Boundary Conditions, Effective Action, and
Virasoro Algebra for AdS 3
by
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
AUG 13 2010
Achilleas P. Porfyriadis
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Bachelor of Science in Physics
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
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@ Achilleas P. Porfyriadis, MMX. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis document
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Author .....T
Certified
Author........__
Department of Physics
May 7, 2010
_
_
_
_
, ........................
Frank Wilczek
Herman Feshbach Professor of Physics
Thesis Supervisor
Accepted by.................................
David E. Pritchard
Senior Thesis Coordinator, Department of Physics
2
Boundary Conditions, Effective Action, and Virasoro
Algebra for AdS 3
by
Achilleas P. Porfyriadis
Submitted to the Department of Physics
on May 7, 2010, in partial fulfillment of the
requirements for the degree of
Bachelor of Science in Physics
Abstract
We construct an effective action of General Relativity for small excitations from
asymptotic transformations and use it to study conformal symmetry in the boundary of AdS 3. By requiring finiteness of the boundary effective action(s) for certain
asymptotic transformations, we derive the well known Virasoro algebra and central
charge associated with the boundary of AdS 3 . Our Virasoro generating transformations are asymptotic symmetries of appropriately defined new asymptotically AdS 3
spaces which are relaxed compared to the standard Brown-Henneaux ones but which
yield the same asymptotic symmetry group and central charge. Thus one may view
the effective action approach proposed in this thesis as a method for deriving boundary conditions for an asymptotic symmetry group. However, most importantly, we
believe that the effective action approach is by itself an alternative independent way
of obtaining and studying asymptotic conformal symmetry in the boundary of certain
space-times based on well-grounded requirements of finite action.
Thesis Supervisor: Frank Wilczek
Title: Herman Feshbach Professor of Physics
4
Acknowledgments
This work would not have been possible without the guidance and insight of my
supervisor, Prof. Frank Wilczek. He has supported me throughout with his patience
and knowledge while allowing me the room to work in my own way. It is a pleasure
and an honor for me to work with him.
I would also like to express my gratitude to Dr. Sean Robinson for countless
discussions we have had on physics relevant, for this thesis and beyond. I am grateful
to Prof. Marin Soljacic for his advice and support throughout my undergraduate
years at MIT.
Lastly, I would like to thank my parents, Pavlos and Maria, for their continuous
support and encouragement.
f6
Contents
1
9
Introduction and Outline
13
2 Brown-Henneaux-Strominger
.....................
2.1
Hamiltonian formulation of GR ......
2.2
Asymptotically AdS 3 spaces and Virasoso ....
2.3
Central charge and entropy calculation .....................
...............
13
18
20
3
Relaxing the boundary conditions
23
4
Effective action for GR
27
31
5 Small asymptotic transformations
5.1
Subleading Lie derivatives of AdS 3
. . . . . . . . . . . . . . . . . . .
1
] . . . .. . . . .. .. . . .. .
5.2 Finite first order effective action S')[
. . .. . . .. .. . . .. .
5.3 Finite second order effective action S()[
31
34
35
6 Asymptotic conformal Killing vectors
37
7 New asymptotically AdS 3 spaces
41
8
Conclusion and Discussion
43
R
Chapter 1
Introduction and Outline
In 1986, Brown and Henneaux [1] studied asymptotic symmetries of three-dimensional
Anti-de-Sitter space (AdS 3 ) and found they form two copies of the Virasoro algebra
with central charge c = 31/(2G), where l is the AdS 3 radius and G is the three dimensional Newton constant. This implies that any consistent quantum theory of
gravity on AdS 3 is dual to a two-dimensional conformal field theory (CFT) with
this central charge.
Within string theory this result is viewed as a special case
of the AdS/CFT correspondence [2, 3], but the original calculation by Brown and
Henneaux utilizes only the canonical formulation of General Relativity (GR). Using
the Brown-Henneaux central charge, in 1998, Strominger [4] derived the BekensteinHawking entropy of BTZ [5, 6] (and related) black holes microscopically by counting
the asymptotic growth of states via the Cardy formula. Recently, Strominger and
collaborators [7, 8] have extended these results to the extremal Kerr (and related)
cases in what is known as the Kerr/CFT correspondence. In Kerr/CFT, asymptotic
symmetries of the Near-Horizon-Extremal-Kerr (NHEK) metric [9] form one copy of
the Virasoro with central charge c = 12J (where J the angular momentum) and the
Cardy formula gives the Bekenstein-Hawking entropy of the original extremal Kerr
black hole.
In all cases, however, asymptotic symmetries are determined by choosing somewhat arbitrarily the fall off conditions at infinity. These boundary conditions, which
are then part of the definition of the theory, are imposed subject only to a few con-
sistency requirements which by themselves are not sufficient to completely fix them.
In this thesis, we derive the boundary conditions from an effective action approach.
For that purpose, we use the effective action of GR for small excitations obtained by
acting with the generators of asymptotic transformations. By requiring finiteness of
the boundary effective action(s) we obtain Virasoro generating transformations which
also happen to be asymptotic symmetries of new asymptotically AdS 3 spaces with relaxed boundary conditions (compared to Brown-Henneaux). Consistency within the
Hamiltonian formulation and the central charge at the Dirac bracket level reproduce
the Brown-Henneaux findings. However, we also stress in this thesis that we need not
confine ourselves in considering asymptotic symmetries of boundary conditions in the
first place as our independent derivation(s) of the Virasoro algebra in the asymptotics
of AdS 3 are well-grounded by themselves on physical arguments (finite action). The
results presented in this thesis will soon appear in [10].
The rest of the thesis is organized as follows. In Chapter 2, we review the BrownHenneaux-Strominger methods for obtaining the asymptotic symmetry group (ASG)
of AdS 3 , central charge of the corresponding two-dimensional CFT, and entropy of
the BTZ black hole. Then, in Chapter 3, we give a relaxed set of boundary conditions
for AdS 3 and motivate the need for deeper understanding than Brown-Henneaux. In
Chapter 4, we construct (to second order) the effective action of GR for small excitations from asymptotic transformations. Here we also derive the corresponding
equation of motion which is an extremely beautiful equation that is still under investigation. In the next chapter, we derive Virasoro generating vectors by requiring
"small" asymptotic transformation of exact AdS 3 with finite effective actions. Hence
the Virasoro manifests itself in the asymptotics of AdS 3 even without appropriately
chosen boundary conditions and an ASG. With the conformal group prominently
settled in the asymptotics of AdS 3 we decide to study, in Chapter 6, asymptotic conformal Killing vectors of AdS 3 , always subject to the requirement of finite effective
action(s). We find that they are also given by the Virasoro generating vectors obtained
in Chapter 5. Then acting with these vectors on AdS 3 we define, in Chapter 7, new
asymptotically AdS 3 space-times which are relaxed compared to Brown-Henneaux
and for which we find that their asymptotic symmetries are given entirely by these
same vectors used to define them. The corresponding ASG is still well defined and
is again the conformal group in two dimensions with central charge c = 31/(2G).
Chapter 8 is devoted to conclusions and discussions.
12
Chapter 2
Brown-Henneaux-Strominger
In this chapter we review the canonical formulation of GR, derivation of the BrownHenneaux central charge, and Strominger's calculation of entropy of the BTZ black
hole.
2.1
Hamiltonian formulation of GR
The calculation of the Brown-Henneaux central charge, c = 31/(2G), passes through
the canonical formulation of GR. Key papers include [11, 12, 13] and in this section
we review relevant results from these papers.
Note that in the recent papers on
Kerr/CFT a more modern formalism by Barnich, Brandt, and Compere [14, 15] was
employed, but here we review the original canonical approach as used in [1] and [4].
Consider the Einstein-Hilbert action of GR with a cosmological constant,
S=J dx
where our signature is (-
g (R - 2A),
(2.1)
+ + - ) and we use units such that G = 1/(161r). On a
manifold M with topology [0, 1] x E, where E is an (n - 1)-dimensional surface such
that {0} x E and {1} x E are both spacelike, there is a natural foliation into constant
time surfaces Et:
ds 2 = -N
2
dt 2 + gy (dx' + Nidt)(dxi + Ni dt),
(2.2)
where gij is the induced metric on Et. The above is called the ADM decomposition
of a metric where goo, goi are traded for N, N". In the ADM formalism spatial indices
i, j,... are lowered and raised with the spatial metric ggj
and its inverse 92.
Note
that ga' is different from the spatial components of the full n-dimensional g":
N'
-1
_
NJ
(2.3)
N2
N2
ij
_N'N-7
Using the ADM decomposition and utilizing Gauss-Codazzi equations, the EinsteinHilbert action (2.1) may be written in the form:
S=
( (n -)R - 2A + KiK' - K 2 ) + boundary terms.
dtI dn-1x N
(2.4)
Here, g is the determinant of the (n - 1)-dimensional spatial metric gij, which is also
used to build (n-'R, and K,, is the extrinsic curvature of Et:
1
VjNi),
(2.5)
where Vi is the covariant derivative with respect to gij.
Note that in (2.4) time
Kj
I (atgij -
=
ViNj -
derivatives occur only through the Btggj in Kj, so the canonical momenta are:
7r/
0-
-
=rfy(K' - g'jK) ,
(2.6)
where we take No =_ N. Using Dirac's terminology for constraint Hamiltonian systems
[16, 17], the equations -rl' = 0 are our primary constrains. On the other hand, the
equations for -r' may be inverted for K", which essentially amounts to inverting for
the velocities,
7ra -
K'j=
V/5
n - 2
guir
,
(2.7)
so there are no other primary constraints. Indeed, plugging (2.7) into (2.4) the action
takes its canonical form:
S
J
dt
j
3 g - NH - NH),
d-IX (7rir
(2.8)
where,
R-
=
(
=
-2/7Vj
y
-
-(2A),
-j(n-)R
7r
(2.9)
(2.10)
.
In the above, we have dropped all boundary terms and we shall forget about them
until we get to the final Hamiltonian.
Once we have the bulk piece of the final
Hamiltonian for GR we will supplement it with the necessary boundary terms so that
it has well defined variational derivatives.
From (2.8) we read off the canonical Hamiltonian for GR, that is to say, the
Hamiltonian on the primary constraint surface r' = 0:
H =
dn-
(2.11)
(NN + NiHj).
Following Dirac's recipe for constraint Hamiltonian dynamics we form the total Hamiltonian, HT = H +
f. d"~x APIr", and demanding
served in time, irY
=
the primary constraints to be pre-
0 e {rIF, HT} = 0, we obtain the secondary constraints:
(2.12)
H-,=0 ,
where we take No
NH. Note that all constraints are first class. We then eliminate
the redundant degrees of freedom by identifying N" with arbitrary functions and
imposing TrA = 0 as strong equations, so that the Poisson bracket is
{F,G}P.B. =
dn-lx (
1g5
15
j 6,su
ogig ogru
.G
(2.13)
and the final Hamiltonian for GR is given by (2.11). Note that this Hamiltonian
vanishes on account of the remaining first class constraints 'H, = 0. However, it is
now time to include the necessary boundary terms which contribute the (nonzero)
energy of GR.
For the Hamiltonian H to correctly reproduce Einstein's field equations in Hamiltonian form it must have well defined variational derivatives 6H/5gij and JH/Jiris.
That is, if we make arbitrary independent variations
ogij, owru
of the canonical vari-
ables the induced variation of the Hamiltonian should take the form:
d n- 1 x (A' 2 ogij + BijJw,'j).
JH =
(2.14)
By construction, we know that if this is the case then the variational derivatives
A'j= JH/6gij and Bij = JH/Jiri' will be such that Hamilton's equations
6H
gij =-
,
-'r
6H
gij
(2.15)
are equivalent to Einstein's field equations. Thus any surface terms arising in the
variation of (2.11) must vanish on the boundary. For a closed space this is easy but
for open spaces, such as asymptotically AdS 3 space-times, it is not necessarily so.
Thus we need to add to the Hamiltonian appropriate surface terms whose variation
will cancel the surface terms arising in the variation of (2.11). Keeping all terms, the
variation of (2.11) is given by:
6H =
dlx (Az'6gij + Bi6ris})
-
j
j
dj-2 si
Gijki (Nogijj;k - Nk Sgij)
(2.16)
di- 2s [2Nk 6rkl + (2Nkffil - N1 rjk)6ggjk]
where the semicolon denotes covariant differentiation with respect to gij, and
1
2
Gikl= --
(gik gjl + gii gjk - 2gij gk ).
(2.17)
Thus the Hamiltonian (2.11) must be supplemented with a surface term E[g9j] whose
variation 6E precisely cancels the surface terms in (2.16). Note that this defines E
only up to the addition of an arbitrary constant. Hence the correct Hamiltonian for
GR is given by H + E and the energy of GR is just the (nonzero) numerical value of
E (for recall that upon imposing the initial value equations H,, = 0 we have H = 0).
and Bj in (2.16) are not needed for our purposes;
The precise expressions of A'
however, they give the Hamiltonian equations of motion for GR which we list for
completeness:
aj
=
i
=
2Ng- 1/ 2
i)
r-n
kl _
1Ng1/2ij (r
2
-Ng
1 2
+g 1 /2 Vk (-1/
2
N kii)
2Ng1/ 2 (,ik~j
1
,2)
(n-
R - 2A) gi)
-
n -2
(n-1)Rj -
(2.18)
+ 2 V(iNj),
-
+ g 1/ 2 ('ViN
n-
n
Wi2
2
- gd3 [IN)
(2.19)
- 21k(iVkNi).
We conclude with the general theory of canonical generators of diffeomorphisms.
Consider a space-time vector ( which generates infinitesimal diffeomorphisms via the
Lie derivative:
Jg
9
, = Egg,,. In the Hamiltonian formulation of GR, the correspond-
ing canonical generator which generates the same transformation via the Poisson
bracket, 6 gij = {gij, H[ ]} PB., is given by:
H[]=
-
1
Xed R1, +
Q[].
(2.20)
Here, the canonical or surface deformation vector (e is related to the space-time
diffeomorphism vector
via
S= N$O,
and
Q[
=
+ NWgt
(2.21)
] is again a surface term whose variation precisely cancels the surface terms
produced by the variation of the bulk integral in (2.20), so that the total generator
H[ ] has well defined variational derivatives (with respect to gij and -Frj) in the Poisson
brackets, i.e.:
Q
d- 2s1 Gijkl (
6
gij;k -
dn- 2s i [2k
+
'k ogij)
-1-r'+
(2 k 11
-
.i
7
ik) Jgjk] .
(2.22)
Q[
The Q's are often called "charges" of the associated transformations because
gives the global charge associated with
)
. For example, as the Hamiltonian is the
generator of time transformations, corresponding to ( = (1,0,0,...), the energy is
given by E =
2.2
Q[&t].
Similarly, Q[8o] gives the angular momentum.
Asymptotically AdS 3 spaces and Virasoso
The AdS 3 metric is given by:
ds 2
-
dr 2 +r2
dt2 + (i+
(i+
2
.23)
(
Asymptotically AdS 3 space-times are defined as ones which behave similarly to AdS 3
in the limit r
->
o0 with certain falloff conditions. The falloff conditions at infinity
are called boundary conditions. For example, Brown-Henneaux chose:
Ir2
p
+ 0(1),
gtt
=
gtr
=
1
0(7),
g
=
(1),
grr
=
12
7+(-s),
goo=
r2
(2.24)
1
+ O(1).
These falloff conditions at infinity (i.e. the boundary conditions) are such that the
following consistency requirements are met [18]: (i) they are invariant under the
AdS 3 isometry group, (ii) they allow for the asymptotically AdS 3 solutions of physical
interest (the BTZ black hole here), and (iii) they yield finite charges in the canonical
(i.e. Hamiltonian) formalism of GR.
Having chosen the boundary conditions, the asymptotic symmetries are given by
the vector fields
which preserve the metric (2.24). That is, the ('s which transform
any metric of the form (2.24) into another one of the same form: hence the
's
are solutions of the equations £Egt = 0(1), Lgt, = 0(1/r 3 ), £ggto = 0(1), etc.
Studying these Lie transformation equations, one finds the most general solution
may be nicely written as:
t
l(T++T~)+
=
(
-r (T+' + T
~ ~
~
TI+"+T-") + 0()
+0(1),
+
1 =r+-T
(2.25)
-T"+O
Here T+ = T+ (x+) and T- = T- (x-) are arbitrary functions of a single argument
X+ = t/l t
4.
That is to say, T+ depend on t, r, 4 only as T*(t, r, 4) = T+(t11 ± 0).
This arbitrariness in the "asymptotic Killing vectors" (2.25) is expected due to the
arbitrariness in the metric (2.24) whose symmetries they are. It is important for
obtaining the Virasoro too: Expanding into modes T+ -einx±
we find that (with
ai = (lat ± a0)/2) the space-time generators,
(ie"'
B+
inr(2.26)
2 ,(.6
122
2r2
form (under Lie brackets) two copies of the classical centerless Virasoro algebra:
[
i,e = i(n
-
)
(2.27)
Note that in writing (2.26) we have dropped the arbitrary sub-leading terms in (2.25)
and therefore the Virasoro (2.27) is obtained to corresponding leading orders in 1/r
only. As we will see in the next section this is more than enough, as we only need to
close the algebra at leading order in 1/r.
To every consistent set of boundary conditions, such as (2.24), corresponds an
asymptotic symmetry group (ASG) which is defined as the set of allowed symmetry
transformations, such as (2.25), modulo the set of trivial symmetry transformations.
Here, "trivial" means that there is no associated charge and the canonical generators
of these transformations vanish upon imposing the constraints. In other words, the
ASG is defined as the factor group obtained by identifying all asymptotic symmetries
which differ by terms with vanishing Q's. Such were, for example, the arbitrary subleading terms in (2.25) which we dropped in writing (2.26). Therefore, given (2.27),
the asymptotic symmetry group corresponding to the Brown-Henneaux boundary
conditions (2.24) is the conformal group in two dimensions.
2.3
Central charge and entropy calculation
To leading order in 1/r, the Poisson bracket algebra of canonical generators H[ ) is
isomorphic to the Lie bracket algebra of the space-time generators ( up to a possible
central extension:
{H[], H[T]}PB. = H[ [,
]Lie ] + K[ , n].
(2.28)
Thus in view of (2.27) we have:
{ H[],H [FEj] }P.B. = i(n - m) H[ L-n] + K[5 , a],
(2.29)
which passing to the Dirac bracket (i.e. imposing the constraints 'H,, = 0) becomes:
{Q[Q1, Q[
]} D. B. =
+Q
]+
K[(n,
].
(2.30)
The central extension K is most easily obtained using the following trick of BrownHenneaux: Since the Q's are defined only up to a constant (recall that they are
defined via 6Q) we may choose this constant so that
Q=
0 at t = 0 on AdS 3 . Then,
evaluating (2.30) at t = 0 on AdS 3 ,
K[E,, a] = {Q[a], Q[
]}D.B.
Q[
=
(2.31)
.-
Using (2.22), the right hand side of the above is given by
27r
dp {Gijkr [(Fm)o gij-k - ($m) ,k oggi
lim
+2
k
[2 ( )I-
-F r+
-,j
( ±)r _Rk] j
gk}
2.2
which upon evaluation at I = 0 on AdS 3 yields:
] = 2i47rin(n2
K[gm,
With L
=
Q[±j, and passing from the Dirac
-
(2.33)
)5m+n,O.
bracket to the commutator { , }D.B.
-
we get from (2.30) and (2.33) the quantum Virasoro algebra:
[L±, L+] = (m - n)L±+n+ 2'rlm(m2
-
1)6m+n,o.
(2.34)
Finally, following Strominger, in order to obtain the entropy of the BTZ black hole
we use the Brown-Henneaux central charge in co-operation with the (microscopic)
Cardy formula:
S=2r
From (2.34) c+ = c_ =
+2r
L
L
(2.35)
(after restoring G). Also, since the global charges of the
BTZ black hole are M = Q[8t] and J =
M=
Q[ao] we quickly obtain,
J= L+ - 0 L-0~
(L+ + L-),
(2.36)
Solving for L+ and L- and plugging everything into the Cardy formula (2.35) reproduces (microscopically) the Bekenstein-Hawking entropy of the BTZ black hole:
S=r/
l(lM + J)
2G
+
V
l(lM - J)
2G
(2.37)
22
Chapter 3
Relaxing the boundary conditions
From the review of the Brown-Henneaux recipe in the previous chapter, one is naturally left with the question how did Brown-Henneaux come up with the particular
boundary conditions (2.24). Since the emergence of the Virasoro algebra seems to
depend crucially on the choice of boundary conditions, one would like to have a convincing argument before making this, or for that matter any other, particular choice.
Recall from Section 2.2 that boundary conditions are subject to the three consistency requirements (i-iii) set forth by Henneaux and Teitelboim in [18]. For (ii) the
boundary conditions should be weak enough to allow for the interesting excitations
of the theory, but at the same time for (iii) they should be strong enough so as to
yield finite charges. In general, there is a narrow window of consistent boundary
conditions. However, contrary to popular belief, the consistency conditions (i-iii) are
not sufficient to fix the boundary conditions -not
even for quantum gravity on AdS 3.
One way to fulfill requirements (i) and (ii) is to start with the BTZ metric and act
on it with the AdS 3 isometry group in all possible ways. This procedure generates
metric perturbations which behave asymptotically as in (2.24). Interestingly, however, at the time of Brown-Henneaux [1] the BTZ black hole [5, 61 was not known.
Brown and Henneaux actually used a different metric instead (they constructed an interesting one themselves by removing a "wedge" from AdS 3 and introducing a "jump"
via appropriate identifications of points) which also approached AdS 3 in the asymptotic limit. It turned out that the boundary conditions they obtained using the same
procedure starting from their metric, that. is (2.24), accommodated the yet to be
discovered BTZ black hole too. This raises the question: what if they didn't? Is it
possible to relax the Brown-Henneaux boundary conditions while maintaining consistency, in the sense of (i-iii), as well as the Virasoro? Clearly, for a random new
choice of relaxed boundary conditions the Lie transformation equations will change
in such a way that the Virasoro generating vectors (2.25) will no longer be asymptotic Killing vectors of the new asymptotically AdS 3 space-times (as defined by the
new boundary conditions). However, there exist choices of boundary conditions for
which the Virasoro generating vectors (2.25) are included in the asymptotic Killing
vectors of the new asymptotically AdS 3 space-times. Here is an interesting set of such
alternative boundary conditions (relaxing the tr and r# components):
=
9tt
31
,
(
1
gt,
=
0(
goo
=
r2+
r),
O(1).
to this choice of boundary
Studying the Lie transformation equations
r
.9rr corresponding
12
conditions one finds the most general asymptotic Killing vectors are given by:
t
=
I(T++T-)+
r
=
-r
T+
+ T
-- T-
-
(T+11+ T-") + 0()
+)1T+1
0(,)
T+2
(3.2)
- T -"
These are just relaxed versions of the vectors in (2.25). The consistency requirements
about including the AdS 3 isometries and the BTZ black hole are automatically met
since both the metric (3.1) and the asymptotic Killing vectors (3.2) are relaxed ver-
sions of Brown-Henneaux's (2.24) and (2.25) respectively. It only remains to check
finiteness of the charges in which case (i.e. if they are indeed finite) the rest of
the Brown-Henneaux-Strominger approach will lead all the way to the same central
charge, c = 31/2G, and correct Bekenstein-Hawking entropy of the BTZ black hole
again. This is indeed the case: for the relaxed (3.1-3.2) the charges
Q[
] are still
finite and equal to the ones obtained for (2.24-2.25).
Recall that we only need the Virasoro at leading order in 1/r and for that purpose
1
the terms involving T+" and T-" in (2.25) or (3.2) are not important. One naturally
then wonders wether it would be possible to relax further the boundary conditions so
as to replace these terms in the asymptotic Killing vectors by completely arbitrary
ones O(1/r 2 ) while also maintaining finite charges. We will see that this is indeed
possible. However, rather than continuing our guesswork, we will arrive at these
boundary conditions via the effective action approach proposed in this thesis.
1Note, however, that in (3.2) the arbitrary O(1/r
1/r compared to (2.25).
3
) terms spoil the Virasoro at higher order in
26
Chapter 4
Effective action for GR
Our goal in this chapter is to construct (to second order) the effective action of GR for
and derive the corresponding equation of motion.
small excitations g,, -+ ,+Eg
We thus start from the Einstein-Hilbert action of GR (2.1) and putting gg, -> gLv+hyv
we expand to second order in h. We then put h,, = Lglv = V,&, + VV4 and obtain
the desired effective action for (.
We have:
S
-
0 )[h] + S(1)[h] + S(2)[h] + 0(h3 ) ,
SC
(4.1)
where,
SC0 )[h]
-
IM
SM)[h]
-
-
J
-g (R - 2A),
S(2) [h]
-
faM
1
d x
+
d"~ 1x v/ ~n(V"hp, - VPh)
dnx v-g (G" + Ag"v)h,, +
M
d-
1
g hv
[g"VgPc"
-
gPg"El
+ 2g"V"V" - 2gV"V V"
x A/- yn" (2huVVh + 2hVvht, + 3hP"Vuh,
-4h[IVch"" - 2h"V h,, - hV 1 h).
Here indices are raised and lowered using the background metric g, h" - guPgvohp,
and the trace is defined as h -- gvh, .
we get an effective action for (. The first order action
+V
Putting hy, = Vt
for ( reads:
d'x /-
-2
SM
+
J
d-
1
x
(G1"
+ A9gI)V,,v
- yn/(D, + V"V
-
(4.2)
2V/,V,").
Upon integrating by parts the bulk piece above and using the contracted Bianchi
identity,
(4.3)
VGLV = 0 ,
the first order action (4.2) reduces to a boundary term:
S
[] =
j
d"-x V/-'n" (l1
VvVP" + (R - 2A)(,)
-
(4.4)
.
Thus at first order the effective action for ( is just a boundary term. Note that (4.3)
is an identity on curvature and so the first order action (4.4) is obtained without
assuming the Einstein Field Equations (EFE) for the background gi.
This is a
consequence of the diffeomorphism invariance of the Einstein-Hilbert action: the
transformation gY, -> g,
+ V1
+ Vj,
is an infinitesimal diffeomorphism (i.e. a
diffeomorphism to first order in () and therefore, at first order, the Einstein-Hilbert
action changes only by a boundary term.
The second order action for (, after forcing intense simplification on the bulk term,
takes the form:
S(2 []
IMdx y!-g (GI"
JM
+ Ag"") (V,"
V
d"- 1 x -\ yn" {JVVVaV
,
+(V,(
+ Rg,(*(" )
-
'VV"Vvg, +
1
+ Vug,)(1 (V - VVV~J) + IV, [(Vv()
+2(V"V")(VVcr(
-
VpVcifv) +
2
RzvpYPVo(v
(4.5)
_
"Vo [ ,(Glv + Agv) -
j,(Gv, + Agzcy)]
This second order action also reduces to a boundary term if we assume the background
.
g,, satisfies the EFE, G" + Ag""
-
0. Note that this is a consequence of gauge
invariance of the second order action for h: assuming EFE for the background g,,,
S(2 )[h] in (4.1) is invariant (changes only by a boundary term) under h,,, -*
V,(, + V,(, and so plugging h,
hy +
= Vg, + v,( into S(2) [h] is essentially plugging
pure gauge. However, if we don't assume the background solves the EFE then (4.5)
has a non-vanishing bulk effective action for (. Thus using the variational principle
with 6(a such that all boundary terms vanish as needed, the effective action up to
second order yields the remarkably beautiful equation of motion (EOM):
(GI
+ Ag"")((Q;mv - RcpVXy,
(4.6)
) = 0.
Here, as expected, we see again that if the background gi, solves the EFE then there
is no equation. But in (4.6) we also recognize the second parenthesis: it's an equation
satisfied by Killing vectors. For an exact Killing vector we have Vj,
+ V,,
= 0
which, after taking a derivative and playing with the relevant equations a bit, implies
-- Rc,
(a;,
=
0 (see [19] for example). Thus for exact Killing vectors there is
again no equation. This should have been expected since if ( is a Killing vector then
our transformation g,
SM)[(]
-
-+ gy,, + V,(, + V,
is not really a transformation and indeed
0 to all orders n > 1.1
We thus have the following compelling picture. The effective EOM (4.6) is a
contraction of two well known equations:
" GIV + Agv = 0 which is satisfied by exact solutions to Einstein's GR.
" o;ttv - Ral,,(o = 0 which is satisfied by exact Killing vectors.
The effective EOM (4.6) is an equation for approximate solutions to Einstein's gravity
and their approximate symmetries. For example, it is satisfied in the asymptotic limit
by asymptotically AdS 3 space-times (2.24), (3.1) and their corresponding asymptotic
2
'For an exact Killing vector S(1)[ ] is easily seen to vanish in equation (4.2). For S( )[ ] note that,
up to a boundary term, the bulk piece in (4.5) may be written as - fM d'x -g (GI" +Agl")((
2
Rcaglv()(o in which case one finds that the resulting total boundary piece in S( ) [ ] also vanishes
for an exact Killing vector.
Killing vectors (2.25), (3.2), respectively. That is, (2.24-2.25) and (3.1-3.2) solve the
equation:
lim (G" + Ag"")((Q.,,
-
RaV,")= 0.
(4.7)
r-* oc
This corroborates the Henneaux-Teitelboim recipe for finding asymptotic symmetries
for any given set of boundary conditions according to the procedure of [18].
Chapter 5
Small asymptotic transformations
In this chapter we obtain Virasoro generating vectors ( by requiring "small" asymptotic transformation of exact AdS 3. In particular, "smallness" is quantified as follows:
" Subleading Lie derivatives (along () of AdS 3
" Finite first order effective action S(')1
2
" Finite second order effective action S )][
Throughout, we assume power series expansion of the components of ( as follows:
=
Z (t,4)r".
(5.1)
n
We view the above as an expansion in 1/r (i.e. expansion around r = o)
and we
assume that each series truncates for some large N onwards (N may be different for
each component).
5.1
Subleading Lie derivatives of AdS 3
We first require that Eggy, are "subleading" to the AdS 3 components g,, in (2.23),
in the same sense that the particular Brown-Henneaux deviations in (2.24) are subleading. That is, we require the deviations of nonzero metric components of AdS 3
(2.23) to be subleading to AdS 3 :
,Cggtt = O(r),
Lgrr = 0(
),
g
=
(5.2)
O(r) ,
and the rest to remain finite:
Eggt, = 0(1),
Eggt4 = 0(1) ,
(5.3)
-gr4 = 0(1).
Using the expansion (5.1) the above Lie derivative conditions are equivalent to:
( _1 +l2
12 (n +
t
,t +
1) r+1 + (n - 2);_1j = 0 ,
14 r(n+ 1)Qt+1
-
l2(n-
1)4 _1-
(5.6)
(Ert + 212 (n - 1)(t_1 + (n - 3)Qt-3 = 0,
1
0,
-
22
(5.5)
n>2
n>2
=0,
n-1 + n-2P
(5.4)
n > 2
= 0,
(n-
3)-
3
n > 3
(5.8)
n > 1
+l2
(5.7)
0,
(5.9)
n >3
Consider equation (5.5) and recall that we assume the series for (r truncates (i.e.
(n = 0 for n > N). Using backwards induction starting with a large enough even and a
large enough odd n, we obtain from (5.5) that (Em+1 = 0, m > 1 and
respectively. Thus:
1
= r(t, #)r + (G(t, #) + 0(-).
m
= 0, m > 1,
(5.10)
r
Then in equations (5.7) and (5.9) the (n terms drop out altogether and using the
same arguments as before we derive from (5.7) and (5.9):
dm0,m>1
2m+i = 0 , m > 0 and
and, d'm+1 = 0, m > 0 and m = 0, m > 1, respectively. Thus:
t=
(G(t, #) + (t_ 1 (t, #)-
(4=
f t,#)+
11
+ O(T),
r
r
1
1
.(512
*1t,#)+ ~g
(5.11)
The general form of the vector ( as in (5.10-5.12) exhausts equations (5.5, 5.7, 5.9).
However, from equations (5.4, 5.6, 5.8) we also get the additional relations (n = 2 in
(5.4), n = 2 in (5.6), and n = 1,2 in (5.8)):
+ (0,, = 0
(5.13)
=0
(5.14)
+
2
With ([(t,q#)
1,0 =
-
2(,
-
0
(5.15)
, = 0
(5.16)
(t, #), equations (5.13,
R(t, #), (t(t, #) =lT(t,#), and ((t, #)
5.14, 5.16) are equivalent to:
lT,t(t, #) =
-R(t, #) ,
,,(t,#)
(5.17)
(5.18)
1l@,t (t, #) = T,(t, #) .
The above, which were also derived by Brown-Henneaux [1], imply that T and D
satisfy the conformal Killing equations in two dimensions with an indefinite metric
and that R is fully determined once we solve the conformal equations as follows:
T(t, #) = T+(t/l + #) + T-(t/l - #) ,
(5.19)
0(t, $) = T+(t/l + #) - T -(t /l - #0),
(5.20)
where T+ , T- are arbitrary functions of a single variable. Therefore, we find that the
most general ('s which give subleading perturbations of AdS 3 are given by:
=
l(T+ + T~) + (t1(t,
#)- + O( ),
r
r
T+ - T- +
rwith,
1(t,
+0
#
r
(T+' + T-') r + 0(1) ,
l2$j,
-
=
0.
1() ,
r2
(5.21)
(5.22)
(5.23)
(5.24)
Finite first order effective action S(
5.2
The first order effective action S)
[
1
)[
]
] (4.4) is a boundary term: an integral over the
boundary at r = oc. We require this to be finite in the sense that the integrand
is finite everywhere on the boundary. That is, we are not going to be concerned
with divergences due to infinite time range or any possible poles in the integrand
expression.
Plugging the expansion (5.1) into the integrand expression in (4.4) we find that
S)
[ ] is
finite if and only if:
(n - 3) (
+ 212 (n - 2) (
n-5, + ($_5,O)
-+1 4 (n +
nt-3
+ (2-3,0) +1 l(n
_4 + l4 'r-2, tt + 412-
_+
-
- 1(-
-
i$
= ,
m >4
For the general form of the vector ( as in (5.10-5.12) the above impose the following
conditions:
n=4:
n =t j
5t
+01
2([ = 0,
+(
1
so that the most general ('s of the form (5.10-5.12) which leave S( )[ ] finite are:
(
i(t,
=
(4=((t, 1
-
4)+/ c
~ (ft(t,
1 (t,
+
1
1++ O( 1)
r2
r
1)
111+0
0
0) T))
(5.25)
(5.26)
1),
r2
r -
(r
-
1
(t, 4) + (-1(t,
4)) + 0( 1) . (5.27)
Compare (5.25-5.27) with the most general ('s which yield subleading perturbations of AdS 3 given in (5.21-5.24). We first note that the two sets are not mutually
exclusive and the leading order terms in (5.21-5.24) are related precisely as needed
in (5.25-5.27). However, at second to leading order the arbitrary 0(1) piece in (r in
(5.23) needs to be taken as required in (5.27). In the end, the most general 's which
1
yield subleading perturbations of AdS 3 and leave S( )[ ] finite are given by:
t
=
l(T+ + T~) +
(T+' + T')
r
12 s 17t
with,
5.3
-
_1 (t, 4)1 + 0(1) ,
r
r
,+0(1),
-
rr
=
(5.28)
r
(5.30)
(5.31)
0.
Finite second order effective action S( 2 )
Since AdS 3 is an exact solution of EFE, the second order effective action S(2)[(] (4.5)
is also a boundary term (i.e. an integral over the boundary at r = oc). We require
S(2) [] to be finite too in the sense that the integrand is finite everywhere on the
boundary.
Plugging in (4.5) vectors ( of the form (5.28-5.30) we find that the integrand is
given by
((t1)2 _
l2(
i)2
r2/16 + O(r), which implies the additional condition:
1=
With
4ls1
1=
(5.32)
i
1i, and using (5.31) to simplify, the integrand in (4.5) becomes:
T+ + 813 $ OT+' + 121l3
while with *_ = -l 1
OT~' - 4l3
1
T+" + 81(-2
-
8l,1 2) r/1 5 +0(1) ,
and using (5.31) to simplify, the integrand in (4.5) becomes:
+ 8kg 1
±2
Note that in the above we have also involved the so far arbitrary (62 and
2.
(413A
T- - 81 3 $j
T-' - 1213$j0T+'
1
-
41
T -"it + 8T-1
2
r/ 15 +0(1).
These,
as part of the 0(1/r 2 ) terms, are allowed to be arbitrary in (5.28-5.29) and if we
would like to maintain their arbitrariness, then in order to leave S(2) [ ] finite we need
to take (01 = 0 and thus, in view of (5.32), ($1 = 0 too. So for the 's in (5.28-5.31)
to leave S(2 ) [ ] finite they need to actually read:
1(T++ T- )+0(-2),
r2
(+=
(=
r1
T+-T~+O(4),
(5.33)
-T+'+T-')r+O(-r).
With the above 's both S(1 )[6] and S(2 ) [6] remain finite as well as perturbations of
AdS 3 are subleading. Hence the ('s in (5.33) are our final vector fields for small
asymptotic transformations of AdS 3 . Clearly they obey the Virasoro (2.27).
It is important to stress here that the vectors (5.33) were not derived as asymptotic symmetries of asymptotically AdS 3 space-times defined by any set of boundary conditions (recall that in equations (5.2-5.3) the components of g,,, were just
those of exact AdS 3 ). We will see in Chapter 7 though that, as a bonus, they actually are asymptotic symmetries of appropriately defined new asymptotically AdS 3
space-times. Here, however, we need not confine ourselves to considering any asymptotic symmetry group. Instead, we have derived the Virasoro from demanding small
asymptotic transformation of AdS 3 and we may proceed as in Section 2.3 to the corresponding canonical generators to find the same central charge c = 31/(2G) at the
Dirac bracket level and in the end the same quantum Virasoro of a two-dimensional
conformal field theory.
Chapter 6
Asymptotic conformal Killing
vectors
In the previous chapter we showed that the Virasoro manifests itself in the asymptotic
transformations of AdS 3 quite independently of any particular choice of boundary conditions. That is, the asymptotics of AdS 3 are governed by the conformal group in two
dimensions independently of any ASG. This motivates considering directly asymptotic conformal Killing vectors of AdS 3 . In this chapter we show that asymptotic
conformal Killing vectors of AdS 3 which leave the first order effective action finite,
also end up taking the form (5.33).
Assume the vector field (,
" satisfies the conformal Killing equation for AdS3 in the asymptotic limit, and
" maintains a finite first order effective action S(')[p).
Throughout, we use the power series expansion (5.1).
By requiring that ( satisfies the conformal Killing equation for AdS 3 ,
2
V,(v + Vv~,i = 2givV"',
(6.1)
in the limit r -+ o, we obtain the following relations:
212i
1
4
+ 2'
3,t
-
'(n + 1)(; + (4 - n) n-2
+ 212 (n - 1)(t_1 + (n - 3)Qt-3
(n + 1)(±nt
l2$,0 + (n24
-
12(1 - 2l2n)(
+ 2(22
12
o + 12-(n
-
-
1
1
-
3
=
(6.2)
0, n > 1
(6.3)
( ,t= 0, n> 2
(6.4)
> 0
0 ,n
__-2,t
ln$-
-
n + 2) -2 +
,t + (n-3,t + l2 n
l
1
, +
-3,$
=
0,
2
(6.6)
- 1)o_1 + (n- 3)e-3=0, n>2
212_ g - (
-
(6.7)
n > -1
(n - 2)(; = 0,
Now, starting from equation (4.2) and using the conformal Killing equation (6.1)
on the boundary integral (where r = oc), we may simplify S(')[ ] for AdS 3 to the
following:
S
By requiring that
[)
-
3
j
dx
/X
n"VV,".
(6.8)
BM
leaves the above finite (in the sense that the integrand is finite
everywhere on the boundary r = oc), we obtain the relation:
(n + 1)n + Qt_1,t + (no-1,+ = 0, n = 0, 2,3,...
(6.9)
Using (6.9), equation (6.7) is equivalent to,
-
n1,I
2(24
2
-
-
n1,t = 0,
-2,t + 3(_
1 =
202,
3, ... ,
0,
1,+
=0,
(6.10)
(6.11)
(6.12)
and in view of (6.10), equation (6.9) becomes:
n
=0 ,
5 (6.5)
= 0,2,3, ... ,
(6.13)
which in turn simplifies (6.10) to:
n
+
1
,=0,
(6.14)
n = 0, 2,3,....
So equations (6.7, 6.9) are equivalent to (6.11-6.12, 6.13-6.14). Now, using (6.136.14), equation (6.2) is equivalent. to:
1 2 n + (n - 3)(n- 2 = 0, n =2,4,5,...
2l2$,e + 2(*2,t - 2l2([ + 3'_ 1
212($,t + 24 - 412r +
-
1
=
(6.16)
0,
(6.17)
O = 0.
2
We solve (6.15) using backwards induction (2m+1
1(t #)r + ('
-
-
(6.15)
,
=
0, m> 1 and dm
=
0 m> 0):
(6.18)
(t, 4)- + 0( r).
Then in equations (6.3) and (6.6) the (n terms drop out altogether and we solve the
resulting equations via backwards induction ((2m+1 = 0, m > -1 and
1, and similarly, dm+1 =0, m> -1 and
2tm =
0, m >
m =0,m
1):
t= ((t,#)+(L 2 (t, ) 1 +
1
(-),
(6.19)
) + " 2 (t, #) + O(-).
(6.20)
S=((t,
Note that the general form of the vector ( as in (6.18-6.20) exhausts equations (6.136.14) too. Thus so far, the vector ( as in (6.18-6.20) with (6.11-6.12, 6.16-6.17)
exhausts (6.2-6.3, 6.6-6.7, 6.9). Finally, equation (6.5) is also automatically satisfied
by the vector
(6.18-6.20) and equation (6.4) imposes only the additional relations:
2
+
- 12
(-24
-
= 0.
2,t
=
0,
(6.21)
(6.22)
So the most general ('s which satisfy the conformal Killing equation for AdS 3 (6.1) in
the limit r -+ oc while also maintaining a finite first order effective action S 1 )][( (4.4),
are given by equations (6.18-6.20) with the additional relations (6.11-6.12, 6.16-6.17,
6.21-6.22).
Note that. (6.12, 6.17) * (5.13, 5.14)1 whereas (6.22) is just (5.16), and recall from
Section 5.1 that (5.13, 5.14, 5.16) are equivalent to the Brown-Henneaux equations
(5.17-5.18) which are solved using the arbitrary functions T+(t/l +
#), T-(t/l
-
#).
Thus in the end, the most general ('s satisfying the conformal Killing equation for
AdS 3 (6.1) in the limit r -+ oc while also maintaining a finite first order effective
action S 0 )[ ] (4.4), are given by:
l(T+ + T~) + (
2 (t,
= T+-T + 2 (t,
=
with (* 2 ,
2
, (i
-
#)i + 0(1),
)
+(
),
(6.24)
(T+' + T-') r + (' 1(t, #) + 0(
),
(6.25)
satisfying the remaining equations (6.11, 6.16, 6.21),2
12
13
T+' + T-'
(6.26)
0,
(-2,t + 3'1
+&2,t-24
(T+' - T~') -
=
-
l2
Since the equations (6.26-6.28) may be solved for (*-2 , (
constraints on T+ and T(5.33).
(6.23)
,3
,
(6.27)
2,t=
0.
(6.28)
2,
i
without imposing any
we find that our final ('s (6.23-6.28) are again of the form
Hence they form the Virasoro at leading order in 1/r as usual and lead to
the same conclusions as in Chapter 5. Finally, recall from Chapter 5 that the second
order effective action S(2 )[3 (4.5) is also finite for all ('s of the form (5.33).
'In view of (6.18-6.20) equation (6.17) simplifies to: 2,
+
Equation (6.16) simplifies, in view of (6.11, 5.13-5.14), to:
2
3
For example:
-2 = 0
52 =
(T+ - T-),
_1=
-l2(T+'+T-').
( l2
0.
_-2,t
+
-2=
0
Chapter 7
New asymptotically AdS 3 spaces
Having established status for the vectors (5.33) using the effective action approach
in the previous chapters let us here attempt to embed them back into the context
of an asymptotic symmetry group. Define new asymptotically AdS 3 space-times by
perturbing the exact AdS 3 (2.23) using our ('s (5.33):
+ 0(1)
9tt = 1
gtr
=
gto =
0(-
r
0(1),
(7.1)
12 +01)
+0
grr
(
),
1
g
=r
2
+0(1).
Note that these are relaxed compared to both Brown-Henneaux (2.24) and our (3.1).
It is easy to show that the asymptotic symmetries of (7.1) are actually all given by
our ('s (5.33). Thus, provided the corresponding charges are still well defined, the
ASG of AdS 3 corresponding to the boundary conditions (7.1) is again the conformal
group in two dimensions. Indeed, one finds that the charges
Q[
] corresponding to
(5.33, 7.1) are again finite. The central charge of the Virasoro at the Dirac bracket
level is again c = 3l/(2G) and it may be used to calculate the BTZ entropy as usual.
42
Chapter 8
Conclusion and Discussion
In this thesis we demonstrated the emergence of the Virasoro algebra in the asymptotics of AdS3 using the effective action of GR for small excitations from asymptotic
transformations.
To date the Virasoro associated with the boundary of AdS 3 was known to arise
in the context of an asymptotic symmetry group (ASG) corresponding to certain
boundary conditions imposed by Brown and Henneaux. Although these boundary
conditions play a key role in obtaining the Virasoro a la Brown-Henneaux, they are
not entirely dictated by the theory. We have bridged this gap by deriving boundary
conditions from finiteness constraints on the effective action(s) of GR calculated in
this thesis. The boundary conditions so derived are relaxed compared to BrownHenneaux but the corresponding ASG and its central charge are unaltered: it's again
the conformal group in two dimensions with c = 31/(2G).
However, we need not confine ourselves in using boundary conditions and an ASG
in the first place. As emphasized throughout this thesis, the emergence of the Virasoro
in the asymptotics of AdS 3 is independent of any ASG: our derivations of the algebra
are well-grounded by themselves on the physical requirement of finite action.
We have seen in Section 2.3 that the two-dimensional conformal symmetry in
the boundary of AdS 3 is relevant for calculating entropy of the BTZ black hole. We
have also mentioned in the introduction that for appropriate boundary conditions the
ASG of the near horizon metric of extremal Kerr (NHEK) also contains the conformal
group with a central charge that gives entropy of extremal Kerr. Indeed, one may
apply our effective action approach for this case too and we shall report the results
elsewhere soon.
Two-dimensional conformal symmetry is ubiquitous in the physics of realistic Kerr
black holes as well. Strominger and collaborators have recently obtained scattering
amplitudes by near-extremal Kerr from certain correlators of two-dimensional conformal field theories in [20, 21]. Moreover, a new hidden two-dimensional conformal
symmetry in the low frequency scalar wave equation of generic Kerr has also been
found very recently in [22]. These papers signal a duality between Kerr black holes
and two-dimensional conformal field theories in a way though that cannot be captured by an ASG. As a future direction exploring the ideas in this thesis, it may be
possible to obtain a unifying realization of all conformal symmetry associated with
Kerr using small action and energy constraints on relevant transformations in Kerr.
Bibliography
[1] J. D. Brown and M. Henneaux, "Central Charges in the Canonical Realization of
Asymptotic Symmetries: An Example from Three-Dimensional Gravity," Commun. Math. Phys. 104, 207 (1986).
[2] J. M. Maldacena, "The large N limit of superconformal field theories and supergravity," Adv. Theor. Math. Phys. 2, 231 (1998) [Int. J. Theor. Phys. 38, 1113
(1999)] [arXiv:hep-th/9711200].
field
[3] 0. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, "Large N
theories, string theory and gravity," Phys. Rept. 323, 183 (2000) [arXiv:hepth/9905111].
[4] A. Strominger, "Black hole entropy from near-horizon microstates," JHEP 9802,
009 (1998) [arXiv:hep-th/9712251].
[5] M. Banados, C. Teitelboim and J. Zanelli, "The Black hole in three-dimensional
space-time," Phys. Rev. Lett. 69, 1849 (1992) [arXiv:hep-th/9204099].
[6] M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, "Geometry of the (2+1)
black hole," Phys. Rev. D 48, 1506 (1993) [arXiv:gr-qc/9302012].
[7] M. Guica, T. Hartman, W. Song and A. Strominger, "The Kerr/CFT Correspondence," arXiv:0809.4266 [hep-th].
[8] T. Hartman, K. Murata, T. Nishioka and A. Strominger, "CFT Duals for Extreme Black Holes," JHEP 0904, 019 (2009) [arXiv:0811.4393 [hep-th]].
[9] J. M. Bardeen and G. T. Horowitz, "The extreme Kerr throat geometry: A
vacuum analog of AdS(2) x S(2)," Phys. Rev. D 60, 104030 (1999) [arXiv:hepth/9905099].
[10] A. P. Porfyriadis and F. Wilczek, "Boundary Conditions, Effective Action, and
Virasoro Algebra for AdS 3 ," in preparation.
[11] P. A. M. Dirac, "The Theory of gravitation in Hamiltonian form," Proc. Roy.
Soc. Lond. A 246, 333 (1958).
[12] R. L. Arnowitt, S. Deser and C. W. Misner, "The dynamics of general relativity,"
arXiv:gr-qc/0405109.
[13] T. Regge and C. Teitelboim, "Role Of Surface Integrals In The Hamiltonian
Formulation Of General Relativity," Annals Phys. 88, 286 (1974).
[14] G. Barnich and F. Brandt, "Covariant theory of asymptotic symmetries, conservation laws and central charges," Nucl. Phys. B 633, 3 (2002) [arXiv:hepth/0111246].
[15] G. Barnich and G. Compere, "Surface charge algebra in gauge theories and thermodynamic integrability," J. Math. Phys. 49, 042901 (2008) [arXiv:0708.2378
[gr-qc]].
[16] P. A. M. Dirac, "Generalized Hamiltonian dynamics," Can. J. Math. 2, 129
(1950).
[17] P. A. M. Dirac, "Generalized Hamiltonian dynamics," Proc. Roy. Soc. Lond. A
246, 326 (1958).
[18] M. Henneaux and C. Teitelboim, "Asymptotically Anti-De Sitter Spaces," Commun. Math. Phys. 98, 391 (1985).
[19] R. M. Wald, "General Relativity," Chicago, USA: Univ. Pr. (1984) 491p
[20] I. Bredberg, T. Hartman, W. Song and A. Strominger, "Black Hole Superradiance From Kerr/CFT," JHEP 1004, 019 (2010) [arXiv:0907.3477 [hep-th]].
[21] T. Hartman,
W. Song and A. Strominger,
"Holographic Derivation of
Kerr-Newman Scattering Amplitudes for General JHEP 1003, 118 (2010)
[arXiv:0908.3909 [hep-th]].
[22] A. Castro, A. Maloney and A. Strominger, "Hidden Conformal Symmetry of the
Kerr Black Hole," arXiv:1004.0996 [hep-th].