Warped AdS/CFT Correspondence
Hsien-Chung Kao, NTNU
and
Wen-Yu Wen, NTU.
I.)AdS/CFT Correspondence
II.)AdS3/CFT2 Correspondence
III.)Warped AdS3/CFT2 Correspondence
IV.)Discussions
AdS/CFT Correspondence
N = 4 SYM in 4D ' IIB String in AdS5 × S 5
S 5:
X12 + X22 + X32 + X42 + X52 + X62 = R2
in R6
2 − X 2 + X 2 + X 2 + X 2 + X 2 = −R2 in R2,4
AdS5: −X−1
0
1
2
3
4
•
N=4 SYM Conformal Sym. in 4D: SO(4, 2) R-sym.: SU (4)
IIB String
Isometry in AdS5 : SO(4, 2)
Isometry in S 5: SO(6)
’t Hooft coupling: λ = gY2 M N = gsN , R = (4πgsN )1/4ls ∼ λ1/4lpl
gs: string coupling, 2πls2 : inverse string tension.
For λ 1, R lpl , SUGRA limit.
• The holographic principle.
• CFT calculation of BH entropy, BH greybody factor.
• PP wave limit.
1
Let X−1 = R cosh ρ cos τ,
X0 = R cosh ρ sin τ,
X1 = R sinh ρ sin θ1 sin θ2 cos φ, X2 = R sinh ρ sin θ1 sin θ2 sin φ, (1)
X3 = R sinh ρ sin θ1 cos θ2 ,
2
Global coordinate: ds = R
Solid
Cylinder
2
h
2
− cosh ρdτ
2
2
2
+ dρ + sinh ρdΩ2
3
i
(2)
Massive
Particle
S3
ρ=0
X4 = R sinh ρ cos θ1.
Light
Ray
Time
ρ=∞
(a)
(b)
(c)
Figure 1. Penrose diagram and the AdS boundary (S 3 × R).
Define dx = dρ/ cosh ρ, tan(x/2) = tanh(ρ/2), ρ → ∞ ∼ x = π/2.
⇒ − cosh2 ρdτ 2 + dρ2 = sec2x(−dτ 2 + dr2).
2
Let X−1 + X4 = R/z, Xµ = Rxµ/z for µ = 0, · · · 3.
Poincare coordinate: ds2 = R2
2 + dx2 + dx2 + dz 2
−dx2
+
dx
0
1
2
3
z2
!
(3)
1
Warp Factor W ~ __
Z
4
R
Boundary
Z
Z=1
Horizon
Z=0
Figure 2. AdS boundary: S 3 × R.
3
S3
Operator
R
States of CFT on S 3
=
Operators of Euclidean CFT on R4
R
4
Zbulk [φ(~
x)] = he d xφ0(~x)O(~x) iField Theory
x, z)|z=0 = φ0(~
Scalar field of mass m: S = N 2
Z
(4)
dx4dz 2
2 + m2R2φ2]
[z
(∂φ)
z5
(5)
1
m2R2
3
2
⇒ z ∂z ( 3 ∂z φ) − p φ −
φ=0
2
z
z
h
i
φ = z 2 A+Iν (pz) + A−Kν (pz) , with ν =
(6)
q
4 + m2R2.
(7)
Near z = 0, φ ∼ B+z 2+ν + B−z 2−ν .
(8)
Boundary condition: φ(x, z)|z= = 2−ν φr0 (x), φr0(x) fixed for → 0.
4
The rescaling xµ → λxµ , z → λz is a isometry in AdS,
φ does not get rescaled, φr0 has dimension 2 − ν,
⇒ the correponding operator O has dimension ∆ ≡ 2 + ν.
Bulk Green function in AdS5: G∆(z, xµ, x0µ ) =
x1
z∆
(9)
[(x − x0)2 + z 2]∆
x2
x3
Figure 4. Three point functions.
With a cubic term in the action,
hO1O2O3i ∼
Z
d4xdz
G∆(z, x, x1)G∆ (z, x, x2)G∆ (z, x, x3).
z5
(10)
5

 0 > m2R2 ≥ −4, 2 − ν < 0, φ induces relevant pertubation;

m2R2 > 0,
2 − ν > 0, φ induces irrelevant pertubation.

 (EFT) = 1 (Eproper );
z

(size)FT = z(proper size).

 z → 0 ∼ spatial infinity, UV;

z → ∞ ∼ horizon,
IR.
IR/UV Correspondence: UV in field theory ∼ IR in gravity .
Blackhole ⇒ Hawking temperature(TH ) ⇒ thermal effect.
Simplest black hole in Poicare coordinate (S 5 supressed):

R2 
2
ds = 2 −
z

!
!
4 −1
z4
z
2
2
1 − 4 dt + d~
x + 1− 4
z0
z0
z
For z ' z0, ds2 ∝ −16 1 −
z0
!2

dz 2  . (11)

dt2 + dz 2 + . . . .
(12)
6
Period in imaginary time: β = πz0/2.
π2 2 3
(Area)
SBH
Bekenstein Hawking entropy: SBH =
=
N T .
⇒
4GN
V
2
(13)
4 π2 2 3
SF T
=
N T .
Weakly coupled field theory entropy:
V
32
π2
2
S
N 2T 3V
(14)
4
3
1
g2 N
1
Figure 5. Field theory entropy vs. Bekenstein Hawking entropy.
g 2N correction to SF T and R4 to SBH go in the right direction.
7
AdS3/CFT2 Correspondence
CFT interpretation of BH QMN
The BTZ blackhole (R = 1), part of SUGRA:
2
ds = −
"
(r2
−
2 )(r 2
r+
r2
−
2)
r−
#
"
#
h
r+ r− i2
dt +
dr + r dθ − 2 dt .(15)
2 )(r 2 − r 2 )
(r2 − r+
r
−
2
r2
2
2
2 + r2 ), J = 2r r .
M = (r+
+ −
−
2 − r2 )/(2πr ), A = 2πr , Ω = J/(2r2 ).
TH = (r+
+
H
+
H
−
+
Locally equivalent to AdS3 :
ds2 = − sinh2 µ(r+dt − r−dφ)2 + dµ2 + cosh2 µ(r+dφ − r−dt)2 . (16)
The dual CFT on the boundary is (1+1)-D.
Two independent copies of CFT with
TL = (r+ − r−)/(2π), TR = (r+ + r−)/(2π), 1/TL + 1/TR = 2/TH .
8
Operators O in (1+1)-D CFT are charaterized by
the conformal weights: (hL, hR):
hL + hR = ∆,


 Scalar:
hR − hL = ±s.
∆=
(17)
q
1 + m2;

 Spinor and vector: ∆ = 1 + |m|.
Retarded Green function: Dret(x, x0) = iθ(t − t0)h[O(x), O(x0 )]iT , (18)
= iθ(t − t0)D̄(x, x0).
D̄(x, x0) = D+(x, x0 ) − D−(x, x0 ), x± = t ± σ.
D+(x) =
(πTR )2hR
(πTL )2hL
;
sinh2hR (πTR x− − i) sinh2hL (πTL x+ − i)
(πTL )2hL
(πTR )2hR
.
D−(x) =
2h
2h
−
+
R
L
sinh
(πTR x + i) sinh
(πTL x + i)
(19)
(20)
9
With k± = (ω ∓ k)/2, Fourier transform of D̄(x):
ik+
ik−
ik+
ik−
Γ hR +
Γ hL −
Γ hR −
.
D̄(k+ , k−) = Γ hL +
2πTL
2πTR
2πTL
2πTR
(21)
Two sets of poles in the lower complex plane of ω:
ωL =
k − 4πiTL(n + hL);
ωR = −k − 4πiTR(n + hR).
(22)
n = 0, 1, 2, . . . . They characterize they decay of the perturbation.
These poles coincide precisely with the QMN in the BTZ backgound!

 1
Scalar perturbation satisfies  q ∂µ (
|g|

q

|g|g µν ∂ν ) − m2 Φ = 0.
(23)
+
−
Let Φ = e−i(k+x +k−x ) f (µ), x± = r±t − r∓φ, z = tanh2 µ,
k+ + k−
r+ − r− = ω − k;

k+ − k−
2
k+

2
k−
r+ + r− = ω + k.

m2 
z(1 − z)f (z) + (1 − z)f (z) +
−
−
f˜(z) = 0.
4z
4
4(1 − z)
˜00
˜0
(24)
10
f˜(z) = z α(1 − z)βs 2F1(as, bs; cs; z),
α = −ik+/2, βs = 1 −
q
(25)
1 + m2 /2
as = −i(k+ − k−)/2 + βs, bs = −i(k+ + k−)/2 + βs, cs = 1 + 2α.
QNM: vanishing Dirichlet condition at infinity.
2
√ 1 ∗
Γ(cs)Γ(cs − as − bs) ∗
f ∂µf − f ∂µf ∝ Flux: F = g
.
2i
Γ(cs − as)Γ(cs − bs) (26)
F vanishes if

 cs − as = −n
 c − b = −n
s
s
⇒ i(k+ ± k−)/2 = n + 1 +
q
1 + m2 /2.
(27)
11
For spinors,
βf = − (m + 1/2) /2, cf = 1/2 + 2α.
af = −i(k+ − k−)/2 + βf + 1/2, bf = −i(k+ + k−)/2 + βf .
For vectors,
βv = m/2, cv = 1 + 2α.
av = −i(k+ − k−)/2 + βv , bv = −i(k+ + k−)/2 + βv .
12
CFT interpretation of BH Greybody Factor
2
ds = −
"
(r2
−
2 )(r 2
r+
r2
−
2)
r−
#
"
#
h
r+ r− i2
dt +
dr + r dθ − 2 dt .(28)
2 )(r 2 − r 2 )
(r2 − r+
r
−
2
r2
2
2
Consider massless scalar Φ. Let Φ = e−iωt+imθ Rω,m (r),


2r
1
2r 
2

∂r Rω,m (r) + − + 2
+ 2
∂r Rω,m (r)
2
2
r
r − r+
r − r−
"
#
4
2
r
M m − Jωm
2
2
+ 2
ω −m +
Rω,m (r) = 0.
2 )2(r2 − r2 )2
2
r
(r − r+
+
2 )/(r2 − r2 ), A = ω−mΩH
Let z = (r2 − r+
1
−
4πT
H
2
, B1 = −
(29)
2 −mΩ r 2
ωr−
H +
4πTH r+r−
A1
00
0
+ B1 R̃ω,m (z) = 0.
z(1 − z)R̃ω,m (z) + (1 − z)R̃ω,m (z) +
z
!2
(30)
z = 0 ∼ horizon, z → ∞ ∼ spatial infinity.
13
In-going into BH at horizon:
√
α
R̃ω,m (z) = z 2F1(a, b; c; z), α = i A1,
a + b = 2α, ab = −A1 − B1, c = 1 + 2α. At horizon, z = 0,
Flux: F = 2AH (ω − mΩH ).
(31)
BTZ ∼ AdS3, hard to tell in-coming from out-going.
c
in
Rω,m = Ai 1 − i 2 ,
r
⇒ Fin = 8πc|Ai|2;
c
out
Rω,m = Ao 1 + i 2 .
r
σ=
F
.
Fin
(32)
(33)
AH |Γ(a + 1)Γ(b + 1)|2
σ m=0
In low energy limit ω → 0, σabs '
=
. (34)
2
ω
πc
|Γ(a + b + 1)|
Choose c so that σabs|ω=0 = AH .
14
Decay rate:
σ
Γ = ω/Tabs
H −1
e
!
!2
ω
ω
2
−1
−ω/(2T
)
H Γ 1 + i
= 4π ω TLTR e
Γ 1+i
.
4πTL
4πTR From CFT, Γ =
Z
×
Z
dσ − e−iω(σ− −i)
dσ
+ −iω(σ+ −i)
e
"
"
2TL
sinh(2πTLσ −)
(35)
#2
2TR
sinh(2πTR σ +)
#2
.
(36)
The result is identical to that obtained from the gravity side.
15
Warped AdS3/CFT2 Correspondence
Warped BTZ Black Hole
The warped BTZ black hole is a solution of TMG:
Z
1
3 √
2
IT M G =
d x −g R + 2/`
16πGZ M
√
`
2
σ τ
+
d3x −gελµν Γrλσ ∂µΓσ
rν + Γµτ Γνr .
96πGν M
3
√
m `
τ
σu
= +1/ −g, ν = 3g .
where ε
(37)
For ν = 1/3, critical chiral gravity theory.
1
` αβ
1
EOM: Gµν − 2 gµν +
ε
∇α Rβν − gβν R = 0.
`
3ν µ
4
(38)
16
`4dr2
2
2
2
2
2
2
,
(39)
ds = −N (r)dt + ` R (r)[dθ + Nφ(r)dt] +
2
2
4R (r)N (r)
q
r
R2(r) ≡
3(ν 2 − 1)r + (ν 2 + 3)(r+ + r−) − 4ν r+r−(ν 2 + 3) ,
4
q
2 + 3)
2(ν 2 + 3)(r − r )(r − r )
2νr
−
r
r
(ν
`
−
+
−
+
, Nθ (r) ≡
.
N 2(r) ≡
4R2
2R2


ν2 + 3 
TH =
4π 

r+ − r−
2νr+ −
q



, AH = π 2νr+ −

(ν 2 + 3)r+r− 
q
r+r−(ν 2 + 3) .
ν > 1, streched AdS3 , ν = 1, BTZ limit,
ν < 1, squashed AdS3 , closed time-like curve.
t, r, θ differnent from the BTZ case.
Warped AdS3 (locally equvalent to warped BTZ):
`2
"
4ν 2
#
(du + sinh σdτ )2 .
− cosh2 σdτ 2 + dσ 2 + 2
ds2 = 2
ν +3
ν +3
(40)
17
Greybody Factor in Warped BTZ
Consider scalar Φ with mass µ. Let Φ = e−iωt+imθ φ(r),
2r − r+ − r− dφ(r)
(αr2 + βr + γ)
d2φ(r)
−
+
φ = 0,
2
2
2
dr
(r − r+)(r − r−) dr
(r − r+) (r − r−)
q
(41)
2 + 3)
m
−
ω
r
r
(ν
4m
2
2
2
2
−
+
µ2`2r+r−
3ω (ν − 1)
µ `
, γ=−
,
α=−
+ 2
+
2
2
2
2
2
(ν + 3)
ν +3
(ν + 3)
ν +3
q
ω 2(ν 2 + 3)(r+ + r−) − 4ν ω 2 r+r−(ν 2 + 3) − 2mω
µ2`2(r+ + r−)
.
+
β=−
2
2
2
(ν + 3)
ν +3
r−r
z = r−r+ ,
−
z = 0 ∼ horizon, z = 1 ∼ spatial infinity.
A
B
00
0
+
+ C φ̃(z) = 0;
z(1 − z)φ̃ (z) + (1 − z)φ̃ (z) +
(42)
z
1−z
2
2
+
m)
4(ωΩ−1
4(ωΩ−1
+
− + m)
, B = −α, C = −
.
A=
2
2
2
2
2
2
(r+ − r−) (ν + 3)
(r+ − r−) (ν + 3)
18
Ω−1
+ = νr+ −
q
r+r−(ν 2 + 3)
2
Ω−1
− = νr− −
√
1
,
φ(z) = z p(1 − z)q u(z), with p = −i
A, q =
2
q
r+r−(ν 2 + 3)
2
1−
p
In-going at horizon: u(z) = z p (1 − z)q 2 F1(a, b; c; z).
(43)
(44)
iη −h∗+
iη −h∗+
−h∗−
− r
+ Aout r
+ r
.
At r → ∞: φ ' Ain
π
π
r
√
12ω2 (ν 2−1)
4µ2`2
1
∗
∗
∗
h± = 2 (1 ± ∆ ), ∆ = 1 + 4α = 1 − (ν 2+3)2 + ν 2+3 .
−h∗−
r
1 + 4α ,
⇒ z(1 − z)u00(z) + {c − (a + b + 1)z}u0(z) − abu(z) = 0,
√
√
with
a = p + q + C, b = p + q − C, c = 2p + 1.
.
(45)
η is chosen so that σabs|ω=0 = AH .
19
Retaining ω dependence,
2
√
√
1
ω
Γ 1 (1 + 1 + 4α) − i ω
Γ 2 (1 + 1 + 4α) − i
2
4π
T̃
4π
T̃
R
L h
i h√
i
σabs ∝ , (46)
ω
Γ 1 − i 2πT Γ 1 + 4α
H
(ν 2 + 3)(r+ − r−)
ν2 + 3
.
, T̃R =
T̃L =
q
8πν
8π ν(r+ + r−) − r+r−(ν 2 + 3)
(47)
2/TH = 1/T̃L + 1/T̃R .
(48)
Waped AdS3, Φ = e−iωg τ +iku φg (σ) :
2
2
2
2
dφg
(ν + 3)k
µ `
g
2
+
tanh
σ
sechσ
+
k
tanh
σ)
−
−
φg = 0.
+
(ω
g
2
2
2
dσ
dσ
4ν
ν +3
d2φ
(49)
20
(ωg +ik)/2
Define φg = zg
00
(1 − zg )(ωg −ik)/2f (zg ), zg ≡
h
1 + i sinh σ
.
2
i
zg (1 − zg )f + cg − (1 + ag + bg )zg f 0 − ag bg f = 0
(50)
3k2(ν 2 − 1)
µ2`2
ag bg = ωg (ωg + 1) +
, ag + bg = 1 + 2ωg ,
− 2
4ν 2
ν +3
cg = 1 + ωg + ik.
h
h
For zg → ∞, φg (zg ) → C+zg + + C−zg − ,
v
with
1
h± = (1 ± ∆),
2
∆≡
(51)
u
u
t
3k2(ν 2 − 1)
4µ2`2
.
1−
+ 2
2
ν
ν +3
Boundary of warped AdS3 : σ → ∞.
x = e−σ and t = τ /2, local patch:
2
2 − 1)
2
2
`
4ν
8ν
3(ν
2
2
2
2
ds2 → 2
+
dx
+
du
+ 2
dt
x
xdtdu .
(ν + 3)x2 ν 2 + 3
ν2 + 3
ν +3
(52)
21
On the boundary,
!
√
1
√ ∂x −gg xx∂x + ∂u g uu∂u − m2 K(x, u) = 0.
−g
K(x, u) = e
iku h+
x
⇒ K(x, u, t) = e
Bulk field: φ(x, t, u) ∼
Z
.
(54)
dt0du0Kb(x, t, u, t0, u0)φ0(t0, u0).
(55)
∂
φ(x, t, u) ∼ xh+−1
In the limit x → 0,
∂x
x
|x2 − t2|
iku
!h
+
(53)
Z
0
eik(u−u )φ0 (t0, u0)
0
0
dt du
. (56)
2h
0
+
|t − t |
Z
√
1
Sef f = lim −
dt du −gg xxφ∂x φ
x→0
2
Z
ik(u−u0)
e
1
0, u0).
∼
dt du dt0 du0
φ
(t,
u)φ
(t
0
0
2
|t − t0|2h+
(57)
22
* R
From the AdS/CFT dictionary: e−Sef f (φ) =
Correlator:
*
O∗(u+, u−)O∗(0, 0)
*
O∗(u+, u−)O∗(0, 0)
+L
+R
∼
eiωu+
|u−|
∼
2h∗+
e φ0 O .
(58)
,
(59)
, u± ≡ u ± t/2c.
(60)
eiωu−
|u+|
+
2h∗+
Assigning T̃L and T̃R to the left and right sectors.
O∗(u+ , u− )O∗ (0, 0)
∼ (2h+ − 1)eiω(u++u−)
T
πT̃R
sinh(πT̃Ru+ )
2h∗+ πT̃L
sinh (π T̃Lu− )
2h∗+
.
(61)
∗
O∗(x+, x− ) with conformal dimension h+ ≡ h+
,
k=k∗ ≡ 22ν ω
ν +3
σabs ∼
Z
*
dx+ dx−e−iω(x+ +x−) O∗(x+, x− )O∗(0, 0)
+
.
(62)
T
23
Discussions
• There is indeed a warped AdS3/CFT2 correspondence.
•Gravity side:
2
√
√
1 (1 + 1 + 4α) − i ω
Γ 1 (1 + 1 + 4α) − i ω
Γ
2
2
4π
T̃
4π
T̃
R
L h
i
h
i
σabs ∝ √
,
ω
Γ 1 − i 2πT Γ 1 + 4α
H
ν2 + 3
(ν 2 + 3)(r+ − r− )
h
i , 2/TH = 1/T̃L + 1/T̃R.
T̃L =
, T̃R =
p
2
8πν
8π ν(r+ + r−) − r+r−(ν + 3)
24
• CFT side:
*
O∗(u+, u−)O∗(0, 0)
*
O∗(u+, u−)O∗(0, 0)
+L
+R
∼
eiωu+
|u− |
∼
2h∗+
,
eiωu−
|u+|
2h∗+
.
Assigning T̃L and T̃R to the left and right sectors.
O∗(u+ , u− )O∗ (0, 0)
∼ (2h+ − 1)eiω(u++u−)
2h∗+ πT̃R
πT̃L
sinh(πT̃R u+ )
sinh (π T̃Lu− )
T
∗
O∗(x+ , x−) ∼ O∗(u+, u−) with h+ ≡ h+
,
k=k∗≡ 22ν ω
2h∗+
.
ν +3
σabs ∼
Z
*
dx+ dx−e−iω(x+ +x− ) O∗(x+, x− )O∗(0, 0)
+
,
T
2
√
√
1 (1 + 1 + 4α) − i ω
Γ 1 (1 + 1 + 4α) − i ω
Γ
2
2
4π
T̃
4π
T̃
R
L h
i
h
i
∝
√
.
ω
Γ 1 − i 2πT Γ 1 + 4α
H
25
• ν > 1, superradiant modes appear when
ν2 + 3 2 2
(ν 2 + 3)2
2
+
µ ` .
ω >
2
2
12(ν − 1)
3(ν − 1)
Conformal weight h∗± becomes complex. ω ∼ angular velocity.
The effective mass of scalar is below the B-F bound
2
2 − 1)ω 2
12(ν
4µ
1
2
µef f ≡ 2
−
< − 2.
ν +3
(ν 2 + 3)2`2
`
• Tortoise coordinate r∗: φ∗(r∗) ≡ z(r)φ(r), r∗ = f (r).
Choose f (r), z(r) s.t.
"
−
d2
#
2 + U ∗(r∗) φ(r∗) = 0.
−
ω
dr∗2
In spatial infinity, the effective potential
(ν 2 + 3)(ν 2 + 3 + 4µ2`2)
∗
∗
lim U (r) → U∞ =
.
2
r→∞
12(ν − 1)
∗ > 0.
Superrandiance: ω 2 − U∞
26
• Comparing the poles of bulk QNM and boundary retarded
Green function.
• Dual theory of a rotating background.
27