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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
AdS/CFT corerspondence
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)
AdS/CFT corerspondence
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
Gravity dual of a Wilson loop:
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