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

qq

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!