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

12-Mar-07 Journal Club

AdS / CFTaka

Anti de Sitter (space) / Conformal Field Theory

W.A. ZajcColumbia University


Explaining the ConnectionExplaining the Connection Maldacena’s extraordinary conjecture

3) Strongly 3) Strongly Coupled Coupled

(Conforma(Conformal) gauge l) gauge

Field Field Theories Theories

(CFT)(CFT)

1) Weakly Coupled 1) Weakly Coupled (classical) gravity in (classical) gravity in Anti-deSitter Space Anti-deSitter Space

(AdS)(AdS)


All You Need To Know About All You Need To Know About StringsStrings


All You Need To Know About D-branesAll 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”

ppp LMTeBraneVolumdTbraneofMass /][)(~

1

11~ p

SSp g

TTheoryString


““Stack” of Stack” of NN D3-branes D3-branesThese shown as 2-d These shown as 2-d

slices of 3-volumesslices of 3-volumes

D3-brane properties:D3-brane properties:

Mass ~ 1/gMass ~ 1/gSS

Source gauge quantum numberSource gauge quantum number

Open strings end on themOpen strings end on them

This direction This direction has no has no meaning, meaning, branes are branes are really really coincidentcoincident


String Interactions on D3-branesString Interactions on D3-branesThis direction This direction

has no has no meaning, meaning, branes are branes are really really coincidentcoincident

D3-branes shown as D3-branes shown as ~1-d slices of 3-volumes~1-d slices of 3-volumes

One string One string “indexed” on “indexed” on green + anti-redgreen + anti-red

SU(N) gauge SU(N) gauge theory of theory of gluon gluon interactionsinteractions

String worldString world

Gauge worldGauge world


Gauge Gauge Gravity GravityThese shown as 2-d These shown as 2-d slices of slices of coincidentcoincident

3-volumes3-volumes

D3-brane properties:D3-brane properties:

Mass ~ 1/gMass ~ 1/gSS

Source gauge quantum numberSource gauge quantum number

Open strings end on themOpen strings end on them

Mass ~ N/gMass ~ N/gSS

Sources gravitySources gravity

Curves spaceCurves space

Generates Generates (sort of) an(sort of) anAnti de SitterAnti de Sitterspacetimespacetime


The Gravity SolutionThe Gravity Solution

“Towards a gravity dual of RHIC Collision”, Sang-Jan Sin, http://him.phys.pusan.ac.kr/PDS_HIM/HIM/2005-11/3_shin.pdf

Where’s Where’s my my

AdS ?AdS ?

There it There it is!is!


The CorrespondenceThe Correspondence Q. Where

do the N D3-branes live?

A. On the boundary of an Anti de Sitter space (that they create!)

~ Ess

entially

~ Ess

entially

flat space

flat space

Curvature Curvature matters matters

!!

This direction This direction ( r ) ( r ) has has meaning; meaning; ~ energy ~ energy scalescale


So What’s the CFT Part ?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)

24

4

4

4

2 1

1

1dr

r

Rdxdx

rR

ds

2222

2

22

2

22

1

1)1(

ndrdr

Rr

dtR

rds

""022

2

2

22 Boundaryrfordr

r

Rdxdx

R

rds

rRzafterdzdxdxz

Rds /)( 22

2

22

""2222

22

2

22 Boundaryrfordrdr

r

Rdt

R

rds


The Conformal PartThe Conformal Part Note that this metric

has no scale, that is, is invariant under (x,z) (x, z) 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

)( 22

22 dzdxdx

z

Rds

)(

254.1)(

2

2

r

NgCcouplingweak

r

NgrV

CYM

CYMqq


The Icky PartThe 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:

More recently: “More on a holographic dual of QCD”, T. Sakai and S. Sugimoto, http://arxiv.org/abs/hep-th/0507073

Horizon


We Don’t Care About ConfinementWe 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


Black Hole ThermodynamicsBlack 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 = dQ,feels like chemical potential

So why not dM = T dSBH + 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:

G

AckSBH 4

)(3

PLANCKL

Ak

24)(


BH Radiation BH Radiation BH’s are Unstable BH’s are Unstable Starting from this:

it’s easy to compute

Black Hole entropy:

Black Hole temperature:

Black Hole lifetime(assuming Stefan-Boltzmann)

22

2

4)(

c

GMRand

L

AkS BH

PLANCKBH

masssolarforkc

GMkSBH

772

10~4

)(

masssolarforK

GMk

c

S

MTBH

810~

8

1

masssolarfors

MBH

70

3

10~

~


Black Holes in Higher DimensionsBlack 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 !

)2(

16)(

2

3

D

MGR

D

DDBH


Black Holes in AdSBlack 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 = 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

322125

2

2225

2

22 ])()(1[])()(1[ drdr

r

MG

R

rdt

r

MG

R

rds


This is Your BraneThis 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 !


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

then defined as

Dimensional estimate:

Viscosityincreases withtemperature

Large cross sections small viscosityThe gauge/string duality is one that maps

strongly coupled gauge fields Weak (semi-classical) gravity

Viscosity PrimerViscosity Primer

yv

AF xx

σmkT

η)(

σσ1

)()(η

gasidealnearlyafor

pn

pnmfppn

pathfreemeandensitymomentum


Ideal HydrodynamicsIdeal 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

11 so )(

1Forces DragForces Inertial

Number Reynolds Hydro Ideal

mfpL

mfpvLV

mfpv

LV

thermal

BULKthermal

BULK

s

4

)densityentropy (4


Why Does This Work??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 N=4 SUPERSYMMETRIC YANG-MILLS PLASMA., G. Policastro, D.T. Son , A.O. Starinets, Phys.Rev.Lett.87:081601,2001 hep-th/0104066

yv

AF xx

h

A

A


The ResultThe Result Viscosity = “Area”/16G

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/0405231

Infinite “Area” !

4

1)(

ks


Isn’t This Result “Just” Quantum Isn’t This Result “Just” Quantum Mechanics?Mechanics?

Recall from previous discussion:

= energy density = 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

~~ mfppn

BBB knknks

1)(

1~


How Perfect is “Perfect”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:

Where do “ordinary” fluids sit wrt this limit?

RHIC “fluid” mightbe at ~2-3 on this scale (!)

sDensityEntropy

4

)(4

T=10T=101212 KK


Water Water RHIC RHIC Water Water RHIC 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

“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

/

s


QCD Critical PointQCD Critical Point


A Loophole To The Bound?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


Entropy of MixingEntropy of Mixing It’s “in” the Sackur-Tetrode equation:

V/NA V/NA 2V/2NA

}2

5])

2(log[{ 2/3

22

3 Nh

mU

N

VNkS

]log[A

A N

VN ]log[

AA N

VN ]log[2]

2

2log[)(

AAAA N

V

N

VNN

V/NA 2V/NA+2V/NB

]log[A

A N

VN ]log[

BB N

VN

2log)(]log[2

]log[]log[ 22

BAAN

VA

BN

VB

AN

VA

NNN

NN

V/NB


Entropy For Distinguishable ParticlesEntropy For Distinguishable Particles


Incorporating IndistinguishabilityIncorporating Indistinguishability


Incorporating Multiple SpeciesIncorporating Multiple Species


Cohen’s Scaling ParameterCohen’s Scaling Parameter


The Scaling RegimeThe Scaling Regime


How Low Can It Go?How Low Can It Go?


Not DiscussedNot Discussed Counter-counter arguments:

Bousso’s entropy bound on spacetime regions?

Counter-counter-counter arguments: Residual entropy ?


Suggested ReadingSuggested 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.)


A Spooky ConnectionA 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)

RHICRHIC DensityEntropyityVis )(4

)cos(

Color Screening

cc

MULTIPLICITY

Entropy Black Hole Area

DISSIPATION

Viscosity Graviton

Absorption


Spooky Connection at a DistanceSpooky 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:

(??)1.0~densityentropy

viscosity

s

4

34

1

SB

s


New Dimensions in RHIC PhysicsNew 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 Our 4-d worlworl

dd

String String theorist’theorist’

s 5-d s 5-d worldworld

The stuff The stuff formerly formerly known as QGPknown as QGP

Heavy Heavy quark quark

moving moving through through

the the mediummedium

Energy loss Energy loss from from string string dragdrag

Jet Jet modificationmodifications from wake s from wake

fieldfield


The Way ForwardThe 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


Landau Knew ItLandau 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)


But We’re Not Quite Done Making MistakesBut 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(!)

TTgTTaTnl

Smfp )(

1~

)/)((1

~1

223

TTgTTaTnl

Smfp )(

1~

)/)((1

~1

223

But wait, it get worse… Even the

classical coupling parameter

is not well-defined relativistically(!)

Energy Kinetic

Energy PotentialΓ

Energy Kinetic

Energy PotentialΓ


A Way OutA 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 is

coupling is not well defined… but s / is Note:

Short mean free paths small viscosity


This is Your BraneThis 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 !

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

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