the coset element kBPS is expanded by the coset algebra basis aαA as
kBPS = aαACAzα.
Note that kBPS has five parameters (CA, z) where A = 1, . . . , 4. As will be shown
later, z can actually be determined by (CA) and moduli at infinity. So the geodesic
generated by kBPS is indeed a four-parameter family. It is easy to show that kBPS is
null, but more importantly, it is nilpotent:
k3BPS = 0 . (4.138)
As will be shown later, kBPS indeed gives the correct BPS attractor flow.
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 104
Constructing kNonBPS
To construct the non-BPS attractor flow, one needs to find an element in the coset
algebra distinct from kBPS that satisfies:
k3NonBPS = 0 . (4.139)
The hint again comes from the BPS generator. Note that kBPS = aαAPAzα can be
written as:
kBPS = e−zL−h k0BPSezL−h , (4.140)
where k0BPS spans only the right four coset generators a1A:
k0BPS = a1ACA . (4.141)
That is, kBPS is generated by starting with the element spanning the four generators
annihilated by the horizontal SL(2) raising operator L+h , then conjugating with the
horizontal SL(2) lowering operator L−h . And it is very easy to show that (k0BPS)3 = 0
which proves (kBPS)3 = 0.
In G2(2)/SL(2,R)2, there are two third-degree nilpotent generators in total [38].
And since there are only two SL(2,R)’s inside H, a natural guess for a non-BPS
solution is to look at vectors with fixed properties under the second SL(2,R) group.
An interesting condition is to have positive charge under some rotation of L3v, i.e.
an SL(2,R) rotation of∑
A=1,2 aαACαA. Therefore, this suggests us to start with
the element spanning the four generators annihilated by the square of the vertical
SL(2,R) raising operator (L+v )2 and then conjugate with the vertical SL(2,R) low-
ering operator L−v :
kNonBPS(z) = e−zL−v k0NonBPSezL−v , (4.142)
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 105
where
k0NonBPS = aαaC
αa where α, a = 1, 2. (4.143)
And one can show that: (k0NonBPS)3 = 0 which proves (kNonBPS)3 = 0. Moreover,
(kNonBPS)2 is rank two.
As long as one can pick the coefficients CαA and the twistor z that describes the
SO(1, 2) direction to be such that the time NUT charge is zero, this generator will
give nice non-BPS extremal black holes. All the known non-BPS solutions may be
recovered this way, and more, as this construction gives absolute freedom to pick the
charges and moduli at infinity for the black hole (clearly for certain values of charges
and moduli the solution will crash into a naked singularity, but this is to be expected
from comparison with the BPS case)
4.5.2 Properties of flow generators
Properties of kBPS
We now turn to solving for va and wa in (4.125) in terms of CA and z. First,
from (4.124) we know that the null space of k2 is five-dimensional and the va span the
two-dimensional complement of this null space. For kBPS = aαACAzα the null space
of (kBPS)2 does not depend on CA. Therefore, the va depend only on the twistor
z = z2/z1.
Recall that we are using the basis where k has the form (4.113). From inspection
of k2BPS, we find that (v1, v2) can always be chosen to have the form:5
v1 = (V1,−η1V1, 0) v2 = (−V2, η1V2,√
2) (4.144)
5When solving for (v, w), there are some freedom on the choice of (v1, v2) and (w1, w2): firstly, a
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 106
where η1 is a 3D metric of signature (1,−1,−1), and V1, V2 are two three-vectors with
V1 · V1 = 0, V1 · V2 = 0, V2 · V2 = −1. (4.145)
Since any linear combination of (v1, v2) forms a new set of (v1, v2), this means in
particular that any v2 + cv1 gives a new v2. Looking at the forms of (v1, v2), we see
that V2 is defined up to a shifting of V1 as V2 = V 02 − cV1.
An explicit computation shows that V1 and V2 are given by the twistor z and u as
V1 =
(z1)2 + (z2)2
(z1)2 − (z2)2
2z1z2
, V2 =1
z1u2 − z2u1
z1u1 + z2u2
z1u1 − z2u2
z1u2 + z2u1
, (4.146)
where the twistor u = u2
u1 is related to c by
u = −1 + 2cz
1− 2czz. (4.147)
rotational freedom
(v1, v2) → (v1, v2)(
R11 R12
R21 R22
)and (w1, w2) → (w1, w2)
(R11 R12
R21 R22
)
where R is orthogonal: RRT = 1. Secondly, a rescaling freedom:
va → rva and wa → 1rwa.
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 107
The twistor representation6 of V1 and V2 are
V αβ1 = 2zαzβ V αβ
2 = zαuβ + zβuα (4.148)
where we have used the rescaling freedom to set z1u2 − z2u1 to be 1. Note that for
the BPS case, the twistor u is totally arbitrary.
Now we solve for wa. The condition that wa are orthogonal to va dictates that
they have the form:
w1 = (W1, η1W1, 0) w2 = (W2, η1W2, 0) (4.149)
where W1 and W2 are linearly independent, and are related to the charges by wa ·wb =
cab:
W1 ·W1 =1
2c11 W1 ·W2 =
1
2c12 W2 ·W2 =
1
2c22 . (4.150)
Recall that V2 is defined up to a shift by V1: V2 = V 02 − cV1. The consequence is that
W1 is defined up to a shift by W2: W1 = W 01 + cW2. Note that the numerical factors
in front of V1 and W2 are opposite. Write down (W 01 ,W2) in terms of (CA, z):
W 01 =
1
4z
(C2 + C4) + (C1 + C3)z
(C2 − C4) + (C1 − C3)z
2C3 + 2C2z
, W2 =1
2
−(C2 + C4) + (C1 + C3)z
−(C2 − C4) + (C1 − C3)z
−2C3 + 2C2z
.
(4.151)
6With the inner product of three-vectors defined as
va · vb = vTa η1vb.
The twistor representation of a three-vector v = (x, y, z) is
σv = xσ0 + yσ3 + zσ1 =(
x + y zz x− y
).
Its length isvT η1v = det(σv) = x2 − y2 − z2.
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 108
The twistor representations of W1 and W2 are
W1 =
C1u2 − C2u1 C2u2 − C3u1
C2u2 − C3u1 C3u2 − C4u1
, W2 =
C1z2 − C2z1 C2z2 − C3z1
C2z2 − C3z1 C3z2 − C4z1
.
(4.152)
Define the totally symmetric Pαβγ :
P 111 = C1, P 112 = C2, P 122 = C3, P 222 = C4. (4.153)
Then the three-vectors (W1,W2) span the four dimensional space
(Wα1 ,W α
2 )BPS = (Pαβγuγ, Pαβγzγ) . (4.154)
Properties of kNonBPS
The form of va for the non-BPS case is only slightly different from the BPS case:
the two vectors va can be chosen to have the form:
v1 = (V1, η1V1, 0), v2 = (V2,−η1V2,√
2) (4.155)
where V1, V2 are two three-vectors satisfying the same condition as the BPS ones
(4.145). Again, the vectors V1 and V2 can be written as (4.146), and the twistor
representations are given in (4.148) with one major difference: u is no longer arbitrary,
but is determined by CαA as:
u =u2
u1=
C22
C12. (4.156)
The form of (w1, w2) are also slightly different from the BPS one (4.149)
w1 = (W1,−η1W1, 0) , w2 = (W2, η1W2, 0) . (4.157)
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 109
The (W1,W2) can be written in terms of (Cαa, z) thus:
and there is one extra degree of freedom to be fixed later.
8The twistor z can be left unfixed because we will not specify the asymptotic values of the scalarswith translational invariance, namely, (a,AI , BI). Fixing them can fix the twistor z.
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 113
To evaluate [k, g], we first use the commutation relation (4.97) to obtain
[a1ACA, a1BGB] = 〈C, G〉(−4L+h ) , (4.172)
where the product between CA and GA is defined as 〈C, G〉 ≡ C1G4 − 3C2G3 +
3C3G2 − C4G1. Then twisting Eq (4.172) with the twistor z as in (4.140) gives the
commutator of k and g with the same twistor z:
[k, g] = [aαAzαCA, aβBzβGB] = 〈C, G〉Θ , (4.173)
where Θ is defined as Θ ≡ − 41+z2 e
−zL−h L+h ezL−h . On the other hand, using (4.128),
[k, g] = (v2vT1 − v1v
T2 )S0(w2 ·m1 − w1 ·m2) . (4.174)
Θ can also be written as Θ = (v2vT1 − v1v
T2 )S0, and we can check that (w2 ·m1 −w1 ·
m2) = 〈C, G〉.
First, separate from GA the piece which has the same dependence on (h, z) as CA
on (Q, z):
GA = GAh + EA with GA
h ≡ CA(Q → h, z) . (4.175)
That is, g contains two pieces:
g = gh + Λ with gh = aαAzαGAh and Λ = aαAzαEA. (4.176)
We need to solve for EA.
There are three constraints from (4.171). The (x0, y0) and u0 are extracted from
the symmetric matrix S = egS0 via (4.117) and (4.118). On the other hand, requiring
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 114
(4.171) gives (x0, y0, u0) in terms of h:
x0 = −(g2hS0)35
(g2hS0)33
,
y0 =
√(g2
hS0)33(g2hS0)55 − (g2
hS0)235
(g2hS0)33
, (4.177)
u0 =1√
(g2hS0)33(g2
hS0)55 − (g2hS0)2
35
.
Therefore, defining Π ≡ (eg − g2h
2)S0, Π has to satisfy three constaints:
Π33 = Π35 = Π55 = 0 (4.178)
in order for (4.177) to hold for arbitrary h. Using the unfixed degree of freedom in
h’s to set 〈C, Gh〉 = 0, (4.169) becomes
Q = S0(k +1
2[k, Λ])S0 . (4.179)
The zero Taub-NUT charge condition in (4.179) imposes the fourth constraint on Λ:
the (3,2)-element of S0(k + 12[k, Λ])S0 for arbitrary k has to vanish. Combining with
(4.178), we have 4 constraints to fix EA to be:
E1 = −E3 = − 1
1 + z2, E2 = −E4 =
z
1 + z2. (4.180)
The remaining 4 conditions in the coupled equations (4.179) determine CA in the
BPS generator kBPS = aαAzαCA to be
C1 =√
2−q0 − q1z − 3p1z2 + p0z3
(1 + z2)2,
C2 =√
2− q1
3− (2p1 − v0)z + (p0 + 2 q1
3)z2 + p1z3
(1 + z2),
C3 =√
2−p1 + (p0 + 2 q1
3)z + (2p1 − v0)z
2 − q1
3z3
(1 + z2),
C4 =√
2p0 + 3p1z − q1z
2 + q0z3
(1 + z2)2. (4.181)
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 115
The GAh are then determined by GA
h = aαAzACA(Q → h, z). Using the solution of
CA and GAh , we see the product 〈CA, GA
h 〉 is proportional to the symplectic product
of (pI , qI) and (hI , hI):
〈CA, GAh 〉 =
2
1 + z2< Q, h > where < Q, h >≡ p0h0 + p1h1 − q1h
1 − q0h0 .
(4.182)
The condition 〈CA, GAh 〉 = 0 is then the integrability condition on h:
< Q, h >= p0h0 + p1h1 − q1h1 − q0h
0 = 0 . (4.183)
Substituting the expressions of CA and GA in terms of (pI , qI) into (4.165), we
obtain the BPS attractor flow in terms of the charges (pI , qI). In particular, the
attractor values are
x∗BPS = − p0q0 + p1 q1
3
2[(p1)2 + p0 q1
3]
, y∗BPS =
√J4(p0, p1, q1
3, q0)
2[(p1)2 + p0 q1
3]
, (4.184)
where J4(p0, p1, q1, q0) is the quartic E7(7) invariant:
J4(p0, p1, q1, q0) = 3(p1q1)
2 − 6(p0q0)(p1q1)− (p0q0)
2 − 4(p1)3q0 + 4p0(q1)3 , (4.185)
thus J4(p0, p1, q1
3, q0) is the discriminant of charge. The attractor values match those
from the compactification of Type II string theory on diagonal T 6, with q1 → q1
3. The
attractor value of u is
u∗BPS =1√
J4(p0, p1, q1
3, q0)
. (4.186)
The constraint on h from u0 = 1 is then J4(h0, h1, h1
3, h0) = 1.
Now we will show that the geodesic we constructed above indeed reproduces the
attractor flow given by replacing charges by the corresponding harmonic functions in
the attractor moduli. Using the properties of Λ, we have proved that, in terms of
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 116
k and g, the flow of (x, y) can be generated from the attractor value by replacing k
where all the ki’s and gh have the same value for the twistor z, we get
Qi = S0(ki+ < Qi, h > Θ +1
2[ki, Λ] +
∑j
< Qi, Qj >
|~xi − ~xj| Θ)S0 . (4.217)
Just as in the single-centered case, the solution of ki and the form of Λ guarantee
that
Qi = S0(ki +1
2[ki, Λ])S0 . (4.218)
We see that as long as the following integrability condition is satisfied:
< Qi, h > +∑
j
< Qi, Qj >
|~xi − ~xj| = 0 , (4.219)
the ki and g given above indeed produce the correct multi-centered attractor solution.
Just like in the single-centered case, the multi-centered solution flows to the correct
attractor moduli (x∗i , y∗i ) near each center, independent of the value of z. It also
follows that the multi-centered solution can be generated by replacing the charges
inside the attractor value by the multi-centered harmonic function:
xBPS(~x) = x∗BPS(Q →∑
i
Qi
|~x− ~xi|+h) , yBPS(~x) = y∗BPS(Q →∑
i
Qi
|~x− ~xi|+h) .
(4.220)
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 126
The sum of the N equations in the integrability condition (4.219) reproduces
the constraint on h: < Qtot, h >= 0. Thus the remaining N − 1 equations impose
N − 1 constraints on the relative positions between the N centers ~xi with i =
1, · · · , N . From (A.5) and (4.94), we see that ∗dω is given by J23. Defining the
angular momentum ~J by
ωi = 2εijkJj x
k
r3as r →∞ , (4.221)
we see that there exists a nonzero angular momentum given by
~J =1
2
∑i<j
~xi − ~xj
|~xi − ~xj|〈Qi, Qj〉 . (4.222)
Thus we have shown that our multi-centered BPS attractor solution reproduces the
one found in [18].
4.7.2 Non-BPS multi-centered solutions.
For given (z, u) and Cαai , Gαa, the non-BPS multi-centered solution is the same
as the single-centered one as in (4.191) with Hαa(τ) replaced by the multi-centered
harmonic function Hαa(~x) =∑
iCαa
i
|~x−~xi| + Gαa satisfying the constraint
u =H22
i (~x)
H12i (~x)
=C22
i
C12i
=G22
G12. (4.223)
Accordingly, the attractor values at each center is the same as (4.193) with the cor-
responding Cαas . The asymptotic moduli are obtained by extraction from S = egS0.
The equation of motion for the non-BPS multi-centered solution simplifies a great
deal since
[∇K(~x), K(~x)] = 0 (4.224)
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 127
automatically, following from the fact that for the non-BPS system:
(w1)i ·m2 = (w2)i ·m1 = 0 , (4.225)
which are guaranteed by the forms of NonBPS (w1, w2)i and (m1,m2). Therefore, the
5N equations (4.212) decouple into N sets of 5 coupled equations:
Qi = S0(ki)S0 . (4.226)
Equation (4.226) differs greatly from the BPS counterpart (4.212). Firstly, the
generators of the multi-centered non-BPS attractor solution ki and g have 3(N +
1)+2 parameters: the two twistors (z, u) and Cαai , Gαa with the constraint (4.223).
In constrast to the BPS case, g does not enter the equation. Thus we can simply
use the three asymptotic moduli, without invoking the zero Taub-NUT condition, to
determine the 3 Gαa inside g. Secondly, unlike the BPS multi-centered solution, the
position of the centers ~xi do not appear in the equation, therefore there will be no
constraint imposed on them: the centers are free. Last but not least, the remaining
3N + 2 parameters in (z, u) and Cαa are not enough to parameterize a generic N -
centered attractor solution, which has 4N D-brane charges (pIi , qI,i). Accordingly, the
multi-centered non-BPS attractor generated by this ansatz will not have arbitrary
charges. Combining with the fact that ~xi do not appear in the R.H.S of the equation,
we find that all the N vanishing Taub-NUT charge conditions can only act on the
charges on the L.H.S. We conclude that, in total, there will be 2N − 2 constraints on
the allowed charges.
Now we will show in detail the derivation of the constraints. First, like in the
single-centered NonBPS solution, the absence of the Taub-NUT charge at infinity
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 128
fixes z via∑
i
Qi = S0(∑
i
ki)S0 (4.227)
The solution is the same as the solution to (4.199) with the charges replaced by the
total charges of N centers: z = z(Q → ∑i Qi). Since all the N centers share the same
twistor z, the absence of the NUT charge at each center imposes N − 1 constraints
on the allowed charges Qi: all z(Qi) have to be equal.
The remaining 4N equations in (4.226) determine Cαa in terms of z and Qi. Since
the N -centers decouple, (4.226) for each center is the same as the single-center one
(4.197). Thus the solution of Cαai is given by (4.198) with (pI , qI) replaced by (pI
i , qI,i).
Again, since all the centers share the same twistor u, the condition (4.223) imposes
another N−1 constraints on the allowed charges. Solving these 2N−2 constraints, we
see all the charges Qi are the image of a single transformation Γ on a multi-centered
D4-D0 system Q40,i:
Qi = ΓQ40,i . (4.228)
The charges at different centers are all mutually local
〈Qi, Qj〉 = 0 . (4.229)
Except for the constraint on the charges, the N centers are independent, and there
is no constraint on the position of the centers. A related fact is that the the angular
momentum is zero.
Like the non-BPS single-centered case, though the multi-centered solution can be
generated from the attractor value by replacing Cαa with the multi-centered harmonic
function Hαa(~x) =∑
iCαa
i
|~x−~xi| + Gαa under the constraint (4.223), while leaving z
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 129
unchanged as in (4.188), the generic solution cannot be generated via the harmonic
function procedure used in the BPS case, namely, by replacing the charges inside
the attractor value by the corresponding multi-centered harmonic functions. The
reason is again due to the fact that the twistor z, being a function of charges, does
not remain invariant under this substitution of charges by harmonic functions. The
multi-centered non-BPS solutions that can be generated by the harmonic function
procedure are those with Qi, h being the image of a single Γ on the Q40,i, h40 of
a pure D4-D0 system:
xNB(~x) = x∗NB(ΓQ40 →∑
i
ΓQ40,i
|~x− ~xi| + Γh40) ,
yNB(~x) = y∗NB(ΓQ40 →∑
i
ΓQ40,i
|~x− ~xi| + Γh40) . (4.230)
It appears that the existence of a simple linear ansatz for “superimposing” single
center solutions exists in general only for mutually local extremal black holes, and
only in the supersymmetric case does it extend to mutually non-local centers.
To summarize, the non-BPS multi-centered solution is different from the BPS
case because it imposes no constraints on the position of the centers, but instead
on the allowed charges Qi: the choice of charges at each center are restricted to
a three-dimensional subspace, and they are mutually local. The result is that the
centers can move freely, and there is no angular momentum in the system. It does
not have interesting moduli spaces of centers with mutually non-local charges, so it
is as “boring” as the pure gravity case.
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 130
4.8 Conclusion and Discussion
In this chapter, we find exact single-centered and multi-centered black hole so-
lutions in theories of gravity which have a symmetric 3D moduli space. The BPS
and extremal non-BPS single-centered solutions correspond to certain geodesics in
the moduli space. We construct these geodesics by exponentiating different types of
nilpotent elements in the coset algebra. Using the Jordan form of these nilpotent
elements, we are able to write them down in closed explicit form. Furthermore, we
can use a symmetric matrix parametrization to recover the metric and full flow of the
scalars in four dimensions.
We have also generalized the geodesics to find solutions for non-BPS and BPS
multi-centered black holes. The BPS multi-centered solution reproduces the known
solution of Bates and Denef. Given our assumption that the 3D spatial slice is flat,
we find that a non-BPS multi-centered black hole is very different from its BPS coun-
terpart. It is constrained to have mutually local charges at all of its centers and
therefore carries no intrinsic angular momentum. It is possible that if we dropped
this assumption, we could find more general non-BPS multi-centered solutions. Such
configurations would probably be amenable to exact analysis only in the axially sym-
metric case, using inverse scattering methods.
There are many other avenues for future work. One could explore nilpotent ele-
ments in other symmetric spaces, and see whether non-BPS bound states with nonlo-
cal charges exist. In particular, it would be interesting to study E8(8)/SO∗(16), which
is the 3D moduli space for d = 4,N = 8 supergravity. We would also like to find a
way to modify our method so that we can apply it to non-symmetric homogeneous
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 131
spaces, and eventually to generic moduli spaces. We could then study the much larger
class of non-BPS extremal black holes in generic N = 2 supergravities.
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 132
Hx*,y*L
Hx1,y1L
Hx2,y2L
Hx3,y3L
Hx4,y4L
-1.0 -0.5 0.5 1.0 1.5 2.0x
1
2
3
4
y
Figure 4.3: Sample BPS flow. The attractor point is labeled (x∗, y∗). The initialpoints of each flow are given by (x1 = 1.5, y1 = 0.5), (x2 = 2, y2 = 4), (x3 = −0.2, y3 =0.1), (x4 = −1, y4 = 3)
Chapter 4: Non-Supersymmetric Attractor Flow in Symmetric Spaces 133
Hx*,y*L
Hx1,y1L
Hx2,y2L
Hx3,y3L
Hx4,y4L
Hx5,y5L
-0.5 0.5 1.0 1.5x
1
2
3
4
5
y
Figure 4.4: Sample non-BPS flow. The attractor point is labeled (x∗, y∗). The initialpoints of each flow are given by: (x1 = 0.539624, y1 = 5.461135), (x2 = 1.67984, y2 =0.518725), (x3 = −0.432811, y3 = 0.289493), (x4 = 1.28447, y4 = 1.49815), (x5 =−0.499491, y5 = 0.181744)
Chapter 5
Chiral Gravity in Three
Dimensions
5.1 Introduction
In three dimensions, the Riemann and Ricci tensor both have the same num-
ber (six) of independent components. Hence Einstein’s equation, with or without a
cosmological constant Λ, completely constrains the geometry and there are no local
propagating degrees of freedom. At first sight this makes the theory sound too trivial
to be interesting. However in the case of a negative cosmological constant, there are
asymptotically AdS3 black hole solutions [16] as well as massless gravitons which can
be viewed as propagating on the boundary. These black holes obey the laws of black
hole thermodynamics and have an entropy given by one-quarter the horizon area.
This raises the interesting question: what is the microscopic origin of the black hole
entropy in these “trivial” theories?
134
Chapter 5: Chiral Gravity in Three Dimensions 135
In order to address this question one must quantize the theory. One proposal
[147] is to recast it as an SL(2,R)L × SL(2,R)R Chern-Simons gauge theory with
kL = kR. Despite some effort this approach has not given a clear accounting of the
black hole entropy (see however [64] for an interesting attempt). So the situation
remains unsatisfactory.
More might be learned by deforming the theory with the addition of the gravita-
tional Chern-Simons term with coefficient 1µ
[51, 50]. The resulting theory is known
as the topologically massive gravity (TMG) and contains a local, massive propagat-
ing degree of freedom, as well as black holes and massless boundary gravitons. The
addition of the Chern-Simons term leads to more degrees of freedom because it con-
tains three, rather than just two, derivatives of the metric. It is the purpose of this
chapter to study this theory for negative Λ = −1/`2. We will argue that the theory is
unstable/inconsistent for generic µ: either the massive gravitons or BTZ black holes
have negative energy. The exception occurs when the parameters obey µ` = 1, at
which point several interesting phenomena simultaneously arise:
(i) The central charges of the dual boundary CFT become cL = 0, cR = 3`/G.
(ii) The conformal weights as well as the wave function of the massive graviton,
generically 12(3 + µ`,−1 + µ`), degenerate with those of the left-moving weight (2, 0)
massless boundary graviton. They are both pure gauge, but the gauge transformation
parameter does not vanish at infinity.
(iii) BTZ black holes and all gravitons have non-negative masses. Further the
angular momentum is fixed in terms of the mass to be J = M`.
This suggests the possibility of a stable, consistent theory at µ` = 1 which is
Chapter 5: Chiral Gravity in Three Dimensions 136
dual to a holomorphic boundary CFT (i.e. containing only right-moving degrees of
freedom) with cR = 3`/G. The hope — which remains to be investigated — is that
for a suitable choice of boundary conditions the zero-energy left-moving excitations
can be discarded as pure gauge. We will refer to this theory as 3D chiral gravity. As
we will review herein, if such a dual CFT exists, and is unitary, an application of the
Cardy formula gives a microscopic derivation of the black hole entropy [139, 95, 96].
Related recent work [148, 106, 72, 67, 14, 150, 149, 105] has considered an alterna-
tive deformation of pure 3D gravity, locally described by the SL(2,R)L × SL(2,R)R
Chern-Simons gauge theory with kL 6= kR. This is a purely topological theory
with no local degrees of freedom and is not equivalent to TMG. It contains all the
subset of solutions of TMG which are Einstein metrics but not the massive gravi-
tons. It is nevertheless possible that the arguments given in [148] (adapted to the
case kL = 0 as in [106]) which are quite general apply to the chiral gravity dis-
cussed herein. Indeed discrepancies with the semiclassical analysis mentioned in
[148, 106, 72, 67, 14, 150, 149, 105] disappear for the special case kL = 0.1 Moreover
the main assumption of [148] — holomorphic factorization of the partition function
— is simply a consistency requirement for chiral gravity because there are only right
movers.
This chapter is organized as follows. Section 5.2 gives a brief review of the cos-
mological TMG and its AdS3 vacuum solution, and shows that the theory is purely
chiral at the special value of µ = 1/`. Section 5.3 describes the linearized gravitational
excitations around AdS3. Section 5.4 shows how the positivity of energy imposes a
1Although we do not study the Euclidean theory herein the relation M` = J for Lorentzian BTZblack holes in chiral gravity suggests that the saddle point action will be holomorphic.
Chapter 5: Chiral Gravity in Three Dimensions 137
stringent constraint on the allowed value of µ. We end with a short summary and
discussion of future directions.
5.2 Topologically Massive Gravity
5.2.1 Action
The action for topologically massive gravity (TMG) with a negative cosmological
constant Λ is [53]
I =1
16πG
∫d3x
√−g(R− 2Λ) +1
16πGµICS (5.1)
where Ics is the Chern-Simons term
Ics = −1
2
∫d3x
√−gελµνΓρλσ[∂µΓσ
ρν +2
3Γσ
µτΓτνρ] . (5.2)
In this work, we will focus the theory with negative cosmological constant Λ.
We have chosen the sign in front of the Einstein-Hilbert action so that BTZ black
holes have positive energy for large µ, while the massive gravitons will turn out to
have negative energy for this choice. This contrasts with most of the literature which
chooses the opposite sign in order that massive gravitons have positive energy (for
large µ). Note that had we chosen the opposite sign in front of the Einstein action, the
BTZ black holes would have negative masses, which signifies a quantum instability.
Namely, the vacuum would be unstable against decaying into the infinite degeneracy
of zero energy solutions corresponding to negative mass black holes surrounded by
positive energy gravitons.
Chapter 5: Chiral Gravity in Three Dimensions 138
The equation of motion of the theory is
Gµν +1
µCµν = 0 , (5.3)
where Gµν is the cosmological-constant-modified Einstein tensor:
Gµν ≡ Rµν − 1
2gµνR + Λgµν (5.4)
and Cµν is the Cotton tensor
Cµν ≡ εµαβ∇α(Rβν − 1
4gβνR) (5.5)
where ε is Levi-Civita tensor.
Cotton tensor is the three-dimensional analogue of Weyl tensor, which vanishes
identically in 3D. From its definition, it is symmetric, manifestly traceless, and obeys
Bianchi identity.2 Since Cµν is manifestly traceless, its presence does not modify the
value of Ricci scalar solved from the equation of motion. That Cµν obeys Bianchi
identity shows that although the presence of the gravitational Chern-Simons term
manifestly breaks the diffeomorphism invariance of TMG, the equation of motion
of the bulk theory still obeys the diffeomorphism invariance. Finally, the metric
compatibility ensures that Einstein metrics which have Gµν = 0 are a subset of the
general solutions of (5.3).
2The Cotton tensor is symmetric thanks to the Bianchi identity of Gµν :
where, as before, φn are the 4(nV + 1) moduli fields: φn = U, zi, z i, σ, AI , BI, and
gab is the space time metric, gmn is the moduli space metric. Therefore, the moduli
Appendix A: Derivation of the Moduli Space M3D 185
space M3D has metric:
ds2 = dU · dU +1
4e−4U(dσ + AIdBI −BIdAI) · (dσ + AIdBI −BIdAI)
+gij(z, z)dzi · dz j (A.7)
+1
2e−2U [(ImN−1)IJ(dBI +NIKdAK) · (dBJ +N JLdAL)] .
It is a para-quaternionic-Kahler manifold. Since the holonomy is reduced from
SO(4nV +4)) to Sp(2,R)×Sp(2nV +2,R), the vielbein has two indices (α, A) trans-
forming under Sp(2,R) and Sp(2nV + 2,R), respectively. The para-quaternionic
vielbein is the analytical continuation of the quaternionic vielbein computed in [61]:
V αA =
iu v
ea iEa
−iE a ea
−v iu
.
The 1-forms are defined as
u ≡ eK/2−UXI(dBI +NIJdAJ) ,
ea ≡ eai dzi ,
Ea ≡ e−Ueai g
ijeK/2DjXI(dBI +NIJdAJ) ,
v ≡ −dU +i
2e−2U(da + AIdBI −BIdAI) , (A.8)
where eai is the veilbein of the 4D moduli space, and the bar denotes complex conju-
gate. The line element is related to the vielbein by
ds2 = −u · u + gabea · eb − gabE
a · E b + v · v = εαβεABV αA ⊗ V βB (A.9)
Appendix A: Derivation of the Moduli Space M3D 186
where εαβ and εAB are the anti-symmetric tensors invariant under Sp(2,R) ∼= SL(2,R)
and Sp(2nv + 2,R).
The isometries of the M∗3D descends from the symmetry of the 4D system. The
gauge symmetries in 4D gives the shifting isometries of M∗3D:
AI −→ AI + ∆AI ,
BI −→ BI + ∆BI , (A.10)
σ −→ σ + ∆σ + ∆BIAI −∆AIBI .
The conserved currents and charges are given by (4.93) and the discussion thereafter.
Appendix B
Energy-Momentum Pseudotensor
In this appendix we show that the energy defined from the energy momentum
pseudotensor for massive gravitons can be negative. For simplicity we specialize to
Λ = 0: for µ` À 1 the Compton wavelength of the gravitons is much shorter than
the AdS3 radius so the latter can be locally ignored.
Let us first review how the energy-momentum pseudo tensor at quadratic order
is defined without the Chern-Simons term or cosmological constant term. The full
Einstein equation in the presence of matter is
Gµν = 16πGTMµν . (B.1)
Expanding the metric gµν = ηµν + hµν , we have through quadratic order
G(1)µν = −G(2)
µν + 16πGTMµν , (B.2)
where G(1)µν and G
(2)µν are terms linear and quadratic in hµν . The energy momentum
pseudo tensor defined as
tµν = − 1
16πGG(2)
µν , with ∂µtµν = 0 (B.3)
187
Appendix B: Energy-Momentum Pseudotensor 188
sources the Newtonian part of the gravitational potential in the same way that the
matter stress tensor does. When adding the Chern-Simons term, the energy momen-
tum pseudo tensor is similarly defined as
tµν = − 1
16πG(G(2)
µν +1
µC(2)
µν ) . (B.4)
In flat background, the linearized equation of motion (5.25) becomes
∂2hµν +1
µεµ
αβ∂α∂2hβν = 0 (B.5)
under the harmonic plus traceless gauge. For a plane wave solution in the form of
hµν(k) = 1√2k0
e−ik·xeµν(k), the gauge conditions and the equations of motion are
kµeµν(k) = 0 , eµµ(k) = 0 , (B.6)
k2[eµν(k) − ikα
µεµ
αβeβν(k)] = 0 . (B.7)
The k2 = 0 solution is pure gauge. So eµν(k) − ikα
µεµ
αβeβν(k) = 0, which implies
(k2 + µ2)eµν(k) = 0. When kµ = (µ, 0, 0), the positive energy solution is
hµν =e−iµτ
√2µ
0 0 0
0 1 i
0 i −1
. (B.8)
It is convenient to define eµ(~0) = (0, 1, i), such that
eµν(~0) = eµ(~0)eν(~0) , (B.9)
and eµν(~k) = eµ(~k)eν(~k) , (B.10)
where eµ(~k) is obtained from eµ(~0) by a boost. The metric fluctuation can be ex-
panded in Fourier modes,
hµν =
∫d~k2 1√
2k0
α(~k)eµν(~k)e−ik·x + α†(~k)e∗µν(~k)eik·x . (B.11)
Appendix B: Energy-Momentum Pseudotensor 189
To calculate the energy momentum pseudo tensor, we will need the following,
tµν = − 1
16πG(G(2)
µν +1
µC(2)
µν ) , (B.12)
G(2)µν = R(2)
µν −1
2ηµνR
(2) , (B.13)
R(2)µν =
1
2hρσ∂µ∂νhρσ − hρσ∂ρ∂(µhν)σ +
1
4(∂µhρσ)∂νh
ρσ (B.14)
+(∂σhρν)∂[σhρ]µ +
1
2∂σ(hρσ∂ρhµν) , (B.15)
C(2)µν = εµ
αβ∂α(R(2)βν −
1
4ηβνR
(2)) + hµλελαβ∂αR
(1)βν − εµ
αβΓλ(1)αν R
(1)βλ , (B.16)
Γλ(1)να =
1
2ηλρ(∂αhρν + ∂νhρα − ∂ρhαν) , (B.17)
R(1)µν = −1
2∂2hµν . (B.18)
Using the equation of motion (5.34), it simplifies to
G(2)µν = −1
4(∂µhρσ)∂νh
ρσ − µ2
2hρ
µhρν − µ2
8ηµνh
ρσhρσ, (B.19)
C(2)µν =
µ2
4(3µhρµh
ρν + εµ
αβhλα,νhβλ) (B.20)
up to total derivatives. So the energy is
E =
∫d2~x t00 (B.21)
=
∫d2~k
n(k)
16πGk0
(k20 −
µ2
2− k0µ)− 1
2µ2e0(~k)∗e0(~k) . (B.22)
For vanishing spatial momentum n(~k) ∝ δ2(~k), the energy is just −µ times a positive
normalization factor.
Bibliography
[1] Ofer Aharony, Steven S. Gubser, Juan Martin Maldacena, Hirosi Ooguri, andYaron Oz. Large N field theories, string theory and gravity. Phys. Rept.,323:183–386, 2000, hep-th/9905111.
[2] Chang-rim Ahn and Masayoshi Yamamoto. Boundary action of n = 2 super-liouville theory. Phys. Rev., D69:026007, 2004, hep-th/0310046.
[3] Changrim Ahn, Chaiho Rim, and Marian Stanishkov. Exact one-point functionof N = 1 super-Liouville theory with boundary. Nucl. Phys., B636:497–513,2002, hep-th/0202043.
[4] Changrim Ahn, Marian Stanishkov, and Masayoshi Yamamoto. One-point func-tions of N = 2 super-Liouville theory with boundary. Nucl. Phys., B683:177–195,2004, hep-th/0311169.
[5] Changrim Ahn, Marian Stanishkov, and Masayoshi Yamamoto. ZZ-branes ofN = 2 super-Liouville theory. JHEP, 07:057, 2004, hep-th/0405274.
[6] A. N. Aliev and Y. Nutku. A theorem on topologically massive gravity. Class.Quant. Grav., 13:L29–L32, 1996, gr-qc/9812089.
[7] A. N. Aliev and Y. Nutku. Spinor formulation of topologically massive gravity.1998, gr-qc/9812090.
[9] Natxo Alonso-Alberca, Ernesto Lozano-Tellechea, and Tomas Ortin. Geometricconstruction of Killing spinors and supersymmetry algebras in homogeneousspacetimes. Class. Quant. Grav., 19:6009–6024, 2002, hep-th/0208158.
[10] Natxo Alonso-Alberca, Ernesto Lozano-Tellechea, and Tomas Ortin. The near-horizon limit of the extreme rotating d = 5 black hole as a homogeneous space-time. Class. Quant. Grav., 20:423–430, 2003, hep-th/0209069.
190
Bibliography 191
[11] Laura Andrianopoli, Riccardo D’Auria, Sergio Ferrara, and Mario Trigiante.Extremal black holes in supergravity. 2006, hep-th/0611345.
[12] Ignatios Antoniadis, S. Ferrara, R. Minasian, and K. S. Narain. R**4 couplingsin M- and type II theories on Calabi-Yau spaces. Nucl. Phys., B507:571–588,1997, hep-th/9707013.
[13] Dumitru Astefanesei, Kevin Goldstein, Rudra P. Jena, Ashoke Sen, andSandip P. Trivedi. Rotating attractors. JHEP, 10:058, 2006, hep-th/0606244.
[14] Spyros D. Avramis, Alex Kehagias, and Constantina Mattheopoulou. Three-dimensional AdS gravity and extremal CFTs at c=8m. JHEP, 11:022, 2007,arXiv:0708.3386 [hep-th].
[15] Vijay Balasubramanian and Per Kraus. A stress tensor for anti-de Sitter gravity.Commun. Math. Phys., 208:413–428, 1999, hep-th/9902121.
[16] Maximo Banados, Claudio Teitelboim, and Jorge Zanelli. The Black holein three-dimensional space-time. Phys. Rev. Lett., 69:1849–1851, 1992, hep-th/9204099.
[17] Tom Banks, W. Fischler, S. H. Shenker, and Leonard Susskind. M theory as amatrix model: A conjecture. Phys. Rev., D55:5112–5128, 1997, hep-th/9610043.
[18] Brandon Bates and Frederik Denef. Exact solutions for supersymmetric sta-tionary black hole composites. 2003, hep-th/0304094.
[19] Katrin Becker, Melanie Becker, and Andrew Strominger. Five-branes, mem-branes and nonperturbative string theory. Nucl. Phys., B456:130–152, 1995,hep-th/9507158.
[20] Klaus Behrndt, Renata Kallosh, Joachim Rahmfeld, Marina Shmakova, andWing Kai Wong. STU black holes and string triality. Phys. Rev., D54:6293–6301, 1996, hep-th/9608059.
[21] E. Bergshoeff, R. Kallosh, T. Ortin, and G. Papadopoulos. kappa-symmetry,supersymmetry and intersecting branes. Nucl. Phys., B502:149–169, 1997, hep-th/9705040.
[22] Eric Bergshoeff et al. N = 2 supergravity in five dimensions revisited. Class.Quant. Grav., 21:3015–3042, 2004, hep-th/0403045.
[23] M. Bershadsky, S. Cecotti, H. Ooguri, and C. Vafa. Kodaira-Spencer theoryof gravity and exact results for quantum string amplitudes. Commun. Math.Phys., 165:311–428, 1994, hep-th/9309140.
Bibliography 192
[24] Adel Bouchareb and Gerard Clement. Black hole mass and angular momen-tum in topologically massive gravity. Class. Quant. Grav., 24:5581–5594, 2007,arXiv:0706.0263 [gr-qc].
[25] Adel Bouchareb et al. G2 generating technique for minimal D=5 supergravityand black rings. Phys. Rev., D76:104032, 2007, arXiv:0708.2361 [hep-th].
[26] J. C. Breckenridge, Robert C. Myers, A. W. Peet, and C. Vafa. D-branes andspinning black holes. Phys. Lett., B391:93–98, 1997, hep-th/9602065.
[27] Peter Breitenlohner and Daniel Z. Freedman. Positive Energy in anti-De SitterBackgrounds and Gauged Extended Supergravity. Phys. Lett., B115:197, 1982.
[28] Peter Breitenlohner and Daniel Z. Freedman. Stability in Gauged ExtendedSupergravity. Ann. Phys., 144:249, 1982.
[29] Peter Breitenlohner, Dieter Maison, and Gary W. Gibbons. Four-DimensionalBlack Holes from Kaluza-Klein Theories. Commun. Math. Phys., 120:295, 1988.
[30] E. Brezin, V. A. Kazakov, and A. B. Zamolodchikov. Scaling Violation InA Field Theory Of Closed String In One Physical Dimension. Nucl. Phys.,B338:673–688, 1990.
[31] J. David Brown and M. Henneaux. Central Charges in the Canonical Realiza-tion of Asymptotic Symmetries: An Example from Three-Dimensional Gravity.Commun. Math. Phys., 104:207–226, 1986.
[32] I. L. Buchbinder, S. L. Lyahovich, and V. A. Krychtin. Canonical quantizationof topologically massive gravity. Class. Quant. Grav., 10:2083–2090, 1993.
[33] John L. Cardy. Boundary conditions, fusion rules and the Verlinde formula.Nucl. Phys., B324:581, 1989.
[34] Anna Ceresole and Gianguido Dall’Agata. Flow equations for non-BPS extremalblack holes. JHEP, 03:110, 2007, hep-th/0702088.
[35] Anna Ceresole, R. D’Auria, and S. Ferrara. The Symplectic Structure of N=2Supergravity and its Central Extension. Nucl. Phys. Proc. Suppl., 46:67–74,1996, hep-th/9509160.
[36] B. Chandrasekhar, S. Parvizi, A. Tavanfar, and H. Yavartanoo. Non-supersymmetric attractors in R**2 gravities. JHEP, 08:004, 2006, hep-th/0602022.
[37] Gerard Clement. The symmetries of five-dimensional minimal supergravityreduced to three dimensions. 2007, arXiv:0710.1192 [gr-qc].
Bibliography 193
[38] D. H. Collingwood and W. M. McGovern. Nilpotent orbits in semisimple Liealgebras. Van Nostrand Reinhold, New York U.S.A., 1993.
[39] Mirjam Cvetic and Donam Youm. All the Static Spherically Symmetric BlackHoles of Heterotic String on a Six Torus. Nucl. Phys., B472:249–267, 1996,hep-th/9512127.
[40] Mirjam Cvetic and Donam Youm. General Rotating Five Dimensional BlackHoles of Toroidally Compactified Heterotic String. Nucl. Phys., B476:118–132,1996, hep-th/9603100.
[41] Michelle Cyrier, Monica Guica, David Mateos, and Andrew Strominger. Mi-croscopic entropy of the black ring. Phys. Rev. Lett., 94:191601, 2005, hep-th/0411187.
[42] Atish Dabholkar, Ashoke Sen, and Sandip P. Trivedi. Black hole microstatesand attractor without supersymmetry. JHEP, 01:096, 2007, hep-th/0611143.
[43] R. D’Auria, S. Ferrara, and M. Trigiante. Critical points of the black-hole poten-tial for homogeneous special geometries. JHEP, 03:097, 2007, hep-th/0701090.
[44] Sebastian de Haro, Sergey N. Solodukhin, and Kostas Skenderis. Holographicreconstruction of spacetime and renormalization in the AdS/CFT correspon-dence. Commun. Math. Phys., 217:595–622, 2001, hep-th/0002230.
[45] B. de Wit, P. G. Lauwers, and Antoine Van Proeyen. Lagrangians of N=2Supergravity - Matter Systems. Nucl. Phys., B255:569, 1985.
[46] B. de Wit, F. Vanderseypen, and Antoine Van Proeyen. Symmetry structureof special geometries. Nucl. Phys., B400:463–524, 1993, hep-th/9210068.
[47] Frederik Denef, Davide Gaiotto, Andrew Strominger, Dieter Van den Bleeken,and Xi Yin. Black hole deconstruction. 2007, hep-th/0703252.
[48] Frederik Denef and Gregory W. Moore. Split states, entropy enigmas, holesand halos. 2007, hep-th/0702146.
[49] T. Dereli and O. Sarioglu. Topologically massive gravity and black holes inthree dimensions. 2000, gr-qc/0009082.
[50] S. Deser, R. Jackiw, and S. Templeton. Three-Dimensional Massive GaugeTheories. Phys. Rev. Lett., 48:975–978, 1982.
[51] S. Deser, R. Jackiw, and S. Templeton. Topologically massive gauge theories.Ann. Phys., 140:372–411, 1982.
Bibliography 194
[52] S. Deser and J. H. Kay. Topologically Massive Supergravity. Phys. Lett.,B120:97–100, 1983.
[53] S. Deser and Bayram Tekin. Massive, topologically massive, models. Class.Quant. Grav., 19:L97–L100, 2002, hep-th/0203273.
[54] S. Deser and Bayram Tekin. Energy in topologically massive gravity. Class.Quant. Grav., 20:L259, 2003, gr-qc/0307073.
[55] P. Di Vecchia and Antonella Liccardo. D-branes in string theory. II. 1999,hep-th/9912275.
[56] Paolo Di Vecchia and Antonella Liccardo. D branes in string theory. I. NATOAdv. Study Inst. Ser. C. Math. Phys. Sci., 556:1–59, 2000, hep-th/9912161.
[57] M. R. Douglas et al. A new hat for the c = 1 matrix model. 2003, hep-th/0307195.
[58] Tohru Eguchi and Yuji Sugawara. Modular bootstrap for boundary N = 2Liouville theory. JHEP, 01:025, 2004, hep-th/0311141.
[59] Henriette Elvang, Roberto Emparan, David Mateos, and Harvey S. Reall. Su-persymmetric black rings and three-charge supertubes. Phys. Rev., D71:024033,2005, hep-th/0408120.
[60] V. Fateev, Alexander B. Zamolodchikov, and Alexei B. Zamolodchikov. Bound-ary Liouville field theory. I: boundary state and boundary two-point function.2000, hep-th/0001012.
[61] S. Ferrara and S. Sabharwal. Quaternionic Manifolds for Type II SuperstringVacua of Calabi-Yau Spaces. Nucl. Phys., B332:317, 1990.
[62] Sergio Ferrara, Gary W. Gibbons, and Renata Kallosh. Black holes and criticalpoints in moduli space. Nucl. Phys., B500:75–93, 1997, hep-th/9702103.
[63] Sergio Ferrara, Renata Kallosh, and Andrew Strominger. N=2 extremal blackholes. Phys. Rev., D52:5412–5416, 1995, hep-th/9508072.
[64] Jens Fjelstad and Stephen Hwang. Sectors of solutions in three-dimensionalgravity and black holes. Nucl. Phys., B628:331–360, 2002, hep-th/0110235.
[65] Takeshi Fukuda and Kazuo Hosomichi. Super Liouville theory with boundary.Nucl. Phys., B635:215–254, 2002, hep-th/0202032.
[66] Matthias R Gaberdiel. Lectures on non-BPS Dirichlet branes. Class. Quant.Grav., 17:3483–3520, 2000, hep-th/0005029.
Bibliography 195
[67] Matthias R. Gaberdiel. Constraints on extremal self-dual CFTs. JHEP, 11:087,2007, arXiv:0707.4073 [hep-th].
[68] Davide Gaiotto, Aaron Simons, Andrew Strominger, and Xi Yin. D0-branes inblack hole attractors. 2004, hep-th/0412179.
[69] Davide Gaiotto, Andrew Strominger, and Xi Yin. Superconformal black holequantum mechanics. JHEP, 11:017, 2005, hep-th/0412322.
[70] Davide Gaiotto, Andrew Strominger, and Xi Yin. New connections between 4Dand 5D black holes. JHEP, 02:024, 2006, hep-th/0503217.
[71] Davide Gaiotto, Andrew Strominger, and Xi Yin. From AdS(3)/CFT(2) toblack holes / topological strings. JHEP, 09:050, 2007, hep-th/0602046.
[72] Davide Gaiotto and Xi Yin. Genus Two Partition Functions of Extremal Con-formal Field Theories. JHEP, 08:029, 2007, arXiv:0707.3437 [hep-th].
[73] Jerome P. Gauntlett and Jan B. Gutowski. Concentric black rings. Phys. Rev.,D71:025013, 2005, hep-th/0408010.
[74] Jerome P. Gauntlett, Robert C. Myers, and Paul K. Townsend. Black holes ofD = 5 supergravity. Class. Quant. Grav., 16:1–21, 1999, hep-th/9810204.
[75] Paul H. Ginsparg and Jean Zinn-Justin. 2-d Gravity + 1-d Matter. Phys. Lett.,B240:333–340, 1990.
[76] Kevin Goldstein, Norihiro Iizuka, Rudra P. Jena, and Sandip P. Trivedi. Non-supersymmetric attractors. Phys. Rev., D72:124021, 2005, hep-th/0507096.
[77] Michael B. Green and Michael Gutperle. Effects of D-instantons. Nucl. Phys.,B498:195–227, 1997, hep-th/9701093.
[78] David J. Gross and Nikola Miljkovic. A Nonperturbative Solution Of D = 1String Theory. Phys. Lett., B238:217, 1990.
[79] Monica Guica and Andrew Strominger. Wrapped M2/M5 duality. 2007, hep-th/0701011.
[80] Murat Gunaydin, Andrew Neitzke, Oleksandr Pavlyk, and Boris Pioline. Quasi-conformal actions, quaternionic discrete series and twistors: SU(2,1) and G2(2)
. 2007, arXiv:0707.1669 [hep-th].
[81] Murat Gunaydin, Andrew Neitzke, Boris Pioline, and Andrew Waldron. BPSblack holes, quantum attractor flows and automorphic forms. Phys. Rev.,D73:084019, 2006, hep-th/0512296.
Bibliography 196
[82] M. Gurses. Perfect fluid sources in 2+1 dimensions. Class. Quant. Grav.,11:2585–2588, 1994.
[83] G. S. Hall, T. Morgan, and Z. Perjes. Three-Dimensional Space-Times. KFKI-1986-95/B.
[84] S. W. Hawking. Particle Creation by Black Holes. Commun. Math. Phys.,43:199–220, 1975.
[85] M. Henningson and K. Skenderis. The holographic Weyl anomaly. JHEP,07:023, 1998, hep-th/9806087.
[86] Gary T. Horowitz and Andrew Strominger. Black strings and P-branes. Nucl.Phys., B360:197–209, 1991.
[87] Min-xin Huang, Albrecht Klemm, Marcos Marino, and Alireza Tavanfar. BlackHoles and Large Order Quantum Geometry. 2007, arXiv:0704.2440 [hep-th].
[88] C. M. Hull and B. Julia. Duality and moduli spaces for time-like reductions.Nucl. Phys., B534:250–260, 1998, hep-th/9803239.
[89] Nobuyuki Ishibashi. The boundary and crosscap states in conformal field the-ories. Mod. Phys. Lett., A4:251, 1989.
[90] Renata Kallosh, Arvind Rajaraman, and Wing Kai Wong. Supersymmetricrotating black holes and attractors. Phys. Rev., D55:3246–3249, 1997, hep-th/9611094.
[91] Renata Kallosh, Navin Sivanandam, and Masoud Soroush. Exact attractivenon-BPS STU black holes. Phys. Rev., D74:065008, 2006, hep-th/0606263.
[92] Renata Kallosh, Navin Sivanandam, and Masoud Soroush. The non-BPS blackhole attractor equation. JHEP, 03:060, 2006, hep-th/0602005.
[93] Elias Kiritsis and Boris Pioline. On R**4 threshold corrections in type IIBstring theory and (p,q) string instantons. Nucl. Phys., B508:509–534, 1997,hep-th/9707018.
[94] Per Kraus. Lectures on black holes and the AdS(3)/CFT(2) correspondence.2006, hep-th/0609074.
[95] Per Kraus and Finn Larsen. Microscopic black hole entropy in theories withhigher derivatives. JHEP, 09:034, 2005, hep-th/0506176.
[96] Per Kraus and Finn Larsen. Holographic gravitational anomalies. JHEP, 01:022,2006, hep-th/0508218.
Bibliography 197
[97] David Kutasov. D-brane dynamics near NS5-branes. 2004, hep-th/0405058.
[98] Gabriel Lopes Cardoso, Anna Ceresole, Gianguido Dall’Agata, Johannes M.Oberreuter, and Jan Perz. First-order flow equations for extremal black holesin very special geometry. JHEP, 10:063, 2007, arXiv:0706.3373 [hep-th].
[99] Gabriel Lopes Cardoso, Bernard de Wit, and Thomas Mohaupt. Correctionsto macroscopic supersymmetric black-hole entropy. Phys. Lett., B451:309–316,1999, hep-th/9812082.
[100] Sergei L. Lukyanov, E. S. Vitchev, and A. B. Zamolodchikov. Integrable modelof boundary interaction: the paperclip. Nucl. Phys., B683:423–454, 2004, hep-th/0312168.
[101] Sergei L. Lukyanov and Alexander B. Zamolodchikov. Dual form of the paper-clip model. 2005, hep-th/0510145.
[102] Juan Martin Maldacena. The large N limit of superconformal field theories andsupergravity. Adv. Theor. Math. Phys., 2:231–252, 1998, hep-th/9711200.
[103] Juan Martin Maldacena and Andrew Strominger. AdS(3) black holes and astringy exclusion principle. JHEP, 12:005, 1998, hep-th/9804085.
[104] Juan Martin Maldacena, Andrew Strominger, and Edward Witten. Black holeentropy in M-theory. JHEP, 12:002, 1997, hep-th/9711053.
[105] Alexander Maloney and Edward Witten. Quantum Gravity Partition Functionsin Three Dimensions. 2007, arXiv:0712.0155 [hep-th].
[107] John McGreevy and Herman L. Verlinde. Strings from tachyons: The c = 1matrix reloaded. JHEP, 12:054, 2003, hep-th/0304224.
[108] Thomas Mohaupt. Black hole entropy, special geometry and strings. Fortsch.Phys., 49:3–161, 2001, hep-th/0007195.
[109] Karim Ait Moussa, Gerard Clement, and Cedric Leygnac. The black holes oftopologically massive gravity. Class. Quant. Grav., 20:L277–L283, 2003, gr-qc/0303042.
[110] Robert C. Myers. Black holes in higher curvature gravity. 1998, gr-qc/9811042.
[111] Yu Nakayama. Liouville field theory: a decade after the revolution. Int. J. Mod.Phys., A19:2771–2930, 2004, hep-th/0402009.
Bibliography 198
[112] Yu Nakayama, Yuji Sugawara, and Hiromitsu Takayanagi. Boundary states forthe rolling D-branes in NS5 background. JHEP, 07:020, 2004, hep-th/0406173.
[113] Suresh Nampuri, Prasanta K. Tripathy, and Sandip P. Trivedi. On The Stabil-ity of Non-Supersymmetric Attractors in String Theory. JHEP, 08:054, 2007,arXiv:0705.4554 [hep-th].
[114] Andrew Neitzke, Boris Pioline, and Stefan Vandoren. Twistors and Black Holes.JHEP, 04:038, 2007, hep-th/0701214.
[115] Rafael I. Nepomechie. Consistent superconformal boundary states. J. Phys.,A34:6509–6524, 2001, hep-th/0102010.
[116] Y. Nutku. Exact solutions of topologically massive gravity with a cosmologicalconstant. Class. Quant. Grav., 10:2657–2661, 1993.
[117] Y. Nutku. Harmonic map formulation of colliding electrovac plane waves. 1993,gr-qc/9812030.
[118] Serkay Olmez, Ozgur Sarioglu, and Bayram Tekin. Mass and angular momen-tum of asymptotically AdS or flat solutions in the topologically massive gravity.Class. Quant. Grav., 22:4355–4362, 2005, gr-qc/0507003.
[119] Hirosi Ooguri, Andrew Strominger, and Cumrun Vafa. Black hole attractorsand the topological string. Phys. Rev., D70:106007, 2004, hep-th/0405146.
[120] Giorgio Parisi. String Theory On The One-Dimensional Lattice. Phys. Lett.,B238:213, 1990.
[121] Mu-In Park. BTZ black hole with higher derivatives, the second law of ther-modynamics, and statistical entropy. 2006, hep-th/0609027.
[122] Mu-In Park. Can Hawking temperatures be negative? 2006, hep-th/0610140.
[123] Mu-In Park. Thoughts on the area theorem. 2006, hep-th/0611048.
[124] Mu-In Park. Thermodynamics of exotic black holes, negative temperature, andBekenstein-Hawking entropy. Phys. Lett., B647:472–476, 2007, hep-th/0602114.
[125] Mu-In Park. BTZ black hole with gravitational Chern-Simons: Thermodynam-ics and statistical entropy. Phys. Rev., D77:026011, 2008, hep-th/0608165.
[126] R. Percacci, P. Sodano, and I. Vuorio. Topologically Massive Planar UniversesWith Constant Twist. Ann. Phys., 176:344, 1987.
[127] Boris Pioline. Lectures on on black holes, topological strings and quantumattractors. Class. Quant. Grav., 23:S981, 2006, hep-th/0607227.
Bibliography 199
[128] Joseph Polchinski. M-theory and the light cone. Prog. Theor. Phys. Suppl.,134:158–170, 1999, hep-th/9903165.
[129] Alexander M. Polyakov. Quantum geometry of fermionic strings. Phys. Lett.,B103:211–213, 1981.
[130] Bindusar Sahoo and Ashoke Sen. BTZ black hole with Chern-Simons and higherderivative terms. JHEP, 07:008, 2006, hep-th/0601228.
[131] Bindusar Sahoo and Ashoke Sen. Higher derivative corrections to non-supersymmetric extremal black holes in N = 2 supergravity. JHEP, 09:029,2006, hep-th/0603149.
[132] Bindusar Sahoo and Ashoke Sen. alpha’-Corrections to Extremal Dyonic BlackHoles in Heterotic String Theory. JHEP, 01:010, 2007, hep-th/0608182.
[133] Kirill Saraikin and Cumrun Vafa. Non-supersymmetric Black Holes and Topo-logical Strings. 2007, hep-th/0703214.
[134] Nathan Seiberg. Why is the matrix model correct? Phys. Rev. Lett., 79:3577–3580, 1997, hep-th/9710009.
[135] Ashoke Sen. Black hole entropy function and the attractor mechanism in higherderivative gravity. JHEP, 09:038, 2005, hep-th/0506177.
[136] Ashoke Sen. Entropy function for heterotic black holes. JHEP, 03:008, 2006,hep-th/0508042.
[137] Aaron Simons, Andrew Strominger, David Mattoon Thompson, and Xi Yin.Supersymmetric branes in AdS(2) x S**2 x CY(3). Phys. Rev., D71:066008,2005, hep-th/0406121.
[138] Sergey N. Solodukhin. Holography with gravitational Chern-Simons. Phys.Rev., D74:024015, 2006, hep-th/0509148.
[139] Andrew Strominger. Black hole entropy from near-horizon microstates. JHEP,02:009, 1998, hep-th/9712251.
[140] Andrew Strominger and Cumrun Vafa. Microscopic Origin of the Bekenstein-Hawking Entropy. Phys. Lett., B379:99–104, 1996, hep-th/9601029.
[141] Leonard Susskind. Another conjecture about M(atrix) theory. 1997, hep-th/9704080.
[142] Yuji Tachikawa. Black hole entropy in the presence of Chern-Simons terms.Class. Quant. Grav., 24:737–744, 2007, hep-th/0611141.
Bibliography 200
[143] Tadashi Takayanagi and Nicolaos Toumbas. A matrix model dual of type 0Bstring theory in two dimensions. JHEP, 07:064, 2003, hep-th/0307083.
[144] David Mattoon Thompson. Descent relations in type 0A / 0B. Phys. Rev.,D65:106005, 2002, hep-th/0105314.
[145] Prasanta K. Tripathy and Sandip P. Trivedi. Non-supersymmetric attractorsin string theory. JHEP, 03:022, 2006, hep-th/0511117.
[146] Robert M. Wald. Black hole entropy is the Noether charge. Phys. Rev.,D48:3427–3431, 1993, gr-qc/9307038.
[147] Edward Witten. (2+1)-Dimensional Gravity as an Exactly Soluble System.Nucl. Phys., B311:46, 1988.
[148] Edward Witten. Three-Dimensional Gravity Revisited. 2007, arXiv:0706.3359[hep-th].
[149] Xi Yin. On Non-handlebody Instantons in 3D Gravity. 2007, arXiv:0711.2803[hep-th].
[150] Xi Yin. Partition Functions of Three-Dimensional Pure Gravity. 2007,arXiv:0710.2129 [hep-th].