AdS / CFT

1

• AdS / CFT
• aka
• Anti de Sitter (space) / Conformal Field Theory
• W.A. Zajc
• Columbia University

2
Explaining the Connection

• Maldacenas extraordinary conjecture

1) Weakly Coupled (classical) gravity in
Anti-deSitter Space (AdS)
3) Strongly Coupled (Conformal) gauge Field
Theories (CFT)
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All You Need To Know About Strings
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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 endpointsconstrained to live on the
brane
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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

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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
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Gauge ? Gravity
These shown as 2-d slices of coincident3-volumes

• Mass N/gS
• Sources gravity
• Curves space
• Generates (sort of) anAnti de Sitterspacetime

• D3-brane properties
• Mass 1/gS
• Source gauge quantum number
• Open strings end on them

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The Gravity Solution
Wheres 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

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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
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So Whats 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)

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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
  thearea fall as 1/r

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The Icky Part

• Icky, that is, if you want to use this
  correspondence to study QCD
• Conformal
• no scale
• Its 1/r all the way down
• No confinement !
• One way out (Witten, hep-th/9803002)
• Modify space to have a horizon
• More recently More on a holographic dual of
  QCD, T. Sakai and S. Sugimoto,
  http//arxiv.org/abs/hep-th/0507073

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We Dont Care About Confinement

• The duality, as described, applies to
• More accurately
• Q. How to thermalize the theory?
• A. Shine a black hole on it (!)

T0 CFT in flat 31 spacetime
Gravity in curved 41 AdS spacetime
?
(Strongly coupled)T0 CFT in flat 31
spacetime
(Classical)Gravity in curved 41 AdS
spacetime
?
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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

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BH Radiation ? BHs are Unstable

• Starting from this
• its easy to compute
• Black Hole entropy
• Black Hole temperature
• Black Hole lifetime(assuming Stefan-Boltzmann)

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Black Holes in Higher Dimensions

• Apply same basic formalism starting from
  D-dimensional result for Schwarzschild radius
• Show that higher-dimensional BHs
• Have a temperature
• And therefore radiate
• And therefore have finite lifetime
• Unless the background spacetime is curved !

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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 ltlt 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

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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 gt R are STABLE !

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Viscosity Primer

• Remove your organic prejudices
• Dont 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
• Viscosityincreases withtemperature
• Large cross sections ? small viscosity
• The gauge/string duality is one that maps
  strongly coupled gauge fields ? Weak
  (semi-classical) gravity

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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/0405231

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Why Does This Work??

• The easy 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 N4
  SUPERSYMMETRIC YANG-MILLS PLASMA., G. Policastro,
  D.T. Son , A.O. Starinets, Phys.Rev.Lett.8708160
  1,2001 hep-th/0104066

hmn
Am
An
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The Result

• 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.94111601, 2005, hep-th/0405231

Infinite Area !
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Isnt 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

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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 mightbe at 2-3 on this scale (!)

T1012 K
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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.97152303,2006, nucl-th/0604032
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QCD Critical Point
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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/0702136

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Entropy of Mixing

• Its in the Sackur-Tetrode equation

?
?
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Entropy For Distinguishable Particles
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Incorporating Indistinguishability
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Incorporating Multiple Species
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Cohens Scaling Parameter
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The Scaling Regime
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How Low Can It Go?
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Not Discussed

• Counter-counter arguments
• Boussos entropy bound on spacetime regions?
• Counter-counter-counter arguments
• Residual entropy ?

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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.)

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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
• were calculated using black hole physics (in
  10 dimensions)

MULTIPLICITY Entropy ? Black Hole
Area DISSIPATION Viscosity ?
Graviton Absorption
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Spooky Connection at a Distance

• Weve 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

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New Dimensions in RHIC Physics

• The stress tensor of a quark moving through N4
  thermal plasma, J.J. Friess et al.,
  hep-th/0607022

Jet modifications from wake field
Our 4-d world
The stuff formerly known as QGP
Heavy quark moving through the medium
String theorists 5-d world
Energy loss from string drag
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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

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Landau Knew It

• Landau (1955) significant extension of Fermis
  approach
• Considers fundamental roles of
• hydrodynamic evolution
• entropy
• The defects of Fermis 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)

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But Were 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(!)

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A Way Out

• 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

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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 gt R are STABLE !