### Self-Dual Warped AdS

_{3}

### Black Holes

### Bo Ning

with Bin Chen, 1005.4175 School of Physics, Peking University

NCTS

### April 22, 2011

Motivation Black Hole Holography Summary

### Contents

1

### Motivation

2

### Self-dual warped AdS

_{3}

### black hole Geometry

### Thermodynamics

3

### Holographic description Temperatures

### Asymptotic behavior Scalar perturbation

4

### Summary

### AdS/CFT correspondence

### Quantum gravity (string theory/M-theory) in (D + 1)-dimensional anti-de-Sitter spacetime is equivalent to D-dimensional conformal field theory at AdS boundary.

### A new way to study the strong coupling problems in field theory: AdS/QCD, AdS/CMT, ...

### A tool to study the problems in quantum gravity;

### Relies heavily on string theory technology.

Motivation Black Hole Holography Summary

### AdS/CFT correspondence

### A case not invoking string theory is AdS

_{3}

### /CFT

_{2}

### correspondence:

### quantum gravity asymptotic to AdS

_{3}

### is holographically dual to a 2D CFT.

For 3D pure gravity with a negative cosmological constant, the vacuum
solution is AdS_{3}, with the isometry groupSL(2; R) SL(2; R);

The analysis of the symmetry of perturbations at asymptotic boundary give rise to a Virasoro algebra with a central term;

The entropy of the BTZ black hole could be read from Cardy formula, which relates the asymptotic density of states in a 2D CFT and the symmetry algebra.

### Topological Massive Gravity

Pure 3D gravity: contains no local propagating degrees of freedom, thus hard to explain the microscopic origin of the BTZ black hole entropy.

Topological Massive Gravity: with an additional gravitational Chern-Simons term, resulting a local, massive propagating degree of freedom.

IT MG= 1 16G

Z
d^{3}xp

g (R + 2=`^{2}) + 1

ICS

; ( > 0; G > 0) (1) ICS= 12

Z
d^{3}xp

g"^{ }_{}

@

+ 23 ^{}^{} ^{}^{}

(2)
TheAdS_{3} vacuain TMG are generally perturbatively unstable except at
the chiral point` = 1 , where a consistent quantum theory of gravity is
conjectured to exist and be dual to a chiral CFT.

Motivation Black Hole Holography Summary

### Topological Massive Gravity

TMG possess other vacua for generic values of, namelywarped AdS_{3}, admit
isometry groupU(1) SL(2; R). Some of them are stable.

Spacelike:

ds^{2}= `^{2}
(^{2}+ 3)

cosh^{2}d^{2}+ d^{2}+ 4^{2}+ 3^{2} (du + sinh d)^{2}

(3)

stable for^{2}> 1 (stretchedcase), where = `=3.

Timelike:

ds^{2}= `^{2}
(^{2}+ 3)

cosh^{2}du^{2}+ d^{2} 4^{2}

^{2}+ 3(d + sinh du)^{2}

(4)

Null: solution to TMG only for^{2}= 1

ds^{2}= `^{2}

"

du^{2}

u^{2} + dx^{+}dx
u^{2}

dx
u^{2}

_{2}#

(5)

### Topological Massive Gravity

Quotients: Just as BTZ black holes are discrete quotients of ordinary AdS_{3},
there are black holes solutions as discrete quotients of warped AdS_{3}.

Spacelike stretched black hole:

ds^{2}

`^{2} = dt^{2}+ dr^{2}

(^{2}+ 3)(r r+)(r r )+

2r p

r+r (^{2}+ 3)
dtd

+ r4

3(^{2} 1)r + (^{2}+ 3)(r++ r ) 4p

r+r (^{2}+ 3)

d^{2} (6)
identifying points along isometry@such that + 2.

Self-dual solution:

ds^{2} = `^{2}

^{2}+ 3

~x^{2}d~^{2}+ d~x~x^{2}^{2} + 4^{2}+ 3^{2} d ~ + ~xd~2
(7)

with identification ~ ~ + 2.

Motivation Black Hole Holography Summary

### Warped AdS/CFT correspondence

It is conjectured that the > 1 quantum TMG with asymptotical spacelike
stretched AdS_{3} geometry is holographically dual to a 2D CFT with central
charges

cL= 4`

G(^{2}+ 3); cR= (5^{2}+ 3)`

G(^{2}+ 3) (8)

Entropy of spacelike stretched black hole could be reproduced through Cardy formula

S = ^{2}`

3 (cLTL+ cRTR) (9) Analysis of theasymptotic symmetrylead to a central extended Virasoro algebra with above central charges.

Warped AdS/CFT correspondence is essential inKerr/CFT correspondence, in
the sense that the warped AdS_{3} structure appears in the near-horizon geometry
of Kerr black hole.

### Kerr/CFT correspondence

NHEK: near-horizon geometry of the extreme Kerr black hole
ds^{2} = 2J ()

r^{2}dt^{2}+ drr^{2}^{2} + d^{2}+ ()^{2}(d + rdt)^{2}

(10)
with + 2. A slice of NHEK geometry at fixed polar angle is locally
aself-dual warped AdS_{3}.

Conjecture: quantum gravity in NHEK is dual to a 2D CFT, with only left-moving temperature.

Near-NHEK: near-horizon geometry of the near-extreme Kerr black hole

ds^{2}= 2J ()

r(r + 4TR)dt^{2}+ dr^{2}

r(r + 4TR)+ d^{2}

+ ()^{2}(d + (r + 2TR)dt)^{2} (11)
Conjecture: near-NHEK is dual to the same CFT with non-vanishing right
temperature, since the right-moving sector gets excited.

Motivation Black Hole Holography Summary

### Motivation

### Is there near-NHEK like solutions in TMG?

### If they exist, what’s the holographic description of them?

### Relationship with the warped AdS/CFT correspondence?

It turns out that such solutions do exist in TMG:

self-dual warped AdS_{3} black holes.

Holographic dual to a chiral 2D CFT, with the same left central charge as in warped AdS/CFT, and non-vanishing left- and right-moving temperatures.

Provide another novel support to the conjectured AdS/CFT correspondence.

### Motivation

### Is there near-NHEK like solutions in TMG?

### If they exist, what’s the holographic description of them?

### Relationship with the warped AdS/CFT correspondence?

It turns out that such solutions do exist in TMG:

self-dual warped AdS_{3} black holes.

Holographic dual to a chiral 2D CFT, with the same left central charge as in warped AdS/CFT, and non-vanishing left- and right-moving temperatures.

Provide another novel support to the conjectured AdS/CFT correspondence.

Motivation Black Hole Holography Summary Geometry Thermodynamics

### Self-dual warped AdS

_{3}

### black hole

Self-dual warped AdS_{3} black hole: non-Einstein black hole solution of TMG,
which is asymptotic to spacelike warped AdS_{3}

ds^{2} = `^{2}

^{2}+ 3

(x x+)(x x ) d^{2}+ 1

(x x+)(x x )dx^{2}
+ 4^{2}+ 3^{2} (d + (x x++ x

2 ) d)^{2}

; (12)

where 2 [ 1; 1], x 2 [ 1; 1] and + 2.

There seems to be two horizons atx+andx .

The vacuum is chosen to be given byx+= x = 0 and = 1, which is
the self-dual warped AdS_{3}.

For^{2}> 1, the solution is free of naked CTCs.

### Self-dual warped AdS

_{3}

### black hole

Self-dual warped AdS_{3} black hole is related to the self-dual warped AdS_{3}
through coordinate transformation

~^{} ~ 1~x= tanh

1 4

(x+ x ) ln x xx x^{+}

;

~ = + 12ln

1 (~^{+})^{2}
1 (~ )^{2}

: (13)

Globally, the maximal analytic extension of the self-dual warped black
hole is diffeomorphic to the self-dual warped AdS_{3}.

However, the above coordinate transformations are singular at the boundaryx ! 1, indicating different physics.

The situation here is very similar to the relation between NHEK and
near-NHEK, or between AdS_{2} and AdS_{2} black hole.

Observer at fixedx measure a Hawking temperature proportional to x+ x . The entropy does not depend on x+ x , but the scattering amplitudes do depend onx+ x .

Motivation Black Hole Holography Summary Geometry Thermodynamics

### Self-dual warped AdS

_{3}

### black hole

Self-dual warped black hole is locally equivalent to spacelike warped AdS_{3}
ds^{2}= `^{2}

^{2}+ 3

cosh^{2}d^{2}+ d^{2}+ 4^{2}

^{2}+ 3(du + sinh d)^{2}

(14) through coordinate transformation

= tan ^{1}

"

2p

(x x+)(x x )

2x x+ x sinh_{x}

+ x

2 #

;

= sinh ^{1}

"

2p

(x x+)(x x )

x+ x cosh_{x}

+ x

2 #

;

u = + tanh ^{1}

2x x+ x

x+ x coth_{x}

+ x

2

: (15)

### Self-dual warped AdS

_{3}

### black hole

Killing vectors of spacelike warped AdS_{3}:
J2 = 2@u

J~1 = 2 sin tanh @ 2 cos @+ 2 sin cosh @u

J~2 = 2 cos tanh @ 2 sin @ 2 cos cosh @u

J~0 = 2@

Self-dual warped AdS_{3} black hole solution is free of curvature singularity and
regular everywhere.

Strictly speaking, it does not belong to the category of regular black holes, which require the existence of a geometric or causal singularity shielded by an event horizon.

On the other hand, similar to black holes, this solution have regular thermodynamic behavior, satisfying the first law of thermodynamics.

Motivation Black Hole Holography Summary Geometry Thermodynamics

### Thermodynamics

Conserved charges:

M^{ADT} = 0 ; J^{ADT} = (^{2} 1)`

6G(^{2}+ 3) (16)

Entropy: with the Chern-Simons contribution S = SE+ SCS= 2`

3G(^{2}+ 3) (17)

Hawking temperatureTH and the angular velocity of the horizonh,
TH= x^{+} x

4` ; h= x^{+} x

2` (18)

The first law

dM^{ADT} = THdS + hdJ^{ADT} (19)

is satisfied for a variation of the black hole parameter.

### Thermodynamics

### Remarks:

ADT mass and angular momentum do not depend onx+ andx . EntropyS does not depend on x+ andx either.

Observers at fixedx measure a Hawking temperature proportional to x+ x .

Similar to the case of near-NHEK.

Even with a different choice of vacuum, the first law of thermodynamics still holds.

Motivation Black Hole Holography Summary Temperatures Asymptotic behavior Scalar perturbation

### Temperatures in dual CFT

Since the self-dual warped black hole metric looks much similar to the near-NHEK geometry, it is sensible to define a quantum vacuum in analogy to the Frolov-Thorne vacuum of Kerr.

The construction of the vacuum begins by expanding the quantum fields in eigenmodes of the asymptotic energy! and angular momentum k. For a scalar field,

=X

!;k;l

!kle^{( i!+ik)}Rl(x) (20)

After tracing over the region inside the horizon, the vacuum is a diagonal density matrix in the energy-angular momentum eigenbasis with a Boltzmann weighting factor

exp

~! kH

TH

(21)

### Temperatures in dual CFT

In AdS/CFT dictionary,black hole in AdSis dual tofinite temperature CFT.

Taking black hole as a thermodynamical system, the thermal equilibrium in black hole system could be compared to thermal equilibrium of finite temperature CFT.

2D CFT contains two independent sectors: left-moving one and right-moving one, possibly with different central charges and temperatures.

Motivation Black Hole Holography Summary Temperatures Asymptotic behavior Scalar perturbation

### Temperatures in dual CFT

The Boltzman factor should be identified with that in CFT exp

~! kH

TH

= exp _{n}

L

TL

nR

TR

(22)

The left and right chargesnL; nRassociated to@ and@ are

nL k ; nR ! (23)

This defines the left and right temperatures:

TL= 2`; TR= x^{+} x

4` (24)

### Temperatures in dual CFT

The right temperaturedenotes the deviation from extremality, originates
essentially similar to the case that Rindler observers detect radiations in
the Minkowski vacuum (as discussed for AdS_{2} black hole).

The left temperaturearises from the periodical identification of points in
spacelike warped AdS_{3} along@. Expressing@ in terms of the spacelike
warped AdS_{3} coordinates,

@= 2J2= ` TLJ2 (25) the temperature of the dual 2D CFT can be read from the coefficient of the shift. This periodical identification makes no contribution to the right temperature.

Motivation Black Hole Holography Summary Temperatures Asymptotic behavior Scalar perturbation

### Asymptotic behavior

To acquire the central charge of the dual CFT through the asymptotic symmetry analysis, we impose the following consistent boundary conditions

h = O(1) hx= O(1=x^{3}) h= O(x )
hx = hx hxx= O(1=x^{3}) hx= O(1=x )
h = h hx= hx h= O(1)

!

(26)

whereh is the deviation of the full metric from the vacuum.

These boundary conditions differ from both the ones inspacelike warped AdS_{3}
and theKerr black holes, since the allowed deviationshandh are of the
same order as the leading terms.

### Asymptotic behavior

Theasymptotic symmetry group, preserving the above boundary conditions, contains one copy of the conformal group of the circle generated by

= () @ (27)

Since + 2, it is convenient to define n() = e^{in} and

n= (n). They admit the following commutators

i [m; n] = (m n) m+n (28) and0 generates theU(1) rotational isometry. That is, the U(1)

isometry is enhanced to a Virasoro algebra.

The conserved charge associated with an asymptotic Killing vector represent the asymptotic symmetries algebra via a covariant Poisson bracket, up to acentral term

ifQm; Qng = (m n)Qm+n+ c12^{L}m(m^{2} 2)m+n;0; (29)
where the central charge

cL = 4`

G(^{2}+ 3) (30)

Motivation Black Hole Holography Summary Temperatures Asymptotic behavior Scalar perturbation

### Asymptotic behavior

ThecLis exactly the value of the left central charge conjectured in
warped AdS/CFT correspondence for spacelike warped AdS_{3}.

The entropy of the self-dual warped black hole can be reproduced from the Cardy formula

S = 2`

3G(^{2}+ 3)= ^{2}`

3 4`

G(^{2}+ 3)
2` ^{2}`

3 cLTL (31)

Conjecture: our self-dual warped black hole solution is dual to a chiral 2D CFT with the temperatures and central charges

TL= 2`; TR= x^{+} x

4` ; cL = 4`

G(^{2}+ 3) (32)
Further supported by the real-time correlator of scalar perturbations.

### Scalar perturbation

Consider a scalar field with mass m in black hole background

= e ^{i!+ik}R(x) (33)

The radial wave functionR(x) satisfies the equation d

dx

(x x+)(x x ) ddx

R(x)

^{2}+ 3
4^{2}

k^{2}
^{2} + `^{2}

^{2}+ 3m^{2}

! +^{k}_{}(x ^{x}^{+}^{+x}_{2} )_{2}
(x x+)(x x )

!

R(x) = 0 (34)

Choose ingoing boundary condition at the horizon for calculating the retarded Green’s function, the solution is

R(x) = N

x x+

x x

_{2}^{i} ^{k}_{}_{+} ^{2!}

x+ x x+ x

x x

^{1}_{2} _{}

· F

1

2 i k; 12 i 2!

x+ x ; 1 i

k

+ 2!

x+ x

; x xx x^{+}

(35)

Motivation Black Hole Holography Summary Temperatures Asymptotic behavior Scalar perturbation

### Scalar perturbation

At asymptotical infinity, the radial eigenfunction has the behavior

R(x) Ax ^{1}^{2} ^{}+ Bx ^{1}^{2}^{+} (36)

where

A = N (x+ x )^{1}^{2}^{+} ( 2)
1 i

k+_{x}_{+}^{2!}_{x}

12 i_{}^{k} _{1}

2 i_{x}_{+}^{2!}_{x} (37)

B = A ( ! ) (38)

The real-time retarded Green’s function could be computed using a prescription in terms of the boundary values of the bulk fields. Consider a real > 0 without loss of generality, the retarded correlator is given by

GR AB = (x+ x )^{2} ( 2)
(2)

12 + i^{k}_{} _{1}

2+ i_{x}_{+}^{2!}_{x}

12 i^{k}_{} _{1}

2 i_{x}_{+}^{2!}_{x}
(39)

### Scalar perturbation

Throwing a scalar at the black hole is dual to exciting the CFT by acting with an operatorO. For an operator of dimensions (hL; hR) at temperature (TL; TR), the momentum-space Euclidean Green’s function is determined by conformal invariance

GE(!L;E; !R;E) T_{L}^{2h}^{L} ^{1}T_{R}^{2h}^{R} ^{1}e^{i}^{!L;E}^{2TL}e^{i}^{!L;E}^{2TR} (hL+ !2T^{L;E}L) (hL !L;E

2TL)

· (hR+ !^{R;E}

2TR) (hR !R;E

2TR) (40) the Euclidean correlatorGE(!L;E; !R;E) is related to the value of the retarded correlatorGR(!L; !R) by

GE(!L;E; !R;E) = GR(i!L;E; i!R;E) (41) Comparing the arguments of the Gamma functions among (39) and (41), we find preciseagreementunder the following identification

hL= hR= 12+ ; !L= k=`; !R= !=`; TL= 2`; TR= x^{+} x
4`

Motivation Black Hole Holography Summary

### Summary

### A novel example of warped AdS/CFT correspondence:

### the self-dual warped AdS

_{3}

### black hole is dual to a chiral CFT with non-vanishing left central charge.

### The quantum topological massive gravity asymptotic to the

### same spacelike warped AdS

_{3}