andrei starinets rudolf peierls centre for theoretical physics...
TRANSCRIPT
Transport in strongly coupled QFTsand
gauge/gravity duality
Andrei Starinets
Moscow State UniversityMoscowRussia
22 August 2009
Rudolf Peierls Centre for Theoretical Physics
Oxford University
XIV Lomonosov Conference
Heavy ion collision experiments at RHIC (2000-current) and LHC (2009-??)create hot and dense nuclear matter known as the “quark-gluon plasma”
Evolution of the plasma “fireball” is described by relativistic fluid dynamics
(note: qualitative difference between p-p and Au-Au collisions)
Need to know
thermodynamics (equation of state)kinetics (first- and second-order transport coefficients)in the regime of intermediate coupling strength:
(relativistic Navier-Stokes equations): Landau; Bjorken
initial conditions (initial energy density profile)thermalization time (start of hydro evolution)
freeze-out conditions (end of hydro evolution)
Elliptic flow, jet quenching… - focus on transport in this talk
Quantum field theories at finite temperature/density
Equilibrium Near-equilibrium
entropyequation of state
…….
transport coefficientsemission rates
………
perturbative non-perturbativepQCD Lattice
perturbative non-perturbativekinetic theory ????
Thermodynamics Kinetics
Figure: an artistic impression from Myers and Vazquez, 0804.2423 [hep-th]
Energy density vs temperature for various gauge theories
Ideal gasof quarksand gluons
Ideal gasof hadrons
AdS/CFT correspondence
conjecturedexact equivalence
Generating functional for correlationfunctions of gauge-invariant operators String partition function
In particular
Classical gravity action serves as a generating functional for the gauge theory correlators
Latest test: Janik’08
• Field content:
• Action:
Gliozzi,Scherk,Olive’77Brink,Schwarz,Scherk’77
(super)conformal field theory = coupling doesn’t run
supersymmetric YM theory
Dual to QCD? (Polchinski-Strassler)
coupling
energy
Dual to QCD? (Polchinski-Strassler)
coupling
energy
Dual to QCD? (Polchinski-Strassler)
coupling
energy
At zero temperature, N=4 SYM is obviously a very bad approximation to QCD
At finite temperature it is qualitatively similar to QCD
non-Abelian plasma (with additional d.o.f.)
area law for spatial Wilson loops
Debye screening
spontaneous breaking of symmetry at high temperature
supersymmetry broken
However:
hydrodynamics
Hydrodynamics: fundamental d.o.f. = densities of conserved charges
Need to add constitutive relations!
Example: charge diffusion
[Fick’s law (1855)]
Conservation law
Constitutive relation
Diffusion equation
Dispersion relation
Expansion parameters:
First-order transport (kinetic) coefficients
* Expect Einstein relations such as to hold
Shear viscosity
Bulk viscosity
Charge diffusion constant
Supercharge diffusion constant
Thermal conductivity
Electrical conductivity
Second-order transport (kinetic) coefficients
Relaxation time
Second order trasport coefficient
Second order trasport coefficient
Second order trasport coefficient
Second order trasport coefficient
(for theories conformal at T=0)
In non-conformal theories such as QCD, the total number of second-order transportcoefficients is quite large
M,J,QHolographically dual system
in thermal equilibrium
M, J, Q
T S
Gravitational fluctuations Deviations from equilibrium
????
and B.C.
Quasinormal spectrum
10-dim gravity4-dim gauge theory – large N,
strong coupling
Computing transport coefficients from “first principles”
Kubo formulae allows one to calculate transportcoefficients from microscopic models
In the regime described by a gravity dual the correlator can be computed using
the gauge theory/gravity duality
Fluctuation-dissipation theory(Callen, Welton, Green, Kubo)
Example: stress-energy tensor correlator in
in the limit
Finite temperature, Mink:
Zero temperature, Euclid:
(in the limit )
The pole (or the lowest quasinormal freq.)
Compare with hydro:
In CFT:
Also, (Gubser, Klebanov, Peet, 1996)
First-order transport coefficients in N = 4 SYMin the limit
Shear viscosity
Bulk viscosity
Charge diffusion constant
Supercharge diffusion constant
Thermal conductivity
Electrical conductivity
(G.Policastro, 2008)
for non-conformal theories see Buchel et al; G.D.Moore et al
Gubser et al.
Shear viscosity in SYM
Correction to : Buchel, Liu, A.S., hep-th/0406264
perturbative thermal gauge theoryS.Huot,S.Jeon,G.Moore, hep-ph/0608062
Buchel, 0805.2683 [hep-th]; Myers, Paulos, Sinha, 0806.2156 [hep-th]
Electrical conductivity in SYM
Weak coupling:
Strong coupling:
* Charge susceptibility can be computed independently:
Einstein relation holds:
D.T.Son, A.S., hep-th/0601157
Universality ofTheorem:For a thermal gauge theory, the ratio of shear viscosity to entropy density is equal to in the regime described by a dual gravity theory
Remarks:
• Extended to non-zero chemical potential:
• Extended to models with fundamental fermions in the limit
• String/Gravity dual to QCD is currently unknown
Benincasa, Buchel, Naryshkin, hep-th/0610145
Mateos, Myers, Thomson, hep-th/0610184
Universality of shear viscosity in the regime described by gravity duals
Graviton’s component obeys equation for a minimally coupled massless scalar. But then .Since the entropy (density) is we get
Three roads to universality of
The absorption argumentD. Son, P. Kovtun, A.S., hep-th/0405231
Direct computation of the correlator in Kubo formula from AdS/CFT A.Buchel, hep-th/0408095
“Membrane paradigm” general formula for diffusion coefficient + interpretation as lowest quasinormal frequency = pole of the shear mode correlator + Buchel-Liu theoremP. Kovtun, D.Son, A.S., hep-th/0309213, A.S., 0806.3797 [hep-th],P.Kovtun, A.S., hep-th/0506184, A.Buchel, J.Liu, hep-th/0311175
Chernai, Kapusta, McLerran, nucl-th/0604032
A viscosity bound conjecture
P.Kovtun, D.Son, A.S., hep-th/0309213, hep-th/0405231
Minimum of in units of
?
A hand-waving argument
Gravity duals fix the coefficient:
Thus ?
Shear viscosity - (volume) entropy density ratiofrom gauge-string duality
In particular, in N=4 SYM:
In ALL theories (in the limit where dual gravity valid) :
Y.Kats and P.Petrov: 0712.0743 [hep-th]
M.Brigante, H.Liu, R.C.Myers, S.Shenker and S.Yaida: 0802.3318 [hep-th], 0712.0805 [hep-th].
Other higher-derivative gravity actions
R.Myers,M.Paulos, A.Sinha: 0903.2834 [hep-th] (and ref. therein – many other papers)
for superconformal Sp(N) gauge theory in d=4
Also: The species problem: T.Cohen, hep-th/0702136; A. Dolbado, F.Llanes-Estrada: hep-th/0703132
Shear viscosity - (volume) entropy density ratio in QCD
The value of this ratio strongly affects the elliptic flow in hydro models of QGP
Viscosity-entropy ratio of a trapped Fermi gas
0.2 0.4 0.6 0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
η/s
T/TF
T.Schafer, cond-mat/0701251
(based on experimental results by Duke U. group, J.E.Thomas et al., 2005-06)
Vis
cosi
ty “m
easu
rem
ents
”at R
HIC
01
23
40510152025
p T (
GeV
)
v2 (%)
ST
AR
PH
EN
IX
EO
S Q
EO
S H
h+/−
Vis
cosi
ty is
ON
E o
f the
par
amet
ers
used
in th
e hy
dro
mod
els
desc
ribin
g th
e az
imut
hala
niso
tropy
of p
artic
le d
istri
butio
n
-elli
ptic
flow
for
parti
cle
spec
ies
“i”
Elli
ptic
flow
repr
oduc
ed fo
r
e.g.
Bai
er, R
omat
schk
e, n
ucl-t
h/06
1010
8
Per
turb
ativ
eQ
CD
:
SY
M:
Che
rnai
, Kap
usta
, McL
erra
n, n
ucl-t
h/06
0403
2
01
23
4p T
[GeV
]
05101520253035v2 (percent)
STA
R n
on-f
low
cor
rect
ed (
est)
.ST
AR
eve
nt-p
laneC
GC
η/s=
10-4
η/s=
0.08
η/s=
0.16
η/s=
0.24
Luzu
man
d R
omat
schk
e, 0
804.
4015
[nuc
-th]
Elli
ptic
flow
with
col
or g
lass
con
dens
ate
initi
al c
ondi
tions
01
23
4p T
[GeV
]
0510152025v2 (percent)
STA
R n
on-f
low
cor
rect
ed (
est.)
STA
R e
vent
-pla
neGla
uber
η/s=
10-4
η/s=
0.08
η/s=
0.16
Luzu
man
d R
omat
schk
e, 0
804.
4015
[nuc
-th]
Elli
ptic
flow
with
Gla
uber
initi
al c
ondi
tions
Vis
cosi
ty/e
ntro
py ra
tio in
QC
D: c
urre
nt s
tatu
s
QC
D:R
HIC
elli
ptic
flow
ana
lysi
s su
gges
ts
Theo
ries
with
gra
vity
dua
ls in
the
regi
me
whe
re th
e du
al g
ravi
ty d
escr
iptio
n is
val
id
(uni
vers
al li
mit)
QC
D:(
Indi
rect
) LQ
CD
sim
ulat
ions
H.M
eyer
, 080
5.45
67 [h
ep-th
]
Trap
ped
stro
ngly
cor
rela
ted
cold
alk
ali a
tom
s
Liqu
id H
eliu
m-3
T.S
chaf
er, 0
808.
0734
[nuc
l-th]
[Kov
tun,
Son
& A
.S] [
Buc
hel]
[Buc
hel &
Liu
, A.S
]
Spe
ctra
l sum
rule
s fo
r the
QG
P
0.5
1
123456
0.1
0.2
0.1
0.05
w0.5
11.5
w
2468
10
P.R
omat
schk
e, D
.Son
, 090
3.39
46 [h
ep-p
h]
In N
=4 S
YM
at A
NY
cou
plin
g
Pho
topr
oduc
tion
rate
in S
YM
02
46
8k
�T
0.005
0.01
0.015
0.02
d�Γ�dk�ΑEMNc2T3�
�1
SYM,
Λ�
�
SYM,
Λ0.5
SYM,
Λ0.2
(Nor
mal
ized
) pho
ton
prod
uctio
n ra
te in
SY
M fo
r var
ious
val
ues
of‘t
Hoo
ftco
uplin
g
Oth
er a
venu
es o
f (re
late
d) re
sear
ch
Bul
k vi
scos
ity fo
r non
-con
form
al th
eorie
s (B
uche
l, B
enin
casa
, Gub
ser,
Moo
re…
)
Non
-rel
ativ
istic
gra
vity
dua
ls (S
on, M
cGre
evy,
…)
Gra
vity
dua
ls o
f the
orie
s w
ith S
SB
, AdS
/CM
T (K
ovtu
n, H
erzo
g, H
artn
oll,
Hor
owitz
…)
Bul
k fro
m th
e bo
unda
ry, t
ime
evol
utio
n of
QG
P (J
anik
,…)
Nav
ier-
Sto
kes
equa
tions
and
thei
r gen
eral
izat
ion
from
gra
vity
(Min
wal
la,…
)
Qua
rks
mov
ing
thro
ugh
plas
ma
(Che
sler
, Yaf
fe, G
ubse
r,…)
S. H
artn
oll
“Lec
ture
s on
hol
ogra
phic
met
hods
for c
onde
nsed
mat
ter p
hysi
cs”,
0903
.324
6 [h
ep-th
]
C. H
erzo
g“L
ectu
res
on h
olog
raph
ic s
uper
fluid
ityan
d su
perc
ondu
ctiv
ity”,
0904
.197
5 [h
ep-th
]
M. R
anga
man
i“G
ravi
ty a
nd h
ydro
dyna
mic
s: L
ectu
res
on th
e flu
id-g
ravi
ty c
orre
spon
denc
e”,
0905
.435
2 [h
ep-th
]
New
dire
ctio
ns
THA
NK
YO
U
Hyd
rody
nam
ic p
rope
rties
of s
trong
ly in
tera
ctin
g ho
t pla
smas
in 4
dim
ensi
ons
can
be re
late
d (fo
r cer
tain
mod
els!
)
to fl
uctu
atio
ns a
nd d
ynam
ics
of 5
-dim
ensi
onal
bla
ck h
oles
AdS
/CFT
cor
resp
onde
nce:
the
role
of J
satis
fies
linea
rized
supe
rgra
vity
e.o.
m. w
ith b
.c.
For a
giv
en o
pera
tor
, i
dent
ify th
e so
urce
fiel
d ,
e.g
.
To c
ompu
te
corre
lato
rsof
,
one
nee
ds to
sol
ve th
e bu
lksu
perg
ravi
tye.
o.m
. for
and
com
pute
the
on-s
hell
actio
nas
a fu
nctio
nal o
f the
b.c
.
Then
, tak
ing
func
tiona
l der
ivat
ives
of
giv
es
The
reci
pe:
War
ning
:e.o
.m. f
or d
iffer
ent b
ulk
field
s m
ay b
e co
uple
d: n
eed
self-
cons
iste
nt s
olut
ion
Hol
ogra
phy
at fi
nite
tem
pera
ture
and
den
sity
Non
zero
exp
ecta
tion
valu
es o
f ene
rgy
and
char
ge d
ensi
ty tr
ansl
ate
into
nont
rivia
l bac
kgro
und
valu
es o
f the
met
ric (a
bove
ext
rem
ality
)=ho
rizon
and
elec
tric
pote
ntia
l = C
HA
RG
ED
BLA
CK
HO
LE (w
ith fl
at h
oriz
on)
tem
pera
ture
of t
he d
ual g
auge
theo
ry
chem
ical
pot
entia
l of t
he d
ual t
heor
y
Gau
ge-s
tring
dua
lity
and
QC
D
App
roac
h I:
use
the
gaug
e-st
ring
(gau
ge-g
ravi
ty) d
ualit
y to
stu
dyN
=4 S
YM
and
sim
ilar t
heor
ies,
get
qua
litat
ive
insi
ghts
into
rele
vant
asp
ects
of Q
CD
, loo
k fo
r uni
vers
al q
uant
ities
App
roac
h II:
botto
m-u
p (a
.k.a
. AdS
/QC
D) –
star
t with
QC
D, b
uild
gra
vity
dua
l ap
prox
imat
ion
App
roac
h III
:sol
ve Q
CD
App
roac
h III
a: p
QC
D(w
eak
coup
ling;
pro
blem
s w
ith c
onve
rgen
ce fo
r the
rmal
qua
ntiti
es)
App
roac
h III
b: L
QC
D (u
sual
latti
ce p
robl
ems
+ pr
oble
ms
with
kin
etic
s)
(exa
ct s
olut
ions
but
lim
ited
set o
f the
orie
s)
(unl
imite
d se
t of t
heor
ies,
app
roxi
mat
e so
lutio
ns, s
yste
mat
ic p
roce
dure
unc
lear
)
(will
not c
onsi
der h
ere
but s
ee e
.g. G
ürso
y, K
irits
is, M
azza
nti,
Nitt
i, 09
03.2
859
[hep
-th])
Ove
r the
last
sev
eral
yea
rs, h
olog
raph
ic (g
auge
/gra
vity
dua
lity)
met
hods
wer
e us
ed to
stu
dy s
trong
ly c
oupl
ed g
auge
theo
ries
at fi
nite
tem
pera
ture
and
dens
ity
Thes
e st
udie
s w
ere
mot
ivat
ed b
y th
e he
avy-
ion
colli
sion
pro
gram
s at
RH
ICan
d LH
C (A
LIC
E, A
TLA
S) a
nd th
e ne
cess
ity to
und
erst
and
hot a
nd d
ense
nucl
ear m
atte
r in
the
regi
me
of in
term
edia
te c
oupl
ing
As
a re
sult,
we
now
hav
e a
bette
r und
erst
andi
ng o
f the
rmod
ynam
ics
and
espe
cial
ly k
inet
ics
(tran
spor
t) of
stro
ngly
cou
pled
gau
ge th
eorie
s
Of c
ours
e, th
ese
calc
ulat
ions
are
don
e fo
r the
oret
ical
mod
els
such
as
N=4
SY
Man
d its
cou
sins
(inc
ludi
ng n
on-c
onfo
rmal
theo
ries
etc)
.
We
don’
t kno
w q
uant
ities
suc
h as
for Q
CD
New
tran
spor
t coe
ffici
ents
in
Sou
nd d
ispe
rsio
n:
Kub
o:
SY
M
Per
turb
atio
nth
eory
Latti
ce
Our
und
erst
andi
ng o
f gau
ge th
eorie
s is
lim
ited…
Per
turb
atio
nth
eory
Latti
ce
Con
ject
ure:
sp
ecifi
c ga
uge
theo
ry in
4 d
im =
spec
ific
strin
g th
eory
in 1
0 di
m
Per
turb
atio
nth
eory
Latti
ce
Dua
l gra
vity
In p
ract
ice:
gra
vity
(low
ene
rgy
limit
of s
tring
theo
ry) i
n 10
dim
=4-
dim
gau
ge th
eory
in a
regi
on o
f a p
aram
eter
spa
ce
Can
add
fund
amen
tal f
erm
ions
with
Com
putin
g re
al-ti
me
corr
elat
ion
func
tions
from
gr
avity
To e
xtra
ct tr
ansp
ort c
oeffi
cien
ts a
nd s
pect
ral f
unct
ions
from
dua
l gra
vity
,w
e ne
ed a
reci
pe fo
r com
putin
g M
inko
wsk
ispa
ce c
orre
lato
rsin
AdS
/CFT
The
reci
pe o
f [D
.T.S
on&
A.S
., 20
01]a
nd [C
.Her
zog
& D
.T.S
on, 2
002]
rela
tes
real
-tim
e co
rrel
ator
sin
fiel
d th
eory
to P
enro
se d
iagr
am o
f bla
ck h
ole
in d
ual g
ravi
ty
Qua
sino
rmal
spec
trum
of d
ual g
ravi
ty =
pol
es o
f the
reta
rded
cor
rela
tors
in 4
d th
eory
[D.T
.Son
& A
.S.,
2001
]
Exa
mpl
e: R
-cur
rent
cor
rela
tori
nin
the
limit
Zero
tem
pera
ture
:
Fini
te te
mpe
ratu
re:
Pol
es o
f
=
qua
sino
rmal
spec
trum
of d
ual g
ravi
ty b
ackg
roun
d
(D.S
on, A
.S.,
hep-
th/0
2050
51,
P.K
ovtu
n, A
.S.,
hep-
th/0
5061
84)
Ana
lytic
stru
ctur
e of
the
corre
lato
rs
ωω
− i
8
− i
12
− i
4
T −
> 0
ω
−p
p
p −
> 0
πΤ πΤ
πΤ
− i
8
− i
12
− i
4πΤ πΤ
πΤ
−p
p
Wea
k co
uplin
g: S
. Har
tnol
land
P. K
umar
, hep
-th/0
5080
92ω
−p
p
Stro
ng c
oupl
ing:
A.S
., he
p-th
/020
7133
Com
putin
g tra
nspo
rt co
effic
ient
s fro
m d
ual g
ravi
ty
Ass
umin
g va
lidity
of t
he g
auge
/gra
vity
dua
lity,
al
l tra
nspo
rt co
effic
ient
s ar
e co
mpl
etel
y de
term
ined
by
the
low
est f
requ
enci
esin
qua
sino
rmal
spec
tra o
f the
dua
l gra
vita
tiona
l bac
kgro
und
This
det
erm
ines
kin
etic
s in
the
regi
me
of a
ther
mal
theo
ryw
here
the
dual
gra
vity
des
crip
tion
is a
pplic
able
(D.S
on, A
.S.,
hep-
th/0
2050
51,
P.K
ovtu
n, A
.S.,
hep-
th/0
5061
84)
Tran
spor
t coe
ffici
ents
and
qua
sipa
rticl
esp
ectra
can
als
o be
obta
ined
from
ther
mal
spe
ctra
l fun
ctio
ns
Spe
ctra
l fun
ctio
n an
d qu
asip
artic
les
12
3w
-0.2
0.2
0.4
0.6
0.8
0.5
1
123456
0.1
0.2
0.1
0.05
w
0.5
11.5
w
2468
10
A
B
CA
: sca
lar c
hann
el
B: s
cala
r cha
nnel
-th
erm
al p
art
C: s
ound
cha
nnel
Is th
e bo
und
dead
?Y
.Kat
san
d P
.Pet
rov,
071
2.07
43 [h
ep-th
]“E
ffect
of c
urva
ture
squ
ared
cor
rect
ions
in A
dSon
the
visc
osity
of t
he d
ual g
auge
theo
ry”
supe
rcon
form
alS
p(N
) gau
ge th
eory
in d
=4
M.~
Brig
ante
, H.~
Liu,
R.~
C.~
Mye
rs, S
.~S
henk
eran
d S
.~Y
aida
,``T
he V
isco
sity
Bou
nd a
nd C
ausa
lity
Vio
latio
n,''
0802
.331
8 [h
ep-th
],``V
isco
sity
Bou
nd V
iola
tion
in H
ighe
r Der
ivat
ive
Gra
vity
,'' 07
12.0
805
[hep
-th].
The
“spe
cies
pro
blem
”T.
Coh
en, h
ep-th
/070
2136
, A.D
obad
o, F
.Lla
nes-
Est
rada
, hep
-th/0
7031
32
A v
isco
sity
bou
nd c
onje
ctur
e
P.K
ovtu
n, D
.Son
, A.S
., he
p-th
/030
9213
, hep
-th/0
4052
31
Min
imum
of
i
n un
its o
f
Che
rnai
, Kap
usta
, McL
erra
n, n
ucl-t
h/06
0403
2
Che
rnai
, Kap
usta
, McL
erra
n, n
ucl-t
h/06
0403
2
QC
D
She
ar v
isco
sity
at n
on-z
ero
chem
ical
pot
entia
l
0.1
0.2
0.3
0.4
0.5
1.05
1.1
1.15
1.2
1.25
Rei
ssne
r-N
ords
trom
-AdS
blac
k ho
le
with
thre
e R
cha
rges
(Beh
rnd,
Cve
tic, S
abra
, 199
8)
We
still
hav
e
J.M
asD
.Son
, A.S
.O
.Sar
emi
K.M
aeda
, M.N
atsu
ume,
T.O
kam
ura
(see
e.g
. Yaf
fe, Y
amad
a, h
ep-th
/060
2074
)
Pho
ton
and
dile
pton
emis
sion
fro
m s
uper
sym
met
ricY
ang-
Mills
pla
sma
S. C
aron
-Huo
t, P
. Kov
tun,
G. M
oore
, A.S
., L.
G. Y
affe
, hep
-th/0
6072
37
Pho
tons
inte
ract
ing
with
mat
ter:
Pho
ton
emis
sion
from
SY
M p
lasm
a
To le
adin
g or
der i
n
Mim
icby
gau
ging
glo
bal R
-sym
met
ry
Nee
d on
ly to
com
pute
cor
rela
tors
of th
e R
-cur
rent
s
Now
con
side
r stro
ngly
inte
ract
ing
syst
ems
at fi
nite
den
sity
an
d LO
W te
mpe
ratu
re
Pro
bing
qua
ntum
liqu
ids
with
hol
ogra
phy
Qua
ntum
liqu
idin
p+1
dim
Low
-ene
rgy
elem
enta
ry
exci
tatio
nsS
peci
fic h
eat
at lo
w T
Qua
ntum
Bos
e liq
uid
Qua
ntum
Fer
mi l
iqui
d(L
anda
u FL
T)
phon
ons
ferm
ioni
cqu
asip
artic
les
+bo
soni
cbr
anch
(zer
o so
und)
Dep
artu
res
from
nor
mal
Fer
mi l
iqui
d oc
cur i
n
-3+
1 an
d 2+
1 –d
imen
sion
al s
yste
ms
with
stro
ngly
cor
rela
ted
elec
trons
-In
1+1
–dim
ensi
onal
sys
tem
s fo
r any
stre
ngth
of i
nter
actio
n (L
uttin
gerl
iqui
d)
One
can
app
ly h
olog
raph
y to
stu
dy s
trong
ly c
oupl
ed F
erm
i sys
tem
sat
low
T
L.D
.Lan
dau
(190
8-19
68)
The
sim
ples
t can
dida
te w
ith a
kno
wn
holo
grap
hic
desc
riptio
n is
at fi
nite
tem
pera
ture
T a
nd n
onze
ro c
hem
ical
pot
entia
l ass
ocia
ted
with
the
“bar
yon
num
ber”
dens
ity o
f the
cha
rge
Ther
e ar
e tw
o di
men
sion
less
par
amet
ers:
is th
e ba
ryon
num
ber d
ensi
tyis
the
hype
rmul
tiple
tmas
s
The
holo
grap
hic
dual
des
crip
tion
in th
e lim
it
is
giv
en b
y th
e D
3/D
7 sy
stem
, with
D3
bran
esre
plac
ed b
y th
e A
dS-
Sch
war
zsch
ild g
eom
etry
and
D7
bran
esem
bedd
ed in
it a
s pr
obes
.
Kar
ch&
Kat
z, h
ep-th
/020
5236
AdS
-Sch
war
zsch
ild b
lack
hol
e (b
rane
) bac
kgro
und
D7
prob
e br
anes
The
wor
ldvo
lum
eU
(1) f
ield
c
oupl
es to
the
flavo
r cur
rent
at th
e bo
unda
ry
Non
trivi
al b
ackg
roun
d va
lue
of
c
orre
spon
ds to
non
trivi
al e
xpec
tatio
n va
lue
of
We
wou
ld li
ke to
com
pute
-the
spe
cific
hea
t at l
ow
tem
pera
ture
-the
cha
rge
dens
ity c
orre
lato
r
The
spec
ific
heat
(in
p+1
dim
ensi
ons)
:
(not
e th
e di
ffere
nce
with
Fer
mi
an
d B
ose
sy
stem
s)
The
(reta
rded
) cha
rge
dens
ity c
orre
lato
rha
s a
pole
cor
resp
ondi
ng to
a p
ropa
gatin
g m
ode
(zer
o so
und)
-eve
n at
zer
o te
mpe
ratu
re
(not
e th
at th
is is
NO
T a
supe
rflui
dph
onon
who
se a
ttenu
atio
n sc
ales
as
)
New
type
of q
uant
um li
quid
?
Epi
logu
eO
n th
e le
vel o
f the
oret
ical
mod
els,
ther
e ex
ists
a c
onne
ctio
nbe
twee
n ne
ar-e
quili
briu
m re
gim
e of
cer
tain
stro
ngly
cou
pled
th
erm
al fi
eld
theo
ries
and
fluct
uatio
ns o
f bla
ck h
oles
This
con
nect
ion
allo
ws
us to
com
pute
tran
spor
t coe
ffici
ents
for t
hese
theo
ries
The
resu
lt fo
r the
she
ar v
isco
sity
turn
s ou
t to
be u
nive
rsal
for a
ll su
ch th
eorie
s in
the
limit
of in
finite
ly s
trong
cou
plin
g
At t
he m
omen
t, th
is m
etho
d is
the
only
theo
retic
al to
ol
avai
labl
e to
stu
dy th
e ne
ar-e
quili
briu
m re
gim
e of
stro
ngly
coup
led
ther
mal
fiel
d th
eorie
s
Influ
ence
s ot
her f
ield
s (h
eavy
ion
phys
ics,
con
dmat
)
A h
and-
wav
ing
argu
men
t
Gra
vity
dua
ls fi
x th
e co
effic
ient
:
Thus
Out
look
Gra
vity
dua
l des
crip
tion
of th
erm
aliz
atio
n?
Gra
vity
dua
ls o
f the
orie
s w
ith fu
ndam
enta
l fer
mio
ns:
-pha
se tr
ansi
tions
-hea
vy q
uark
bou
nd s
tate
s in
pla
sma
-tra
nspo
rt pr
oper
ties
Fini
te ‘t
Hoo
ftco
uplin
g co
rrec
tions
to p
hoto
n em
issi
on s
pect
rum
Und
erst
andi
ng 1
/N c
orre
ctio
ns
Pho
nino
Equ
atio
ns s
uch
as
desc
ribe
the
low
ene
rgy
li
mit
of s
tring
theo
ry
As
long
as
the
dila
ton
is s
mal
l, an
d th
us th
e st
ring
inte
ract
ions
are
sup
pres
sed,
this
lim
it co
rres
pond
s to
cla
ssic
al 1
0-di
m E
inst
ein
grav
ity c
oupl
ed to
cer
tain
mat
ter f
ield
s su
ch a
s M
axw
ell f
ield
, p-fo
rms,
dila
ton,
ferm
ions
Val
idity
con
ditio
ns fo
r the
cla
ssic
al (s
uper
)gra
vity
appr
oxim
atio
n
-cur
vatu
re in
varia
nts
shou
ld b
e sm
all:
-qua
ntum
loop
effe
cts
(stri
ng in
tera
ctio
ns =
dila
ton)
sho
uld
be s
mal
l:
In A
dS/C
FT d
ualit
y, th
ese
two
cond
ition
s tra
nsla
te in
to
and
The
chal
leng
e of
RH
IC (c
ontin
ued)
Rap
id th
erm
aliz
atio
n
Larg
e el
liptic
flow
Jet q
uenc
hing
Pho
ton/
dile
pton
emis
sion
rate
s
??
4D b
ound
ary
z 0
The
bulk
and
the
boun
dary
in A
dS/C
FT c
orre
spon
denc
e
5D b
ulk
(+5
inte
rnal
dim
ensi
ons)
UV
/IR: t
he A
dSm
etric
is in
varia
nt u
nder
z pl
ays
a ro
le o
f inv
erse
ene
rgy
scal
e in
4D
theo
ry