Download AdSCFTPrimer - Nevis Laboratories

AdS / CFT
aka
Anti de Sitter (space) / Conformal Field Theory
W.A. Zajc
Columbia University
12-Mar-07
Journal Club
Explaining the Connection
Maldacena’s extraordinary conjecture
1) Weakly Coupled
(classical) gravity in
Anti-deSitter Space (AdS)

12-Mar-07
3) Strongly
Coupled
(Conformal)
gauge Field
Theories
(CFT)
Journal Club
All You Need To Know About Strings
12-Mar-07
Journal Club
All You Need To Know About D-branes

‘D’ = Dirichlet  an extended object that imposes boundary
conditions on (open) string endpoints
String explores
the full space  “the bulk”
String endpoints
constrained to live
on “the brane”

D-branes characterized by


Their dimensionality; Dp-brane lives in p spatial dimensions
Their tension Tp , defined such that
Mass of brane ~  Tp d ( BraneVolum e)  [Tp ]  M / Lp
1 1
String Theory  Tp ~
g S  S p 1
Required, e.g., to open closed strings upon brane contact
D-branes are essential dynamical objects in string theory

12-Mar-07
Journal Club
“Stack” of N D3-branes
These shown as 2-d
slices of 3-volumes
This direction
has no
meaning,
branes are
really
coincident
D3-brane properties:
12-Mar-07

Mass ~ 1/gS

Source gauge quantum number

Open strings end on them
Journal Club
String Interactions on D3-branes
D3-branes shown as
~1-d slices of 3-volumes
String world
One string
“indexed” on
green + anti-red
This direction
has no
meaning,
branes are
really
coincident
Gauge world
SU(N) gauge
theory of
gluon
interactions
12-Mar-07
Journal Club
Gauge  Gravity
These shown as 2-d
slices of coincident
3-volumes
D3-brane properties:

Mass ~ 1/gS

Source gauge quantum number

Open strings end on them
12-Mar-07
Mass ~ N/gS
 Sources gravity
 Curves space
Generates
(sort of) an
Anti de Sitter
spacetime
Journal Club
The Gravity Solution
Where’s
my
AdS ?
There it
is!

“Towards a gravity dual of RHIC Collision”, Sang-Jan Sin,
http://him.phys.pusan.ac.kr/PDS_HIM/HIM/2005-11/3_shin.pdf
12-Mar-07
Journal Club
The Correspondence


Q. Where
do the N
D3-branes
live?
A. On the
boundary
of an Anti
de Sitter
space
(that they
create!)
12-Mar-07
Curvature
matters !
This direction ( r )
has meaning;
~ energy scale
Journal Club
So What’s the CFT Part ?

“Real” AdS
in n spacetime dimensions
2
ds 2  (1 

r
2
)
dt

2
R
1
2
2
2
dr

r
d

n2
2
r
1 2
R
The D-brane induced “almost AdS”
R4 2
ds 
  dx dx  1  4 dr
4
r
R
1 4
r
2

1


Their limits (which are also called AdS):
r 2 2 R2 2 2 2
 “Real” AdS
: ds   2 dt  2 dr  r d for r   " Boundary "
R
r
r2
R2 2
2


for r  0 " Boundary "
 D-brane “almost AdS”: ds  2   dx dx  2 dr
R
r
2

The scaling form of the limit (which is also called
AdS)
R2
2


2
2
ds 
12-Mar-07
z2
(  dx dx  dz ) after z  R / r
Journal Club
The Conformal Part

Note that this metric
2
R
ds 2  2 (  dx  dx  dz 2 )
z
has no scale, that is, is invariant under (x,z)  (lx, lz)
 Potential must scale as 1/r
2
gYM N C
Vqq (r )  1.254
r
2
gYM N C
( weak coupling  C
)
r

AdS interpretation:
Still an area law for
Wilson lines, but the
warp factor 1/z makes the
“area” fall as 1/r
12-Mar-07
Journal Club
The Icky Part


Icky, that is, if you want to use this correspondence to
study QCD
Conformal
 no scale
 “It’s 1/r all the way down”
 No confinement !

One way out (Witten, hep-th/9803002)

Modify space to have a horizon:
Horizon
More recently: “More on a holographic dual of QCD”,
T. Sakai and S. Sugimoto,
http://arxiv.org/abs/hep-th/0507073
12-Mar-07

Journal Club
We Don’t Care About Confinement

The duality, as described, applies to
T=0 CFT
in flat 3+1
spacetime


Gravity
in curved 4+1
AdS spacetime

(~Classical)
Gravity
in curved 4+1
AdS spacetime
More accurately:
(Strongly
coupled)
T=0 CFT
in flat 3+1
spacetime
Q. How to thermalize the theory?
A. Shine a “black” hole on it (!)
12-Mar-07
Journal Club
Black Hole Thermodynamics

~1970, Bekenstein:




Black hole area law “feels like” 2nd law of thermodynamics:
 AMERGED ≥ A1 + A2
Charge for black hole contributes to energy as dM = F dQ,
feels like chemical potential
So why not
dM = T dSBH + F dQ , with SBH ~ Black Hole Area ??
Counter-arguments:
“Black holes have no hair”  no internal d.o.f  no entropy
 Entropy  temperature  radiation, but black holes are black


~1974, Hawking:


Black holes do radiate !
Semi-classical computation allowed determination of entropy:
S BH
12-Mar-07
c3 A
 (k )
4G
 (k )
A
2
4 L PLANCK
Journal Club
BH Radiation  BH’s are Unstable

Starting from this:
S BH  (k )
A
2
4 L PLANCK
and
RBH
2GM
 2
c
it’s easy to compute



4GM 2
Black Hole entropy: S BH  (k )
~ 1077 k for solar mass
c
M c 1
Black Hole temperature:
TBH 

S
k 8GM
~ 108 K for solar mass
Black Hole lifetime
3

~
M
(assuming Stefan-Boltzmann) BH
~ 1070 s
12-Mar-07
for solar mass
Journal Club
Black Holes in Higher Dimensions

Apply same basic formalism starting from Ddimensional result for Schwarzschild radius:
( RBH ) D 3 

Show that higher-dimensional BH’s




12-Mar-07
16GD M
 D  2 ( D  2)
Have a temperature
And therefore radiate
And therefore have finite lifetime
Unless the background spacetime is curved !
Journal Club
Black Holes in AdS

The metric becomes
G5 M 2 2
G M
r2
r2
ds  [1  ( 2 )  (
) ]dt  [1  ( 2 )  ( 5 ) 2 ]1 dr 2  r 2 d3
R
r
R
r
2



The spacetime curvature R introduces a new scale in
the problem
Especially because light reaches the boundary in time
T =  R and is “reflected”
Black hole is in a “box”:



Small black holes: rbh << R  rbh ~ M1/2  Unstable
Large black holes: rbh ~ R  rbh ~ M1/4  STABLE !
In addition, for large black holes:


12-Mar-07
In 5-d spacetime, BH “area” ~ Length3  S ~ M3/4
T ~ M1/4  S ~ T 3 , that is, just like a QGP
Journal Club
This is Your Brane


This is
your
brane
on AdS
Negative
curvature R


Finite time
~R for light to
reach
boundary
and return
Black holes
of lifetime > ~
R are
STABLE !
12-Mar-07
Journal Club
Viscosity Primer

Remove your organic prejudices


Don’t equate viscous with “sticky” !
Think instead of a not-quite-ideal fluid:




“not-quite-ideal”  “supports a shear stress”
Viscosity 
Fx
v x
 
then defined as
A
y
Dimensional
estimate:
Viscosity
increases with
temperature
η ( momentum density )( mean free path )
1
p
 n p mfp  n p

nσ σ
mkT
 for a( nearly ) ideal gas η 
σ
 Large cross sections  small viscosity
The gauge/string duality is one that maps
strongly coupled gauge fields  Weak (semi-classical) gravity
12-Mar-07
Journal Club
Ideal Hydrodynamics

Why the interest in viscosity?
A.) Its vanishing is associated with the applicability of
ideal hydrodynamics (Landau, 1955):
Inertial Forces V BU LK L
Ideal Hydro  Reynolds Number  

 1
Drag Forces

V BU LK L
L
   v t herm al( mfp ) so  
 1 
 1
v t herm al mfp
mfp
B.) Successes of ideal hydrodynamics applied to RHIC
data suggest that the fluid is “as perfect as it can be”,
that is, it approaches the (conjectured) quantum
mechanical limit



4
( entropy density) 
4
s
See “A Viscosity Bound Conjecture”,
P. Kovtun, D.T. Son, A.O. Starinets, hep-th/0405231
12-Mar-07
Journal Club
Why Does This Work??

The easy part:




The hard part:


Fx
v
Recall
  x
that is,
A
y
viscosity ~ x-momentum transport in y-direction ~ Txy
There are standard methods (Kubo relations) to calculate such
dissipative quantities
This calculation is difficult in a strongly-coupled gauge theory
The weird part:



12-Mar-07
A (supersymmetric) pseudo-QCD theory
can be mapped to a 10-dimensional
classical gravity theory on the
background of black 3-branes
The calculation can be performed there
as the absorption of gravitons by the brane
h
A
A
THE SHEAR VISCOSITY OF STRONGLY COUPLED N=4 SUPERSYMMETRIC YANGMILLS PLASMA., G. Policastro, D.T. Son , A.O. Starinets,
Phys.Rev.Lett.87:081601,2001 hep-th/0104066
Journal Club
The Result

Viscosity  = “Area”/16G
Infinite “Area” !

Normalize by entropy (density) S = “Area”/4G

Dividing out the infinite “areas” :



 1
( )
s
k 4
Conjectured to be a lower bound “for all relativistic quantum field
theories at finite temperature and zero chemical potential”.
See “Viscosity in strongly interacting quantum field theories from
black hole physics”, P. Kovtun, D.T. Son, A.O. Starinets,
Phys.Rev.Lett.94:111601, 2005, hep-th/0405231
12-Mar-07
Journal Club
Isn’t This Result “Just” Quantum Mechanics?

Recall from previous discussion:
 ~ np mfp ~ e



e = energy density
 = lifetime of quasiparticle
Entropy density s ~ kB n 

1 e
1
~
 ( )  
s kB n kB n
kB

where last step


12-Mar-07
e
follows from requirement that lifetime of quasiparticle must
exceed ~h/Energy
establishes that the bound is from below
Journal Club
How Perfect is “Perfect”

All “realistic” hydrodynamic calculations for RHIC fluids to
date have assumed zero viscosity
  = 0  “perfect fluid”

But there is a (conjectured) quantum limit:


  ( Entropy Density ) 
s
4
4

Where do
“ordinary”
fluids sit wrt
this limit?

RHIC “fluid” might
be at ~2-3 on this
scale (!)
12-Mar-07
12 K
T=10Journal
Club
Water  RHIC  Water  RHIC

/s

The search for QCD phase transition of course was
informed by analogy to ordinary matter
Results from RHIC are now “flowing” back to
ordinary matter
“On the Strongly-Interacting Low-Viscosity Matter
Created in Relativistic Nuclear Collisions”,
L.P. Csernai, J.I. Kapusta and L.D. McLerran,
Phys.Rev.Lett.97:152303,2006, nucl-th/0604032
12-Mar-07
Journal Club
QCD Critical Point
12-Mar-07
Journal Club
A Loophole To The Bound?

Kovtun, Son and Starinets also note

Cohen seeks to exploit this loophole:

12-Mar-07
“Is there a 'most perfect fluid' consistent with quantum field
theory?”, Thomas D. Cohen, hep-th/0702136
Journal Club
Entropy of Mixing

It’s “in” the Sackur-Tetrode equation:
V/NA
V
N A log[
]
NA
V/NA
V
N A log[
]
NA
12-Mar-07
V/NA
N A log[

V
]
NA
V/NB
V
N B log[
]
NB
S  k N{ log[
2V/2NA
( N A  N A ) log[

V 2mU 3 / 2 5
(
) ] }
N 32 Nh2
2
2V
V
]  2 log[
]
2N A
NA
2V/NA+2V/NB
N A log[ N2V ]  N B log[ N2V ]
A
B
 2 N A log[ NV ]  ( N A  N B ) log 2
A
Journal Club
Entropy For Distinguishable Particles
12-Mar-07
Journal Club
Incorporating Indistinguishability
12-Mar-07
Journal Club
Incorporating Multiple Species
12-Mar-07
Journal Club
Cohen’s Scaling Parameter
12-Mar-07
Journal Club
The Scaling Regime
12-Mar-07
Journal Club
How Low Can It Go?
12-Mar-07
Journal Club
Not Discussed

Counter-counter arguments:


Counter-counter-counter arguments:

12-Mar-07
Bousso’s entropy bound on spacetime regions?
Residual entropy ?
Journal Club
Suggested Reading

November, 2005 issue of Scientific
American



“The Illusion of Gravity”
J. Maldacena
A test of this prediction comes from the
Relativistic Heavy Ion Collider (RHIC) at
BrookhavenNational Laboratory, which
has been colliding gold nuclei at very
high energies. A preliminary analysis of
these experiments indicates the
collisions are creating a fluid with very
low viscosity. Even though Son and his
co-workers studied a simplified version
of chromodynamics, they seem to have
come up with a property that is shared
by the real world. Does this mean that
RHIC is creating small five-dimensional
black holes? It is really too early to tell,
both experimentally and theoretically.
(Even if so, there is nothing to fear from
these tiny black holes-they evaporate
almost as fast as they are formed, and
they "live" in five dimensions, not in our
own four-dimensional world.)
12-Mar-07
Journal Club
A Spooky Connection

RHIC physics clearly relies on


The quantum nature of matter (Einstein, 1905)
The relativistic nature of matter (Einstein, 1905)
but presumably has no connection to


General relativity (Einstein, 1912-7)
Wait ! Both sides of this equation
( Vis cosity )R H IC


( Entropy Density )R H IC
4
were calculated using black hole physics (in 10 dimensions)
MULTIPLICITY
Entropy  Black Hole Area
c
c
DISSIPATION
Viscosity  Graviton
12-Mar-07
Color Screening
Absorption
Journal Club
Spooky Connection at a Distance


We’ve yet to understand
the discrepancy between
lattice results and StefanBoltzmann limit:
The success of naïve
hydrodynamics requires
very low viscosities
viscosity

  ~ 0.1(??)
entropy density s

Both are predicted from
~gravitational phenomena
in N = 4 supersymmetric
theories:   1
4
e
3

e SB 4
s
12-Mar-07
Journal Club
New Dimensions in RHIC Physics
“The stress tensor of a quark moving through N=4
thermal plasma”, J.J. Friess et al., hep-th/0607022

Our 4-d
world
String
theorist’s
5-d world
12-Mar-07
The stuff formerly
known as QGP
Jet modifications
from wake field
Heavy quark
moving
through
the
Energy loss medium
from string
drag
Journal Club
The Way Forward




12-Mar-07
Recall
“ We need to learn to expand in powers of 1 / g(T) ”
For example, the mean free path lmfp
Limit lmfp  0 is hydrodynamics
Journal Club
Landau Knew It


Landau (1955) significant
extension of Fermi’s
approach
Considers fundamental
roles of


hydrodynamic evolution
entropy
“The defects of Fermi’s
theory arise mainly because
the expansion of the
compound system is not
correctly taken into
account…(The) expansion
of the system can be
considered on the basis of
relativistic hydrodynamics.”
 (Emphasis added by WAZ)

12-Mar-07
Journal Club
But We’re Not Quite Done Making Mistakes

Recall our argument for short
mean free paths:
l mfp 

1
n
~
1
T ( aS( T )/T )
3
2
~
1
g ( T )T
2
l mfp 
But this relies on the number
density n , which is not welldefined for a relativistic field
theory at strong coupling(!)
Γ
Potential Energy
Kinetic Energy


1
n
~
1
T ( aS( T )/T )
3
2
~
1
g ( T )T
2
But wait, it get worse…
Even the
classical coupling parameter
Potential Energy
Γ
Kinetic Energy
is not well-defined relativistically(!)
12-Mar-07
Journal Club
A Way Out



12-Mar-07
How can we quantify the coupling properties of our
“plasma” ?
A solution was provided by Dam Son:

n( T ) is not well-defined …
but s(T) is

mean free path not well-defined… but viscosity  is

coupling G is not well defined…
but s /  is
Note:
Short
mean free paths  small viscosity
Journal Club
This is Your Brane



This is
your
brane on
AdS
More seriously:
Negative
curvature R


12-Mar-07
Finite time ~R
for light to
reach
boundary and
return
Black holes of
lifetime > ~ R
are STABLE !
Journal Club