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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 :. scalar field :. - PowerPoint PPT Presentation
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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
2x
xax
xxx 2
22
1)(
x
x
xx
22222 )(
21)(
x
xA
x
xx
x
xA
xxA
222 )()(
x
x
x
xx
I. AdS/CFT correspondence The inversion invariance of a massless field theory in 4D:
scalar field:
vector field:
spinor field:
The conformal group in 4D:
Poincare transformations
10 generators: MP ,
Dilatation = Inversion x inversion
xx 1 generators: D
Special conformal transformation = inversion x translation x inversion:
22
2
21 xaxa
xaxx
4 generators: K
Lie algebra of the 15 generators: DiiMKP
iKKD
KKiKM
iPPD
nspermutatioMiMM
PPiPM
22,
,
,
,
,
,
other commutators vanish.
Under the conformal group:
The conformal symmetry at quantum level requires
0)( g
since )(gT
4
Classical Quantum
massless Yes No
massless QCD Yes No
N=4 Supersymmetric Yang-Mills Yes Yes
The conformal group and O(2,4) O(2,4) = rotation group of M(2,4)
M(2,4) = 6D Minkowski space of signature (-, -, +, +, +, +):
21
24
2 dXdXdXdXds
3 2, 1, ,0
Introduce 14
XX
Xx
4D Lorentz transformation: O(1,3) subgroup among X’s
4D Dilatation: O(2) rotation
coshsinh
sinhcosh
141
144
XXX
XXX
XX
4D Translation (infinitesimal): O(2,4) transformation
XbXX
XbXX
XXbXX
11
44
14 )(
4D Special conformal transformation (infinitesimal): O(2,4) transformation
XaXX
XaXX
XXaXX
11
44
14 )(
AdS5:
Isometry group: O(2,4)
A hyperboloid in M(2,4)
Metric:
24
2
4424
21
24
2
1
XXX
dXXdXXdXdXdX
dXdXdXdXds
or 222
22 1
dzddtz
ds x
where
zzx
zzxX
z
xX
1
2
11
1
2
1
2
1 224
Space of a constant curvature
ggggR
gR 4 20R
221
24 LXXXX
Throughout this lecture, we set the AdS radius L=1.
AdS5-Schwarzschild
4
4
1hz
zf
A black hole at hzz Hawking temperature
hzT
1
(Plasma temperature)
Curvature :
CggggR
ijh
ji z
fC
400 ijh
ji fzC
444
1 3,2,1,,, lkji
40404
3
hzC
The same Ricci tensor and curvature scalar as AdS5
gR 4 20R
222
22 11
dzf
dfdtz
ds x
)(1
4 jkiljlikh
ijkl zC
:55 SAdS
25
2222
2 11
ddz
fdfdt
zds x
The metric:
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
ildSchwarzsch-55 SAdS
25
2222
2 11
ddz
fdfdt
zds x
:55 SAdS
Large Nc field theory: t’Hooft
)]~~~~
Tr()~
d~~
Tr()~~
([Tr~
)Tr()dTr()(Tr~
2
2
lkjiijkl
YMkjiijk
YMiic
lkjiijkl
YMkjiijk
YMii
dgcgddN
dgcgdd
2 couplingHooft t ' theand
)( oftion representaadjoint esupport th ~
where
YMc
ciYMi
gN
NSUg
Power counting of a Feynman diagram at large Nc:
gg
gc
g n
nng
gc
VEc
Fc
V
c
E
c
cc
c
fNcN
gFEV
NNN
N
NN
N
)(serieson Perturbati
22 ltopologica where
~
loops F and verticesV s,propagator E of diagram vacuuma
~ loop a , ~ vertex a ,~ propagator a
22,
22
Dominated by the diagram with lowest g, -------- the planar diagram ~ a string world sheet
Large Nc field theory:
A planar diagram
A non-planar diagram
A handle free world sheet
A world sheet with a handle
handles of no.
sheet worlda of genus the
g
coupling string~1
g string closed a of serieson Perturbati
)( theryfield theof serieson Perturbati
2s
22,
22
sc
g
gg
gg
gc
g n
nng
gc
gN
b
fNcN
Maldacena conjecture: Maldacena, Witten
(x)])0,([
4
)2
1 tension(string
1
bulk in the theory string IIB Type boundary on the YM SUSY 4
0string
)()(
22
04
xZe
gN
gN
N
xOxxd
sc
YMc
actionty supergravi classical][
|)]()0,([
and limit In the
sugra
)()0,(][
0string 0
sugra
I
exxZ
N
xxI
c
------ Euclidean signature, generalizable to Minkowski signature
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
N_c 3branes
AdS_5 X S^5 bulk
AdS boundary z=0
z
25
2222
2 11
ddz
fdfdt
zds x
Matching the symmetries
For most applications:
Minkowski signature:
4
44440
5
4
5
lim16
1)2(
16
1gggxd
GRgdzxd
GII zGHEH
The role of the Gibbons-Hawking term
gRgRggdzxdG
Ihz
EH 2
1
16
1
0
4
5
sugrasugra iII ee
. 0,1,2,3,4 , ,6
2constant nalgravitatio 10d e with thconstant nalgravitatio 5d where
lim16
1 termHawking-Gibbons][
)2(16
1actionHilbert -Einstein][
;
2
4
10310
5
44444
05
0
4
5
mattersugra
c
zGH
z
EH
GHEH
NG
GG
gggxdG
I
RgdzxdG
I
IIII
h
Example:
Recipe for calculating stress tensor correlators:
--------- Write dxdxzxhdz
fdfdt
zds ),(
11 222
2
2x
--------- Solve the 5d Einstein equation subject the boundary condition
--------- Expand in power series of 0h
--------- Extract the coefficients.
EHI
)()0,( 0 xhxh
operator tensor stress)( n fluctuatio metric),( xOzx
Near the black hole horizon:
Euclidean signature Minkowski signature
Decaying mode only Incoming mode for retarded correlatorsOutgoing mode for advanced correlators
II Some applications to N=4 SUSY YM Plasma:
Equation of state in strong coupling: Plasma temperature = Hawking temperature
22
222
22
2
14
1 )1(
xdz
dtz
dds
zz
hh
h
Near Schwarzschild horizon
Continuating to Euclidean time, it
hhh zd
zd
zdds
2 , scoordinatepolar 2d
14 22
222
22 x
To avoid a conic singularity at 0 , the period of hz
Recalling the Matsubara formulation
hzT
1
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 atThe EH action of AdS-Schwarzschild:
z
44
5
3
053
5
11
8)1220(
16
10
h
z
EH zG
V
z
dzdtV
GI
h
The EH action of plain AdS
4
5
3
053
5
)0( 1
8)1220(
16
10
G
V
z
dzdtV
GI EH
----- To eliminate the conic singularity,----- To match the proper length in Euclidean time
nz
00 2
1 )0(4
4
GHGHh
IIz
f
Plasma free energy:
342
2
45
3)0(
0 81600lim
1VTN
zG
VIIF c
hEHEH
Plasma entropy:
332
2
23
VTNT
FS c
V
Bekenstein-Hawking entropy:
8
areahorizon
4
1unitsPlanck in measured area)(horizon
4
1
PBH l
S
8
1
length Planck d10 where GlP
------ The metric on the horizon :
3365
33
25
22
2
) of angle solid the( areahorizon The
1
VTSz
V
ddz
ds
h
h
x
------ The gravitational constant of the dual: 2
48
10 2 cP N
lG
plasmacBH SVTNS 3322
2
1
agree with the entropy extraced from the gravity action.
Gubser, Klebanov & Pest, PRD54, 3915 (1996)
The ratio 3/4:
The plasma entropy density at and cN322
3 2
1/ TNVSs c
The free field limit:
322322
30
7
240
78 TNTN cc
the contents of N=4 SUSY YM number entropy density
gauge potential 1
real scalars 6
Weyl spinors 4
322322
5
1
30
16 TNTN cc
322
30
1TN c
222)0(
3
2TNs c
The lattice QCD yields
75.04
3)0(
s
s
.8.00
s
s
Shear viscosity in strong coupling:
Kubo formula
Gravity dual: the coefficient of the
y
x
y
vf x
The friction force per unit area
Policastro, Son and Starinets, JHEP09, 043 (2002)
22
2222
2
4
1du
fudfdt
u
Tds x
where 10 1 22
2
uufz
zu
h
2xyh term of the gravity action
)0(),()(),(
)0,(Im1
lim
,
,0
xyxyxitiR
xyxy
Rxyxy
TxTtedtdG
G
qxqwhere
The metric fluctuation
Classification according to O(2) symmetry between x and y
No mixing between and others!xyh
Substituting into Einstein equation 04 gR and linearize
The Laplace equation of a scalar field
dxdxuzthdu
fudfdt
u
Tds ),,(
4
1 22
2222
2 x
in the axial gauge, 0uh
dxdyuzthdufu
dfdtu
Tds xy ),,(2
4
1 22
2222
2 x
xyxy h
T
uh
xg
xg 22 where 0
1
0-spin ; ;
1-spin , ; ,
2-spin ,
yyxxzztt
zyzxtytx
yyxxxy
hhhh
hhhh
hhh
Calculation details:
zyxjiuuf
f
f
f
ufTuT
ijiuj
tut
uuuij
uij
utt
,,, 2
1
1
2
1
2
2
1 2 12 22422
------ Nonzero components of the Christofel (up to symmetris):
fuR
u
TRf
u
TR uuijijtt 2
2222 1
4
4
------ Nonzero components of the Ricci tensor:
uyxu
xyuz
yxz
xyz
yxt
xyt
uuxyz
zxy
txy ufT
f
,2
1 ,
2
1
2
1
,2 ,2
1
2
1
:)symmetries to(up components nonzero with
22
Linear expansion:
4,2,2
1
component nonzeroonly the with
32
uzzyx
u
f
uuu
f
u
Tr
rRR
x
ggxgu
f
uuu
f
u
Thr uzzy
xy
x
2
1,2,
2
14 3
2
The solution:
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: 1u
correlator advanced waveoutgoing )-(1
correlator retarded waveincoming )1(~),,(
)(ˆ
2
)(ˆ
2
tqzii
tqzii
eu
euuzt
The incoming solution at low energy and zero momentum:
tii
eOu
iuuzt
)ˆ(
2
1ln
2
ˆ1)1(),,( 2
ˆ2
T
Teuuuuzt tqzi
i
2ˆ
2ˆ where)()1()1(),,( )(
ˆ2
1ˆ
2
04
ˆˆ
2
1ˆˆˆ11ˆ)1(1)1(
2222
2
22
uii
qdu
duiui
du
duu
32, 8
)0,( TNi
G cR
xyxy
32
8
1TN c
Viscosity ratio: 08.04
1
s
Elliptic flow of RHIC:
Lattice QCD: noisy
1.0s
V_4 = 4d spacetime volume
)0,(2
1
16
lim8
1
8
1
)()( of termquadratic The
,432
4
0422
4
1
0
24422
Rxyxyc
ucc
GHEH
GVTNVi
uu
fTNV
uu
fxdduTN
II
III. Remarks:N=4 SYM is not QCD, since1). It is supersymmetric2). 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:
2222
2
2
4
5
1
fielddilaton thewhere
1216
1
dzddtz
ds
cz
RegxddzG
I EH
x
----- Regge behavior of meson spectrum ---- confinement;----- Rho messon mass gives ----- Lack of string theory support.
MeV; 338c
Karch, Katz, Son & Stephenov
Deconfinement phase transition: Herzog, PRL98, 091601 (2007)
Hadronic phase:
Plasma phase:
2222
2
4
5hadronic
1with
1216
1 2
dzddtz
ds
RegxddzG
I czEH
x
dzzTddtzTz
ds
RegxddzG
I czEH
1444224442
2
4
5plasma
111
with
1216
1 2
x
Hawking-Page transition:
---- First order transition with entropy jump
MeV1914917.0
plasmahadronic
cT
II
c
EHEH
2cN
---- Consistent with large N_c QCD because of the liberation of quark-gluon degrees of freedom.
Thank You!