Boundary conditions, effective action, and Virasoro algebra for AdS₃

Boundary Conditions, Effective Action, and

Virasoro Algebra for AdS

3

by

Achilleas P. Porfyriadis

Submitted to the Department of Physics

in partial fulfillment of the requirements for the degree of

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

AUG 13 2010

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Bachelor of Science in Physics

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June 2010

@

Achilleas P. Porfyriadis, MMX. All rights reserved.

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Author...__

_ _ _ _

Department of Physics

May 7, 2010

Certified , ...

Herman Feshbach

Frank Wilczek

Professor of Physics

Thesis Supervisor

Accepted by...

David E. Pritchard

Senior Thesis Coordinator, Department of Physics

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 bound-ary of AdS3. 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 AdS3. Our Virasoro generating

transforma-tions are asymptotic symmetries of appropriately defined new asymptotically AdS3

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 bound-ary 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

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.

Contents

1 Introduction and Outline 9

2 Brown-Henneaux-Strominger 13

2.1 Hamiltonian formulation of GR ... ... 13

2.2 Asymptotically AdS3 spaces and Virasoso .... ... 18

2.3 Central charge and entropy calculation ... 20

3 Relaxing the boundary conditions 23

4 Effective action for GR 27

5 Small asymptotic transformations 31

5.1 Subleading Lie derivatives of AdS3 . . . . 31

5.2 Finite first order effective action S')[

1

_{] . . . .}

₃₄

5.3 Finite second order effective action S()[

. . . .

35

6 Asymptotic conformal Killing vectors 37

7 New asymptotically AdS3 spaces 41

Chapter 1

Introduction and Outline

In 1986, Brown and Henneaux [1] studied asymptotic symmetries of three-dimensional Anti-de-Sitter space (AdS3) and found they form two copies of the Virasoro algebra

with central charge c = 31/(2G), where

l

is the AdS3 radius and G is the three

di-mensional Newton constant. This implies that any consistent quantum theory of gravity on AdS3 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 Bekenstein-Hawking 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 some-what 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 AdS3 spaces with

re-laxed 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 AdS3 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 Brown-Henneaux-Strominger methods for obtaining the asymptotic symmetry group (ASG) of AdS3, 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 AdS3 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 ex-citations from asymptotic transformations. Here we also derive the corresponding equation of motion which is an extremely beautiful equation that is still under in-vestigation. In the next chapter, we derive Virasoro generating vectors by requiring "small" asymptotic transformation of exact AdS3 with finite effective actions. Hence

the Virasoro manifests itself in the asymptotics of AdS3 even without appropriately

chosen boundary conditions and an ASG. With the conformal group prominently settled in the asymptotics of AdS3 we decide to study, in Chapter 6, asymptotic

con-formal Killing vectors of AdS3, 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 AdS3 we define, in Chapter 7, new

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 2

Brown-Henneaux-Strominger

In this chapter we review the canonical formulation of GR, derivation of the Brown-Henneaux 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

g

(R

-

2A),

(2.1)

where our signature is (-

+

+ - ) 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:

ds2 _{= -N} 2

dt2 + 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":

-1 N'

_ N2 N2 (2.3)

NJ ij _N'N-7

Using the ADM decomposition and utilizing Gauss-Codazzi equations, the Einstein-Hilbert action (2.1) may be written in the form:

S = dtI dn-1x N ( (n-)R - 2A + KiK' - K2_{) + boundary terms. (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

Kj = I (atgij - ViNj - VjNi), (2.5)

where Vi is the covariant derivative with respect to gij. Note that in (2.4) time derivatives occur only through the Btggj in Kj, so the canonical momenta are:

7r/ 0- -=r

fy(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,

K'j= 7ra - 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

d-IX (7rir

3

_g

_-

_{NH -}

_NH),

_(2.8)

where,

= ( y - -j(n-)R -(2A), (2.9)

R- = -2/7Vj 7r . (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- (NN + NiHj). (2.11)

Following Dirac's recipe for constraint Hamiltonian dynamics we form the total Hamil-tonian, HT = H +

f.

d"~x APIr", and demanding the primary constraints to be

pre-served in time, irY = 0 e {rIF, HT} = 0, we obtain the secondary constraints:

H-, =0 ,

(2.12)

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

(

.G (2.13)

1g5 j 6,su

ogig ogru

15

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 Hamil-tonian 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:

JH

=

d

n-1

x

(A'2ogij

+

BijJw,'j). (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

6 H 6 H

gij =- , -'r 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 AdS3 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:

6

H

=

dlx

(Az'6gij + Bi6ris})

d-

j

j-2 si Gijki (Nogijj;k - Nk Sgij) (2.16)

-

j

di- 2_{s [2Nk 6rkl + (2Nkffil - N1 rjk)6ggjk]}

where the semicolon denotes covariant differentiation with respect to gij, and 1

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).

The precise expressions of A' and Bj in (2.16) are not needed for our purposes; however, they give the Hamiltonian equations of motion for GR which we list for completeness:

aj

= 2Ng- 1/2 r-n i) + 2 V(iNj), (2.18)

i = 1Ng1/2ij (r kl _ 1 ,2) - 2Ng1/2 (,ik~j - n Wi2

2 n -2 n- 2

-Ng 1 2 (n-1)Rj - (n- R - 2A) gi) + g1/2('ViN - gd3 [IN)

+g1/2 Vk (-1/ 2N kii) - 21k(iVkNi). (2.19)

We conclude with the general theory of canonical generators of diffeomorphisms. Consider a space-time vector ( which generates infinitesimal diffeomorphisms via the

Lie derivative: Jg9 , = Egg,,. In the Hamiltonian formulation of GR, the

correspond-ing canonical generator which generates the same transformation via the Poisson bracket, 6gij = {gij, H[ ]} PB., is given by:

H [ ] = - 1 X edR1 , +

Q[].

(2.20)

Here, the canonical or surface deformation vector (e is related to the space-time diffeomorphism vector via

S= N$O , = + NWgt (2.21)

and

Q[

] 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

brackets, i.e.:

Q

d-2_s

1 Gijkl

(

6gij;k - 'k ogij)

+ dn-2_{s i}

_[2k

_-1-r'+

₍₂

_{k 11 - .i}

7ik) Jgjk] . (2.22)

The Q's are often called "charges" of the associated transformations because

Q[

) 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

=

Q[&t].

Similarly,

Q[8o]

gives the angular momentum.

2.2

Asymptotically AdS

3

spaces and Virasoso

The AdS3 metric is given by:

ds

2 _-

_(i+

_dt

2 ₊

_(i+

_dr

2

_+r2

2_.23)

₍

Asymptotically AdS3 space-times are defined as ones which behave similarly to AdS3

in the limit r -> o0 with certain falloff conditions. The falloff conditions at infinity are called boundary conditions. For example, Brown-Henneaux chose:

Ir2 gtt = p + 0(1), 1 gtr = 0(7), g = (1), (2.24) 12 1 grr = 7+(-s), goo= r2+ 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

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/r3), £ggto = 0(1), etc.

Studying these Lie transformation equations, one finds the most general solution may be nicely written as:

t l(T++T~)+ TI+"+T-") + 0()

= -r (T+' + T +0(1), (2.25)

(

~ ~

~

1 =r+-T + -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,

122 inr(2.26)

(i e"' B+ 2r2 2 ,(.6

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 sub-leading 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

Brown-Henneaux: 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 AdS3. Then,

evaluating (2.30) at t = 0 on AdS3,

(2.31)

K[E,,

a]

=

{Q[a], Q[ ]}D.B.

= Q[ .

-Using (2.22), the right hand side of the above is given by

27r

lim dp {Gijkr [(Fm)o gij-k - ($m) ,k

oggi

+2

k -F r+

[2 ( )I-

-,j ( ±)r _Rk] j gk} 2.2

which upon evaluation at I = 0 on AdS3 yields:

K[gm, ] = 2i47rin(n2 - )5m+n,O. (2.33)

With L =

Q[±j,

and passing from the Dirac 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 L

+2r

L (2.35)

From (2.34) c+ = c_ = (after restoring G). Also, since the global charges of the

BTZ black hole are M = Q[8t] and J =

Q[ao]

we quickly obtain,

M= (L+ + L-), J= L+ - L-₀ 0~ (2.36)

Solving for L+ and L- and plugging everything into the Cardy formula (2.35) repro-duces (microscopically) the Bekenstein-Hawking entropy of the BTZ black hole:

l(lM

+

J) + l(lM - J)

S=r/ _{2G V}

Chapter 3

Relaxing the boundary conditions

From the review of the Brown-Henneaux recipe in the previous chapter, one is nat-urally 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 con-vincing 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 consis-tency 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 AdS3.

One way to fulfill requirements (i) and (ii) is to start with the BTZ metric and act on it with the AdS3 isometry group in all possible ways. This procedure generates

metric perturbations which behave asymptotically as in (2.24). Interestingly, how-ever, 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 in-teresting one themselves by removing a "wedge" from AdS3 and introducing a "jump"

via appropriate identifications of points) which also approached AdS3 in the

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 con-sistency, 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 asymp-totic Killing vectors of the new asympasymp-totically AdS3 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 AdS3 space-times. Here is an interesting set of such

alternative boundary conditions (relaxing the tr and r# components):

9tt = ( , 31

1

gt, = 0( r),

goo = r2+ O(1).

Studying the Lie transformation equations corresponding to this choice of boundary conditions one finds the most general asymptotic Killing vectors are given by:

t = I(T++T-)+ (T+11+ T-") + 0()

r = -r + T + 0(,) ) 1T+1 (3.2)

.9rr r

12

T+ -- T- - T+2 - T -"

These are just relaxed versions of the vectors in (2.25). The consistency requirements about including the AdS3 isometries and the BTZ black hole are automatically met

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 the terms involving T+" and T-" in (2.25) or (3.2) are not important.1 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.

1_{Note, however, that in (3.2) the arbitrary O(1/r} 3

) terms spoil the Virasoro at higher order in 1/r compared to (2.25).

Chapter 4

Effective action for GR

Our goal in this chapter is to construct (to second order) the effective action of GR for small excitations g,, -+ ,+Eg and derive the corresponding equation of motion.

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

-

SC

0

)[h] + S(

1)[h] +

S(2)[h] + 0(h

3

)

,

(4.1)

where, SC0)[h] -

IM

_{-g (R - 2A),}

SM)[h]

- -

J

dnx

v-g (G" +

Ag"v)h,, + d"~1

x

v/

~n(V"hp,

-

VPh)

M faM S(2) [h] - d x 1 g hv [g"VgPc" - gPg"El + 2g"V"V" - 2gV"V V"

+ d- 1x A/-

yn"

(2huVVh + 2hVvht, + 3hP"Vuh,

-4h[IVch""

-

2h"V

h,, -

hV1h).

Here indices are raised and lowered using the background metric g, h" - guPgvohp,

Putting hy, = Vt

+V

we get an effective action for (. The first order action for ( reads:

S M -2 d'x /- (G1" + A9gI)V,,v

+

J

d- 1x -

yn/(D,

+ V"V - 2V/,V,"). (4.2)

Upon integrating by parts the bulk piece above and using the contracted Bianchi identity,

VGLV = 0 , (4.3)

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, + Vj, + V1 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

[]

IM

dx

y! -g

(GI" + Ag"") (V,"

V

+ Rg,(*(" )

JM d"-1x -\ yn" {JVVVaV , - 'VV"Vvg, + RzvpYPVo(v (4.5)

1

+(V,( + Vug,)(1 (V

-

VVV~J) + IV,

[(Vv()

2

_

+2(V"V")(VVcr( - VpVcifv) + "Vo [ ,(Glv + Agv) - j,(Gv, + Agzcy)] .

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,,,} -* hy + V,(, + V,(, and so plugging h, = 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, ) = 0. (4.6)

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

(a;, -- Rc, = 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, -+ gy,,

+

V,(, + V, is not really a transformation and indeed

SM)[(] - 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 AdS3 space-times (2.24), (3.1) and their corresponding asymptotic

'For an exact Killing vector S(1)[ ] is easily seen to vanish in equation (4.2). For S(2)[ ] note that, up to a boundary term, the bulk piece in (4.5) may be written as - fM d'x -g (GI" +Agl")((

Rcaglv()(o in which case one finds that the resulting total boundary piece in S(2)

[

] 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" asymp-totic transformation of exact AdS3. In particular, "smallness" is quantified as follows:

" Subleading Lie derivatives (along () of AdS3

" Finite first order effective action

S(')1

" Finite second order effective action S [2)]

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 AdS3 components g,, in (2.23),

in the same sense that the particular Brown-Henneaux deviations in (2.24) are sub-leading. That is, we require the deviations of nonzero metric components of AdS3

(2.23) to be subleading to AdS3:

,Cggtt

=

O(r),

Lgrr = 0(

)

,

g

=

O(r) ,

(5.2)

and the rest to remain finite:

Eggt, = 0(1), Eggt4 = 0(1) , -gr4 = 0(1). (5.3)

Using the expansion (5.1) the above Lie derivative conditions are equivalent to:

( _1 +l2 ,t + t = 0, n > 2 (5.4) 12 (n + 1) r+1 + (n - 2);_1j = 0 , n>2 (5.5) n-1 + n-2P

=0,

n>2

(5.6) 14 r(n + 1)Qt+1 - 1 (Ert + 212 (n - 1)(t_1 + (n - 3)Qt-3 = 0, n > 3 (5.7) 22 - 0, n > 1 (5.8) l2(n- 1)4 _1- (n- 3)- 3 +l2 0,

n >

3 (5.9)

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 m = 0, m > 1,

respectively. Thus:

1

= _{r(t, #)r + (G(t, #) + 0(-). (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): 2m+i = 0 , m > 0 and dm0,m>1 and, d'm+1 = 0, m > 0 and m = 0, m > 1, respectively. Thus:

11

t= (G(t, #) + (t_1(t, #)- + O(T), _(5.11)

r r

1 1

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(,

-

₁

,0

=

0

(5.15)

2 - , = 0 (5.16)

With ([(t,q#) R(t, #), (t(t, #) =lT(t, #), and ((t, #) (t, #), equations (5.13, 5.14, 5.16) are equivalent to:

lT,t(t, #) = ,,(t, #) -R(t, #) , (5.17)

1l@,t (t, #) = T,(t, #) . (5.18)

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 AdS3 are given by:

= l(T+ + T~) + _(t 1(t, #)- + O( ), (5.21) r r T+ - T- + 1(t, # +0 1() , (5.22) r r2 r- (T+' + T-') r + 0(1) , (5.23) with, l2$j, - = 0. (5.24)

5.2

Finite first order effective action

S(

1

)[

]

The first order effective action S)

[

] (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) ( n-5, + ($_5,O) + 212 (n - 2) ( nt-3 + (2-3,0) +1 l(n - 1(-+14

(n +

-- _+ _4 + l4_'r-2,

tt + 412- - -

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

5t j +( +01 2([ = 0,

so that the most general ('s of the form (5.10-5.12) which leave S(1)[ ] finite are:

1 ( = i(t, 1+ + O( 1) (5.25) r r2 / 111 (4 =((t, 1 4)+ c 1 (t, 1) +0 1 ) , (5.26) (r r2 -

~

(ft(t, + 0) T)) 0 r - - 1 (t, 4) + (-1(t, 4)) + 0( 1) .(5.27)

Compare (5.25-5.27) with the most general ('s which yield subleading perturba-tions of AdS3 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

yield subleading perturbations of AdS3 and leave S(1)[ ] finite are given by: t = l(T+ + T~) + _1(t,

4)1

+ 0(1) , (5.28) r r r (T+' + T') r - ,+0(1), (5.30) rr with, 12

_s

-17t = 0 . (5.31)

5.3

Finite second order effective action

S(

2

)

Since AdS3 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

((t

1)2 _ l2( i)2 r2/16 + O(r), which implies the additional condition:

1= i (5.32)

With 1 = 1i, and using (5.31) to simplify, the integrand in (4.5) becomes:

4ls1 T+ + 813$ OT+' + 121l3 O T~' - 4l3 1T+" + 81(-2 - 8l,1 2) r/15+0(1) ,

while with *_ = -l 1 and using (5.31) to simplify, the integrand in (4.5) becomes:

(413A T- - 813$j T-' - 1213$j0T+' 1 - 41 T -" + 8T-

it

_{1 2} +

8kg

₁

±

2 r/ 1

5_+0(1).

Note that in the above we have also involved the so far arbitrary (62 and 2. These,

as part of the 0(1/r2) 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}

AdS3 are subleading. Hence the ('s in (5.33) are our final vector fields for small

asymptotic transformations of AdS3. Clearly they obey the Virasoro (2.27).

It is important to stress here that the vectors (5.33) were not derived as asymp-totic symmetries of asympasymp-totically AdS3 space-times defined by any set of

bound-ary conditions (recall that in equations (5.2-5.3) the components of g,,, were just those of exact AdS3). We will see in Chapter 7 though that, as a bonus, they

ac-tually are asymptotic symmetries of appropriately defined new asymptotically AdS3

space-times. Here, however, we need not confine ourselves to considering any asymp-totic symmetry group. Instead, we have derived the Virasoro from demanding small asymptotic transformation of AdS3 and we may proceed as in Section 2.3 to the

cor-responding 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 AdS3 quite independently of any particular choice of boundary

con-ditions. That is, the asymptotics of AdS3 are governed by the conformal group in two

dimensions independently of any ASG. This motivates considering directly asymp-totic conformal Killing vectors of AdS3. In this chapter we show that asymptotic

conformal Killing vectors of AdS3 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 AdS3,

2

in the limit r -+

o,

we obtain the following relations: 212i

+

2' 3,t - '(n + 1)(; + (4 - n) n-2 - ln$- ₁ - 3 = 0, n > 1 (6.2) 14(n + 1)(±nt + 212 (n - 1)(t_1 + (n - 3)Qt-3 - 1 ( ,t= 0, n> 2 (6.3) l2$,0 + (n24 - __-2,t 0 ,n > 0 (6.4) 12(1 - 2l2n)( + 2(22 - n + 2) -2 + l ,t + (n-3,t + l2 n 1, + -3,$ = 0, 2 5 (6.5) 12 o

+

12-(n

- 1)o

_1

+ (n - 3)e-3=0,

n>2

(6.6)

212_ g - ( - (n - 2)(; = 0, n > -1 (6.7)

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 AdS3 to the

following:

S [) -

j

dx /X n"VV,". (6.8)

3

BM

By requiring that 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 - n1,t = 0, 202, 3, ... , (6.10)

2(24 - -2,t + 3(_ 1 = 0, (6.11)

2 - 1,+ =0, (6.12)

and in view of (6.10), equation (6.9) becomes:

which in turn simplifies (6.10) to:

n +

1

,=0,

n = 0, 2,3,....

(6.14)

So equations (6.7, 6.9) are equivalent to (6.11-6.12, 6.13-6.14). Now, using (6.13-6.14), equation (6.2) is equivalent. to:

12

n + (n - 3)(n-2 = 0, n =2,4,5,... , (6.15)

2l2$,e + 2(*2,t - 2l2([ + 3'_ 1 - - = 0, (6.16)

212($,t + 24 - 412r + - 2 O = 0. (6.17)

We solve (6.15) using backwards induction (2m+1 = 0, m> 1 and dm = 0 m> 0):

1(t #)r + (' 1

(t,

4)- + 0( r). (6.18)

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 2tm = 0, m >

1, and similarly, dm+1 =0, m> -1 and m =0,m 1):

1 1

t= ((t,#)+(L

2

(t,

)

+

(-),

(6.19)

S=((t,

)

+ "

2

(t,

#)

+

O(-).

(6.20)

Note that the general form of the vector ( as in (6.18-6.20) exhausts equations (6.13-6.14) 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

+

(-24 - 2,t = 0, (6.21)

12

- = 0 . (6.22)

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

AdS3 (6.1) in the limit r -+ oc while also maintaining a finite first order effective

action S0)[ ] (4.4), are given by:

l(T+ + T~)

+

(

2(t, #)i + 0(1),

(6.23)

= T+-T + 2(t, ) +( ), (6.24)

= -

(T+' + T-')

_r

_{+ ('}

₁

_{(t, #) + 0(}

),

_(6.25)

with (*2, 2, (i satisfying the remaining equations (6.11, 6.16, 6.21),2

- (-2,t + 3'1 0, (6.26)

12 T+' + T-' +&2,t-24 = , (6.27)

13 (T+' - T~') - -

l

2 2,t= 0. (6.28)

Since the equations (6.26-6.28) may be solved for (*-2, ( 2, i without imposing any

constraints on T+ and T- ,3 we find that our final ('s (6.23-6.28) are again of the form

(5.33). 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, + ( - 0.

2_{Equation (6.16) simplifies, in view of (6.11, 5.13-5.14), to: l}2

_-2,t + -2= 0

3

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 AdS3 space-times by

perturbing the exact AdS3 (2.23) using our ('s (5.33):

9tt = - + 0(1) 1 gtr = 0(-r gto = 0(1), (7.1) 12 +01) grr +0 ( ), 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 AdS3 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

Chapter 8

Conclusion and Discussion

In this thesis we demonstrated the emergence of the Virasoro algebra in the asymp-totics of AdS3 using the effective action of GR for small excitations from asymptotic

transformations.

To date the Virasoro associated with the boundary of AdS3 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 Brown-Henneaux 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 AdS3 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 AdS3 is relevant for calculating entropy of the BTZ black hole. We

have also mentioned in the introduction that for appropriate boundary conditions the

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 con-formal field theories in [20, 21]. Moreover, a new hidden two-dimensional concon-formal 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 cap-tured 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.

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