Fluctuation Partition Function of a Wilson Loop in a Strongly Coupled N=4 SYM Plasma Defu Hou...

Fluctuation Partition Function of a Wilson Loop in a Strongly Coupled N

=4 SYM Plasma

Defu Hou (CCNU), James T.Liu (U. Michigan)

and

Hai-cang Ren (Rockefeller & CCNU)

USTC， 7、 2008

Contents:

I. AdS/CFT correspondence and Wilson loopsII. Semi-classical expansionIII. Some examplesIV. Remarks

I. AdS/CFT correspondence and Wilson loops (Maldacena; Witten)

AdS/CFT corerspondence

N_c 3-braneson AdS boundary

AdS_5XS^5 bulk

coupling string 4

tensionstring2

1 couplingHooft t '

bulk in the theory string IIB Type boundary on the SYM )SU( 4

2

22

ssc

YMc

c

ggN

LgN

NN


Symmetry matching in AdS/CFT

Field theory symmetry String theory symmetry

SU(N_c) N_c 3-branes

4D conformal group AdS_5 isometry, SO(2,4)

R-symmetry, SU(4) S^5 isometry, SO(6)


0 and 0 and

tysupergravi Classical theory field large coupledStrongly

0

theorystring coupledly Weak theory field Large

sc

c

sc

c

gN

N

gN

N

Leading order results:

Equation of state ( Witten ): )0(322

4

3

2

1sTNs c

Viscosity ratio (Policastro,Son & Starinets ):

4

1

s

Jet quenching (Liu,Rajagopal & Wiederman): 32

3

4

54

3

ˆ Tq

And many others.

AdS/CFT correspondence

But N=4 SYM is not QCD! It is supersymmetric; It is conformal ( no confinement ); No fundamental quarks; It is large N_c.

---- 1, 2 may not be serious issues for QGP;

---- Attempts to add quark flavors; ---- Attempts to introduce IR cutoff.

Gravity dual of a Wilson loop:

][

2expexp][ min CS

iAdxiCW

C

)(xA = the gauge potential of N=4 SYM; C = a loop on the AdS boundary z=0;

CCS by bounded area minimum the.min

The metric of 55 SAdS

0for 1 2

5222

22 Tddzddt

zds x

The metric of 55 SAdS

444

25

2222

2

1 where

0for 11

zTf

Tddzf

dfdtz

ds

x

-Schwarzschild


][

2expexp][ min CS

iAdxiCW

C

C

AdS boundary z=0

AdS bulk

z

x

t


Heavy quark self energy

Quark-antiquark potential J. Madacena; S. J. Rey et. al.

Jet-quenching parameterH. Liu et. al.

x

x

t

t

C on the boundary Implied physical quantity

Comparison with RHIC physics: 65.5

Finite coupling correction:

.

...

][][

2][ln .min

CbCSiCW

---- b[C] comes from the fluctuation of the string world sheet around the one of minimum area. ---- Has been considered by Forste, Ghoshal, Theisen and by Drukker, Gross, Tesytlin at T=0.

---- Generalization to nonzero T.

Finite N_c correction: String interaction, very difficult.

II. Semi-classical expansion:

,2

exp]][[const. where

][2

exp][

)2(

min

XSi

dXdZ

ZCSi

CW

Classical solution:

here). ildSchwarzschS(AdS space target 10din

embeddedsheet world2d a of area the][action Goto-Nambu The

][][

55

.min

XS

XSCS

NG

NG

Target space metric

dXdXXGds )(2

jiijji

ij

XXgddgds

)( )(2

)( XX

World sheet metric

gdS 2NG ]X[

II. Semi-classical expansion:

,2

exp]][[const. where

][2

exp][

)2(

min

XSi

dXdZ

ZCSi

CW

Quadratic fluctuations: 0 , XXX

][][],[ )2()2()2( FB SXSXS

where theta=fermionic coordinate.

FB ZZZ Need to explore the full super multiplet of the world sheet.

extracted from Metsaev-Tseytlin action

Bosonic fluctuations:

]4

1

[2

1

][][][

11

ˆˆˆˆˆ

ˆ2

)2(

klijklijjkiljlik

bajibaaj

ai

ij

NGNGB

gggggggg

EERggd

XSXXSXS

Decompose the X into its eight tangent components and

two longitudinal ones:

0)()( and 0)( log.logtr.

XXXj

tr.tr.tr.)2( ~

2

1][ AXSB operator.matrix 88 a is tr. A

We find that )(det tr.2

1

AZ B

aijjia

ij Eg ˆˆ

1aa )( XE

，

Fermionic fluctuations:

22112)2( jiij

jiij

F DPDPgdiS

jIJ

baab

jjIJIJ

j

ijijij

aa

jj

iD

ggPE

2,

8

1)(

matrix gamma 1616 a with a

-symmetry:

0 2 )2( FIj

jI Si

where is dependent and i

jiji

jij PP 2211

Gauge fixing: 21

1,2I spinor Majorana components 16 where ，I

Fermionic fluctuations:

jiij

baab

jjiij

F

iggdiS

2,

8

12 2)2(

Choose the 10-beins a

Eˆ

’s such that two of them, 0,1 with E

are tangent to the embedding world sheet, )(X

jjjjjj eeE and

where jje and are the world sheet zweibeins and spin connection.

Kjjbaab

jj

,

8

1,

8

1

For the world sheets considered below K does not contribute (not in general!)

---- Write 8182180810 and IIIIi

----Pack into eight 2-component Majorana spinors.

8

13

28

1

)2( 2ˆ2n

nFnnjj

nn

F AiiegdiS

where

，diagjjj .,4

1ˆ

Finally

)2(24

4

11ˆˆdet)(det RAZ j

jFF

III. Examples:

O

x

z

t

Embedding ： 0) 0, 0, 0, 0, , 0, 0, 0, ,(),( X

World sheet metric:

22

22 11

df

fdds with 4

4

1hz

f

Zweibeins:

df

de jj 0

d

fde j

j11

Spin connections:

dz

dh

jj

4

401 1

1

Curvature:

4

4)2( 3

12hz

R

A straight string

Z_h

Transverse fluctuations:

)),( , ),,( ),,( ),,( ,(

)),( , ),,( ),,( ),,( ,(321

321

s

sX

s=5, 6, 7, 8, 9

Substituting into the Nambu-Goto action we find

3

1

9

5

2222222)2( )'()()()'()(2

1

a s

ssaaaNG ggMgggdS

where 33

812 and ,

)2(

4

42 R

zM

h

We have

52

3)2(2

tr.

0

0 3

1

3

8

I

IRA

Partition function: )(det

2

1

3

8det

4

11det

4

11-det

22

5)2(22

3

)2(22)2(22

R

RR

ZZZ FB

A pair of parallel lines:

O z

Embedding: 0) 0, 0, 0, 0, ),( 0, 0, , ,( zX

Z_0

0

022

0

440

4

44440

22

h

h

h

zzz

rz

rz

zzz

zzzz

d

dzz

，，

World sheet metric:

with 4

40

04

4

1 and 1hh z

zf

z

zf

Zweibeins: dz

fde j

j 0 df

z

zde j

j

21 1

1

Spin connections: dz

z

fz

zfd

h

jj

4

4

20

001 12

Curvature: 2

2)2( )32(2)2(4

zf

fzfR

The world sheet tangent vectors: 0) 0, 0, 0, 0, , 0, 0, 1, ,0(

0) 0, 0, 0, 0, 0, 0, 0, 0, ,1(

1

0

z

The transverse bosonic fluctuation:

f

z tan 0

9 8, 6,7, ,5 ,sin , , ,cos- ,0

10

s1321

XX

sfzzzX

2

40

402

22 d

zf

zd

z

fds

Substituting ),(),(),( XXX

into the Nambu-Goto action we find

9

5

22

3

2

2223

222121

2121

2)2(

)'()(

)()'()()()'()(

2

1

s

ss

a

aaa

NG

gg

MggMgg

gdS

3322

2

2223

)2(21

)1(2 with 2 and 24 CC

zf

fzMRM j

jj

j

where

)2(223

21 82 RMM the same as the case without the black hole

We have

52

22

)2(2

tr.

0 0

0 2 0

0 0 24

I

I

R

A

Partition function:

)()det2det(-24det

4

11det

4

11-det

22

52)2(22

1

)2(22)2(22

R

RRZ

IV. Remarks

UV divergence:

------ Quadratic divergence is cancelled between bosons and fermions. Z=1 for zero world sheet curvature and zero target space curvature.------ Logarithmic divergence:

096

1 curvature spaceTarget

al topologic curvaturesheet World )2(2

FFR

Rgd

------ The black hole does not introduce new UV divergences

Analog of an ordinary field theory:A nonzero temperature does not introduce new UV

divergences.

Method for computing the determinant ratio (Kruczenski & Tirziu)

Given

under Dirichlet boundary condition

Generalization to more complicated loops, such asA pair of oblique parallel lines (boosted quark-antiquark potential);A pair of light-like parallel lines (jet-quenching).

2 ,1 )(2

2

jbxaxVdx

dM ij

)(

)(

det

det

2

1

2

1

b

b

M

M

.1)( 0)( with 0 where aaM jjjj

Thank You!

Fluctuation Partition Function of a Wilson Loop in a Strongly Coupled N=4 SYM Plasma Defu Hou...

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