Transcript I. AdS/CFT correspondence

AdS/CFT Correspondence
and Some Applications
An amateur’s point of view
Hai-cang Ren
(Rockefeller & CCNU)
Contents
I. AdS/CFT correspondence
II. Some applications
III. Remarks
I. AdS/CFT correspondence
The inversion invariance of a massless field theory in 4D:

x a
x
x

x
2
2
scalar field:
1
 ( x) 
x
2
 x 

2
x 

vector field:
A ( x ) 
1
x
2

 x  2x x
 x 
A  2  
A

 
2 2
2
(x )
x 
x 
spinor field:

 ( x) 
 x
2
(x )
 x 

2
x 

2

 x x
The conformal group in 4D:
Poincare transformations
10 generators:
P , M

Dilatation = Inversion x inversion
x  x
1 generators: D
Special conformal transformation = inversion x translation x inversion:
x


x  a x

2
1  2 a x  a x
2
4 generators:
2
Lie algebra of the 15 generators:
M , P    i  P   P 
M , M    i  M  permutatio
D , P    iP
M , K    i  K   K 
D , K   iK
P , K   2 iM  2 i  D
























other commutators vanish.
ns
K
Under the conformal group:
Classical
Quantum
Yes
No
massless QCD
Yes
No
N=4 Supersymmetric Yang-Mills
Yes
Yes
massless

4
The conformal symmetry at quantum level requires
 (g)  0

since  T    ( g )
The conformal group and O(2,4)
O(2,4) = rotation group of M(2,4)
M(2,4) = 6D Minkowski space of signature (-, -, +, +, +, +):
ds
2
 dX

dX
Introduce

 dX
 dX
2
4
X
x 
X

 X
4
  0 , 1, 2, 3
2
1
1
4D Lorentz transformation: O(1,3) subgroup among’s X 
4D Dilatation: O(2) rotation

X

X
4
X
1


 X

 X
4
 X
4
cosh   X
1
sinh 
sinh   X
1
cosh 
4D Translation (infinitesimal): O(2,4) transformation

X

X
4
X
1

 X

 X
4

 b ( X
4
 X
1
)

 X
b X


1
b X

4D Special conformal transformation (infinitesimal): O(2,4) transformation
X


 X

 a  ( X 4  X 1 )


X4  X4 a X



X 1  X 1  a X

AdS5:
A hyperboloid in M(2,4)

X X   X 4  X 1   L
2
2
2
Throughout this lecture, we set the AdS radius L=1.
Isometry group: O(2,4)
Metric:
ds

 dX
2
 dX
or
ds
1

2
z
2
dX

dX


 dt
 dX
 dX
2
2
4
 dX
2
4
 dx

2
X
2
1

dX
1 X
 dz
2


4
 X
2
4
X


where
X
X


4

x
z
1 1 2
1 
1 2 
1 

x z 
x  z 
  1 


2 2
z 
2
z 



Space of a constant curvature
R    g  g   g  g 
R    4 g 
R   20

2
 X 4 dX
AdS5-Schwarzschild
ds
2
1 
1
2
2
2 
 2   fdt  d x  dz 
z 
f

A black hole at
z  zh
f 1
4
4
zh
1
Hawking temperature T 
z
z h
(Plasma temperature)
Curvature：
R    g  g   g  g   C 
C 0i0 j 
f
z
4
h
C 0404  
 ij
C 4i4 j  
3
C ijkl  
z
4
h
1
4
h
z f
1
z
4
h
 ij
( ik 
i , j , k , l  1, 2 ,3
jl
  il 
jk
)
The same Ricci tensor and curvature scalar as AdS5
R    4 g 
R   20
AdS 5  S :
5
The metric:
ds
2

1 
1
2
2
2 
2



fdt

d
x

dz

d

5
2

z 
f

The isometry group:
O(2,4) X O(6)
------- O(6) is isomorphic to SU(4), the symmetry group of
the R-charge of N=4 SUSY YM
------- A superstring theory can be established in AdS 5  S 5 :
AdS
5
S
ds
2
5
- Schwarzsch
ild
1 
1
2
2
2 
2
 2   fdt  d x  dz   d  5
z 
f

Large Nc field theory: t’Hooft
 ~ Tr ( d  i d  i )  g YM c
~
Nc

ijk
Tr(  i  j d  k )  g YM d
2
ijkl
Tr(  i  j  k  l )
~ ~
~ ~
~
~ ~ ~ ~
ijk
2
ijkl
[Tr ( d  i d  i )  g YM c Tr(  i  j d  k )  g YM d Tr(  i  j  k  l )]
~
 i  g YM  i support th e adjoint representa tion of SU ( N c )
where
 
and the ' t Hooft coupling
2
N c g YM
Power counting of a Feynman diagram at large Nc:
a propagator
~

,
a vertex ~
Nc
a vacuum
diagram
Nc

,
a loop ~ N c
of E propagator s, V vertices
E
and F loops
V
    Nc 
F

E V
 
~ 
 Nc  Nc 

 Nc    
where
  V  E  F  topologica
Perturbati on series 
N
g
22 g
c
c
n
l  2  2g
 
n
g ,n
N
22 g
c
f g ( )
g
Dominated by the diagram with lowest g,
-------- the planar diagram ~ a string world sheet
Large Nc field theory:
A planar diagram
A handle free world sheet
g  the genus of a world sheet
 no. of handles
A non-planar diagram
A world sheet with a handle
Perturbati on series of the field thery 
N
22 g
c
c
g
Perturbati on series of a closed string 
b
n
g
gs
2g
g
1
Nc
~ g s  string coupling
 
n
g ,n
N
g
22 g
c
f g ( )
Maldacena conjecture: Maldacena, Witten
N  4 SUSY YM on the boundary

  N c g YM

2


Nc
d
 e
In the limit
4
x0 ( x ) O ( x )


Type IIB string theory in the bulk
1

(string tension
2
4 g s
Z string [ ( x , 0 )   0 (x)]
N c   and   
Z string [ ( x , 0 )   0 ( x )]  e
I sugra [ ]  classical
 I su g ra [  ]
supergravi
| ( x , 0 )  0 ( x )
ty action
------ Euclidean signature, generalizable to Minkowski signature

1
2  
)
N_c 3branes
AdS_5 X S^5 bulk
z
AdS boundary z=0
ds
2

1 
1
2
2
2 
2



fdt

d
x

dz

d

5
2

z 
f

Matching the symmetries
N=4 SYM
Type IIB string theory
N_c colors
N_c 3-branes
4d conformal group
AdS_5 isometry group
R-charge SU(4)
S^5 isometry group
For most applications:
I sugra    I EH    I GH    I matter  ;
I EH [ ]  Einstein
I GH [ ]  Gibbons
where
5d gravitatio
  6,
action  
- Hilbert
term 
- Hawking
nal constant
 d x  dz
4
16  G 5
1
16  G 5
G5 
zh
1
G 10

3
g (R  2 )
0
lim
d x
4
z 0

g g

   g
4
with th e 10d gravitatio
44

4 

G 10 
nal constant

4
2
2N c
 ,   0,1,2,3,4 .
The role of the Gibbons-Hawking term
 I EH  
zh
1
4
16  G 5
Minkowski signature:
I EH    I GH   
1
16  G 5
 d x  dz
e

0
 I sugra
 d x  dz
4
g g
 
 e
iI sugra
1

 R   Rg
2


  g  


 
 g (R  2 ) 
1
16  G 5
lim
z 0
d
4

x g g

   g
4
44

4 

Example:
 ( x , z )  metric fluctuatio

n
O ( x )  stress tensor operator
Recipe for calculating stress tensor correlators:
1 
  fdt
2 
z 
--------- Write ds 2 
2
 dx 
2
2 


dz   h  ( x , z ) dx dx
f

1
--------- Solve the 5d Einstein equation subject the boundary condition
h  ( x , 0 )  h  ( x )
0
Near the black hole horizon:
Euclidean signature
Minkowski signature
Decaying mode only
Incoming mode for retarded correlators
Outgoing mode for advanced correlators
0
--------- Expand I EH in power series of h 
--------- Extract the coefficients.
II Some applications to N=4 SUSY YM Plasma:
Equation of state in strong coupling:
Plasma temperature = Hawking temperature
Near Schwarzschild horizon
z  z h (1   )
  1
2
ds
2
 d

2
4
2
 dt
2
2
1

zh
2
dx
2
zh
Continuating to Euclidean time, t   i 
ds
2
 d
2
4

2
zh
 d
2
2

1
2
dx
2
zh
2d polar coordinate

2 
s   ,
 
zh 

To avoid a conic singularity at   0 , the period of    z h
Recalling the Matsubara formulation
T 
1
z h
Free energy = temperature X (the gravity action without metric fluctuations)
E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998), hep-th/9803131.
Consider a 4D Euclidean space of spatial volume V_3 at z  
The EH action of AdS-Schwarzschild:
I EH 0   

1
16  G 5
zh
V 3 (  20  12 )  dt
0
dz
z

5

V3   1
1 



4
4 

8 G 5  
zh 
The EH action of plain AdS
I
(0)
EH
0  


1
16  G 5

V 3 (  20  12 )  dt 
0

dz
z
5

V3   1
8 G 5 
4
----- To eliminate the conic singularity,    z n
----- To match the proper length in Euclidean time
 
4

 

f      1 
4 

2zh 


I GH 0   I GH 0 
(0)
Plasma free energy:
F 
1

lim
 0

I 0   I 0   
(0)
EH
EH
Plasma entropy:

 F 
2
3
S  
N c T V3
 
2
  T  V3
2
V3
16  G 5 z
4
h
 

2
8
2
4
N c T V3
Bekenstein-Hawking entropy:
Gubser, Klebanov & Pest, PRD54, 3915 (1996)
S BH 
1
 (horizon
area) measured
4
in Planck units 
1

horizon
8
4
1
lP
where l P  10 d Planck length G 8
------ The metric on the horizon：
ds
2

1
2
zh
dx  d 5
2
The horizon
2
area 
V3
z
3
h
 ( the solid angle of S )   T V 3
------ The gravitational constant of the dual:
 S BH 
1
2
5
G 10  l
6
8
P
3


2
2
4
2
2N c
 N c T V 3  S plasma
3
agree with the entropy extraced from the gravity action.
area
The ratio 3/4:
The plasma entropy density at N c   and   
s  S / V3 
1
2
Nc T
2
2
3
The free field limit:
the contents of N=4 SUSY YM
number
gauge potential
1
entropy density
1
2
30
real scalars
6
6
Nc T
1
30
Weyl spinors
4
8
7
240
s
(0)
s
s
(0)


2
3
3
Nc T
2
2
2
 0 . 75
4
The lattice QCD yields
s
s0
 0 .8 .
2
3
Nc T
2
2
Nc T
2
2
3

1
5
3

Nc T
7
30
2
2
3
Nc T
2
2
3
Shear viscosity in strong coupling:
Policastro, Son and Starinets, JHEP09, 043 (2002)
y
The friction force per unit area
f  
v x
y
x
Kubo formula
   lim
0
G xy , xy ( , q ) 
where
Gravity dual:
1
R

Im G xy , xy ( , 0 )
R
 dtd x e
i t  iq  x
 ( t )  T xy ( x ), T xy ( 0 )  
2
the coefficient of the h xy term of the gravity action
 T
2
ds
2

u
where
2

u 
fdt
z
2
z
2
h
2
 dx
2

1
2
du
2
4u f
f  1 u
2
0  u 1
The metric fluctuation
 T
2
ds
2

2

u
fdt
2
 dx
2

1
2
du
2
4u f

 h  ( t , z , u ) dx dx

in the axial gauge, h  u  0
Classification according to O(2) symmetry between x and y
h xx  h yy
h xy ,
h tx ,
h ty ;
h zx ,
h zy
spin - 1
h tt ;
h zz ;
h xx  h yy
spin - 0
h xy and others!
No mixing between
 T
2
ds
spin - 2
2

u
2

fdt
2
 dx
2
Substituting into Einstein equation
 


 g x 
1
g
 
 0

x 

1
2
du
2
4u f
 2 h xy ( t , z , u ) dxdy
R   4 g   0
where
The Laplace equation of a scalar field
and linearize
  hy 
x
u
 T
2
2
h xy
Calculation details:
------ Nonzero components of the Christofel (up to symmetris):
 tt  2  T
u
2
2
u
1 f 1


2  f
u
 ut 
t
1
4

 ij  2  T
u



2
 uj  
2
1
i
f  ij
 uu  
u
 ij
2u
12
f 
 

2u
f 
i, j  x, y , z
------ Nonzero components of the Ricci tensor:
4 T
2
R tt 
2
4 T
2
R ij  
f
u
u
2
 ij
R uu  
1
2
u f
Linear expansion:



     
with
1


t
xy

x
yt



2f
y
xt

1


2
R   R   r
r
r
x
y
 4h
y
 
y
 
x
x
1
2  T 
1
2  T
2
2
nonzero
z
xy
 
x
yz

1
2
y
xz
components
,z

1
2
,z
(up to symmetries

u
xy
 2 T

x
yu

2
y
xu

2
):
f   u  , u
1
2

 ,u
with the only nonzero component
 u
    u  , zz
 f
 u
    u  , zz
 f


3   f
  2 u
  , u   4
u  u



1
 

3   f
  2 u
  ,u   



u
u

x
2

g




 gg

 


x 
The solution:
 ( t , z , u )  (1  u )
d 
2
u (1  u )
2
du
2
i
 ˆ
2
(1  u )
1
 ˆ
2
 (u ) e
i ( qz   t )
  1  (1  i )ˆ u  1  1  i ˆ u
where
2

ˆ 

2 T
qˆ 
q
2 T
2
 2

1 i
ˆ
2
  ˆ  qˆ 
ˆ  i
u   0
du
2
4 

d
Heun equation (Fucks equation of 4 canonical singularities)
------trivial when energy and momentum equatl to zero;
------low energy-momentum solution can be obtained perturbatively.
The boundary condition at horizon: u  1
i
 ˆ

i ( qz   t )
2
e
 (1  u )
 (t , z , u ) ~ 
i
ˆ

i ( qz   t )
2
(1
u
)
e


incoming
wave

retarded correlator
outgoing
wave

advanced
The incoming solution at low energy and zero momentum:
 ( t , z , u )  (1  u )
i
 ˆ
2
ˆ 1  u

2   i t
1

i
ln
 O (ˆ )  e

2
2


correlator
The quadratic
term of I EH ( )  I GH ( )
2
f   
1
2
2
4
4
2
2
4
  N c  T  du  d x 
   V 4 N c  T lim
8
u  u 
8
0
1
1
 
i
16
V 4 N T  
2
c
3
1
2
V 4 G xy , xy ( , 0 )
R
 G xy , xy ( , 0 )  
R
 
1
8
N c T
2
i
8
N c T 
2
3
V_4 = 4d spacetime volume
3
Viscosity ratio:

s
Elliptic flow of RHIC:

1
4

s
Lattice QCD: noisy
 f  


u0 
u

u


 0 . 08
 0 .1
III. Remarks:
N=4 SYM is not QCD, since
1). It is supersymmetric
2). It is conformal ( no confinement )
3). No fundamental quarks
---- 1) and 2) may not be serious issues since sQGP is in
the deconfined phase at a nonzero temperature. The
supersymmetry of N=4 SYM is broken at a nonzero T.
---- 3) may be improved, since heavy fundamental quarks
may be introduced by adding D7 branes. ( Krach & Katz)
Introducing an infrared cutoff ---- AdS/QCD:
I EH  
1
16  G 5
 dz  d
4
x ge
where the dilaton field
ds
2

1
z
2
dt
2
 d x  dz
2

  cz
2

 R  12 
2
Karch, Katz, Son & Stephenov
----- Regge behavior of meson spectrum ---- confinement;
----- Rho messon mass gives c  338 MeV;
----- Lack of string theory support.
Deconfinement phase transition:
Hadronic phase:
I EH
 
hadronic
with
ds
2
1
z
Plasma phase:
I EH
with
plasma
ds
 
2

dt
2
1
4
1
2
2
ge
 d x  dz
2
 dz  d x
4
16  G 5
z
 dz  d x
16  G 5
1

Herzog, PRL98, 091601 (2007)
1  
4
4
T z
4
dt
2
2
 R  12 

ge
2
 cz
 cz
2
 R  12 
 d x  1   T z
2
4
4
4

1
dz

Hawking-Page transition:
I EH
hadronic
 I EH
 T c  0 . 4917
plasma
c  191 MeV
2
---- First order transition with entropy jump  N c
---- Consistent with large N_c QCD because of the liberation
of quark-gluon degrees of freedom.
Thank You!