1. Anti de Sitter (space) / Conformal Field Theory
AdS / CFT
aka
Anti de Sitter (space) / Conformal Field Theory
W.A. Zajc
Columbia University
12-Mar-07
Journal Club

2. Explaining the Connection
Maldacena’s extraordinary conjecture
1) Weakly Coupled (classical) gravity in Anti-deSitter Space (AdS)
3) Strongly Coupled (Conformal) gauge Field Theories (CFT)
12-Mar-07
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3. All You Need To Know About Strings
12-Mar-07
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4. All You Need To Know About D-branes
‘D’ = Dirichlet  an extended object that imposes boundary conditions on (open) string endpoints
D-branes characterized by
Their dimensionality; Dp-brane lives in p spatial dimensions
Their tension Tp , defined such that
Required, e.g., to open closed strings upon brane contact
D-branes are essential dynamical objects in string theory
String explores the full space  “the bulk”
String endpoints constrained to live on “the brane”
12-Mar-07
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5. These shown as 2-d slices of 3-volumes
“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:
Mass ~ 1/gS
Source gauge quantum number
Open strings end on them
12-Mar-07
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6. String Interactions on D3-branes
D3-branes shown as ~1-d slices of 3-volumes
This direction has no meaning, branes are really coincident
String world
One string “indexed” on green + anti-red
Gauge world
SU(N) gauge theory of gluon interactions
12-Mar-07
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7. These shown as 2-d slices of coincident 3-volumes
Gauge  Gravity
These shown as 2-d slices of coincident 3-volumes
Mass ~ N/gS
Sources gravity
Curves space
Generates (sort of) an Anti de Sitter spacetime
D3-brane properties:
Mass ~ 1/gS
Source gauge quantum number
Open strings end on them
12-Mar-07
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8. The Gravity Solution Where’s my AdS ? There it is!
“Towards a gravity dual of RHIC Collision”, Sang-Jan Sin,
12-Mar-07
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9. ~ Essentially flat space
The Correspondence
Q. Where do the N D3-branes live?
A. On the boundary of an Anti de Sitter space (that they create!)
~ Essentially flat space
Curvature matters !
This direction ( r ) has meaning; ~ energy scale
12-Mar-07
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10. So What’s the CFT Part ? “Real” AdS in n spacetime dimensions
The D-brane induced “almost AdS”
Their limits (which are also called AdS):
“Real” AdS :
D-brane “almost AdS”:
The scaling form of the limit (which is also called AdS)
12-Mar-07
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11. The Conformal Part Note that this metric
has no scale, that is, is invariant under (xm,z)  (lxm, lz)
Potential must scale as 1/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
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12. 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/ )
Modify space to have a horizon:
More recently: “More on a holographic dual of QCD”, T. Sakai and S. Sugimoto,
Horizon
12-Mar-07
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13. We Don’t Care About Confinement
The duality, as described, applies to
More accurately:
Q. How to thermalize the theory?
A. Shine a “black” hole on it (!)
T=0 CFT in flat 3+1 spacetime
Gravity in curved 4+1 AdS spacetime 
(Strongly coupled) T=0 CFT in flat 3+1 spacetime
(~Classical) Gravity in curved 4+1 AdS spacetime 
12-Mar-07
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14. 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:
12-Mar-07
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15. BH Radiation  BH’s are Unstable
Starting from this:
it’s easy to compute
Black Hole entropy:
Black Hole temperature:
Black Hole lifetime (assuming Stefan-Boltzmann)
12-Mar-07
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16. Black Holes in Higher Dimensions
Apply same basic formalism starting from D-dimensional result for Schwarzschild radius:
Show that higher-dimensional BH’s
Have a temperature
And therefore radiate
And therefore have finite lifetime
Unless the background spacetime is curved !
12-Mar-07
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17. Black Holes in AdS The metric becomes
The spacetime curvature R introduces a new scale in the problem
Especially because light reaches the boundary in time T = p 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:
In 5-d spacetime, BH “area” ~ Length3  S ~ M3/4
T ~ M1/4  S ~ T 3 , that is, just like a QGP
12-Mar-07
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18. 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
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19. 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 h then defined as
Dimensional estimate:
Viscosity increases with temperature
Large cross sections  small viscosity
The gauge/string duality is one that maps strongly coupled gauge fields  Weak (semi-classical) gravity
12-Mar-07
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20. Ideal Hydrodynamics Why the interest in viscosity?
A.) Its vanishing is associated with the applicability of ideal hydrodynamics (Landau, 1955):
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
See “A Viscosity Bound Conjecture”, P. Kovtun, D.T. Son, A.O. Starinets, hep-th/
12-Mar-07
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21. Why Does This Work?? hmn Am An The easy part: The hard part:
Recall
that is, viscosity ~ x-momentum transport in y-direction ~ Txy
There are standard methods (Kubo relations) to calculate such dissipative quantities
The hard part:
This calculation is difficult in a strongly-coupled gauge theory
The weird part:
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
THE SHEAR VISCOSITY OF STRONGLY COUPLED N=4 SUPERSYMMETRIC YANG-MILLS PLASMA., G. Policastro, D.T. Son , A.O. Starinets, Phys.Rev.Lett.87:081601, hep-th/
hmn Am An
12-Mar-07
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22. The Result Infinite “Area” ! Viscosity h = “Area”/16pG
Normalize by entropy (density) S = “Area”/4G
Dividing out the infinite “areas” :
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/
Infinite “Area” !
12-Mar-07
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23. Isn’t This Result “Just” Quantum Mechanics?
Recall from previous discussion:
e = energy density
t = lifetime of quasiparticle
Entropy density s ~ kB n 
where last step
follows from requirement that lifetime of quasiparticle must exceed ~h/Energy
establishes that the bound is from below
12-Mar-07
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24. How Perfect is “Perfect”
All “realistic” hydrodynamic calculations for RHIC fluids to date have assumed zero viscosity
h = 0  “perfect fluid”
But there is a (conjectured) quantum limit:
Where do “ordinary” fluids sit wrt this limit?
RHIC “fluid” might be at ~2-3 on this scale (!)
T=1012 K
12-Mar-07
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25. Water  RHIC  Water  RHIC
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
h / s
“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/
12-Mar-07
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26. QCD Critical Point
12-Mar-07
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27. A Loophole To The Bound? Kovtun, Son and Starinets also note
Cohen seeks to exploit this loophole:
“Is there a 'most perfect fluid' consistent with quantum field theory?”, Thomas D. Cohen, hep-th/
12-Mar-07
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28. Entropy of Mixing   It’s “in” the Sackur-Tetrode equation: V/NA
V/NB
2V/NA+2V/NB 
12-Mar-07
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29. Entropy For Distinguishable Particles
12-Mar-07
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30. Incorporating Indistinguishability
12-Mar-07
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31. Incorporating Multiple Species
12-Mar-07
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32. Cohen’s Scaling Parameter
12-Mar-07
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33. The Scaling Regime
12-Mar-07
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34. How Low Can It Go?
12-Mar-07
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35. Not Discussed Counter-counter arguments:
Bousso’s entropy bound on spacetime regions?
Counter-counter-counter arguments:
Residual entropy ?
12-Mar-07
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36. 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
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37. 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, )
Wait ! Both sides of this equation
were calculated using black hole physics (in 10 dimensions)
MULTIPLICITY
Entropy  Black Hole Area
DISSIPATION
Viscosity  Graviton
Absorption
12-Mar-07
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38. Spooky Connection at a Distance
We’ve yet to understand the discrepancy between lattice results and Stefan-Boltzmann limit:
The success of naïve hydrodynamics requires very low viscosities
Both are predicted from ~gravitational phenomena in N = 4 supersymmetric theories:
12-Mar-07
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39. New Dimensions in RHIC Physics
“The stress tensor of a quark moving through N=4 thermal plasma”, J.J. Friess et al., hep-th/
Jet modifications from wake field
Our 4-d world
The stuff formerly known as QGP
Heavy quark moving through the medium
String theorist’s 5-d world
Energy loss from string drag
12-Mar-07
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40. The Way Forward
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
12-Mar-07
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41. 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
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42. But We’re Not Quite Done Making Mistakes
Recall our argument for short mean free paths:
But this relies on the number density n , which is not well-defined for a relativistic field theory at strong coupling(!)
But wait, it get worse…
Even the classical coupling parameter
is not well-defined relativistically(!)
12-Mar-07
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43. A Way Out Short mean free paths  small viscosity
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 h is
coupling G is not well defined… but s / h is
Note:
Short mean free paths  small viscosity
12-Mar-07
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44. This is Your Brane This is your brane on AdS More seriously:
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