Remarks on the gauge/gravity duality

Juan Maldacena

Field Theory Gravity theory=

Gauge TheoriesQCD

Quantum GravityString theory

Large N and string theory

• SU(N) theory when N ∞ , with g2 N = fixed• Planar diagrams dominate• View them as the worldsheet of a string

• 1/N corrections non-planar diagrams = strings worldsheets with more complicated topologies

+ ==+

‘t Hooft 74

Look for a string theory in 4d not consistent

At least 5 dimensions Polyakov

D-branes in string theory

• Solitons in string theory. Excitations described by open strings

• Lowest energies U(N) gauge fields

• Can also be viewed as black branes

i j

A ij

Polchinski 95

Horowitz Strominger 91

N=4 SU(N) Yang Millstheory in 3+1 dimensions

String theory on AdS5 x S5

i j

A ij

JM 97g2N =³Rls

´4

AdS/CFT

Conformalsymmetry

g2N =³Rls

´4

Duality:

Development of the dictionary

• Correlation functions

• Wilson loops

• Various deformations: masses, marginal deformations

• Many new examples, both conformal and non-conformal

• Continues today…

Gubser, Klebanov, Polyakov, Witten

Lessons for gravity

• Use the field theory as a definition of string theory or quantum gravity (finite N).

• Lessons for black holes.

• Emergent space time

Black holes

Entropy = area = statistical entropy in the field theory

Unitary evolution:Quantum mechanicsand gravity are compatible

Counting supersymmetric black holes

• More detailed black hole counting, subleading corrections

• AdS3/CFT2 examples

• Connections to the topological string

• Connections between matrix models, N=1 quantum field theories and geometry

Gopakumar, Vafa,Dijkgraaf, Vafa

Holography

• Physics in some region described by a theory on the boundary of the region with a number of q-bits that grows like the area of the region in Planck units.

Not clear how to extend the idea to other cases - What are the degrees of freedom - How do we define a ``boundary’’

Emergent space time

Spacetime: like the fermi surface, only defined in the classical limit

Initial singularity

Asymptotic future

De-Sitter

Euclidean conformal field theory

= ?

No explicit example is known!How do we get “emergent” time ?

Is there a dS/CFT ?

Ψ[g] = Z[g]Wave function ofthe universe

Partition function of a Euclidean field theory

WittenStromingerJM

Objections: - dS decays - dS is thermal

Description of the string landscape, and eternal inflation?

What are the futureboundaries?

Who measures that?

• What are the field theory duals of the AdS4

vacua in the landscape (preserving N=1 susy) ?

How do describe the interior of a black hole

• Crunching cosmology• Intrinsically approximate description, up e-S

• There is no obvious “exact” sense in which the interior exists.

• Probably relevant for describing cosmology

• There is probably no other region behind the singularity.

• Can we have crunch-bang transitions ?

Lessons for gauge theories

• We can view the geometry as the solution of the theory in the large ‘t Hooft coupling limit.

• View these as toy model field theories that can be explicitly solved. These theories capture many interesting physical phenomena.

• Explicit examples of confining theories• Thermal aspects are calculable. Transport

properties can be computed. e.g. RHIC applications or as toy model for condensed matter problems.

• Could be describing physics at higher energies (Randall-Sundrum models).

• Examples of constructions of metastable vacua in supersymmetric theories. Use to understand supersymmetry breaking, supersymmetry breaking mediation mechanisms.

• Many examples of conformal N=1 SUSY gauge theories with a geometric description.

• Used to test other dualities.

All coupling• Connect the weak and strong coupling

regimes by computing exactly for all coupling.

• Integrability in N=4 SYM.

• Cusp anomalous dimension known exactly for all values of g2 N.

Future

• More on the dictionary

• Toy model for analyzing field theory dynamics.

• Further exact results in N=4.

Big challenges

• Understand how to describe universes with cosmological singularities

• Clear description of the interior of black holes

• String theory describing the large N limit of ordinary Yang Mills theory, or other large N theories whose duals have stringy curvature.

4 dimensionsExtra dimension

Graviton is localized in the extra dimension

Massive spin 2 particle in 4 dimensions

Gravitational potential in the extra dimension

Jet Physics at strong coupling

Diego Hofman and J. M., to appear

Jet physics at strong coupling

e+

e-

γ

q

q

J ¹ (p)j0i

Why?• New physics at the LHC could involve a strong or

moderately strongly coupled field theory• We would like to understand the transition

between the weak coupling picture of the event where one can qualitatively think in terms of underlying partons and the gravity picture where we do not see the partons in any obvious way.

Weak coupling Strong couling

?

How do we describe the produced state

• Not convenient to talk about partons

• Inclusive observable

• Energy correlation functions.

θ1

θ2

h²(µ1)²(µ2)i

h²(µ1) ¢¢¢²(µn)i

h²(µ1)i

Basham, Brown, Ellis, Love 78

²(µ) =RdtT0ini

Integrated flux of energy at infinity

Three point function

Symmetries determine the three point functionup to two constants

h²(µ)i = h0jj yRT j j0i

h²(µ)i = a+b(cos2µ¡ 13)

weak coupling in QCD:

In any theory with a gravity dual:

In N=4 SYM at weak and strong coupling:

N=1 superconformal field theory, j = R-current:

²(µ) » 1

²(µ) » 1

²(µ) » 1+cos2µ

²(µ) » 1+32a¡ cc

(cos2µ¡13)

(unpolarized e-beam)

θ is angle to beam

Can we get a non-zero b from AdS ?

Bulk couplings between an on shell gravitonand two on shell gluons

- Only two possible couplings in 5 dimensions- a is fixed by the two point function of the currents - The second coupling is a higher derivative correction (which is present in bosonic string theory)- If a ~ b higher derivative correction as important as the leading term)

S5¡ d »RaF 2+bR¹ º±¾F ¹ ºF ±¾

Two point function

Consider a state produced by a scalar operator.

Two point function is a somewhat complicatedfunction of the angle between the two detectors

At very weak coupling is goes like

µ12

h²(µ1)²(µ2)i » g2N 1µ212

; µ12 ¿ 1

Strong coupling computation

x

x

xx

Since we integrate the stresstensor wavefunction localized onan H3 subspace (SO(1,3) symmetry)

h²(µ1)²(µ2)i =RH 3

ª ¤@2¡ ª f (µ1;x)f (µ2;x)

g(x,¢ )

H3

x

²(µ;x) = f (µ;x)

In the gravity approximation the distribution ofenergy depends just on three random variables:a point on H3

1/Δ

General description

Small angle singularity

The small angle singularity is governed by the twist of the operators contributing to the light cone OPE of the stress tensor

Rdx¡ T¡ ¡ (~y)

Rdx¡ T¡ ¡ (0) » y¡ 4+¿n

Rdx¡ On

¡ ¡ ¡

Twist = Δ - S

Non-local operator of spin 3Balitsky Braun 88

h²(µ1)²(µ2)i » g2N 1µ4¡ 2¢12

Single trace and double trace operators.

Double trace operators of the formO¢ O¢

At weak coupling single trace operators have dimension one but at strong couplingthey get large dimension

are the dominant contribution at strong coupling

¿3 = 2+¸ +¢¢¢ for ¸ ¿ 1

¿3 = 21=2¸1=4 ; for ¸ À 1

Related to deep inelastic scattering

It is related to a particular moment of the deepinelastic amplitude of gravitons.

Polchinski Strassler 02

h²(¡ q)²(q)i »R1¡ 1

dsAD I S (s;q2)

Conclusions

• One can define inclusive, event shape variables for conformal theories

• They can be computed at weak and strong coupling• Small angle features governed by the anomalous

dimension of twist two operators • Events are more spherically symmetric at strong coupling,

but not completely uniform. • In a non-conformal theory one would need to face the

details of hadronization. Depending on these details there could be small or large changes.

Strings and Strong Interactions

Before 60s proton, neutron elementary During 60s many new strongly interacting particles Many had higher spins s = 2, 3, 4 …. All these particles different oscillation modes of a string.

This model explained features of the spectrum of mesons.

Rotating String model

constTJm max2 ~ From E. Klempt hep-ex/0101031

Strong Interactions from Quantum ChromoDynamics

3 colors (charges)

They interact exchanging gluons

Electrodynamics Chromodynamics (QCD)

electron

photon

gluong gg

g

Gauge group

U(1) SU(3)

3 x 3 matrices

Gluons carry color charge, sothey interact amongthemselves

Experiments at higher energiesrevealed quarks and gluons

Coupling constant decreases at high energy

g 0 at high energies QCD is easier to study at high energies

Hard to study at low energies

Indeed, at low energies we expect to see confinement

q qFlux tubes of color field = glue

At low energies we have something that looks like a string. There are approximate phenomenological models in terms of strings.

V = T L

How do strings emerge from QCDCan we have an effective low energy theory in terms of strings ?

Gross, Politzer, Wilczek

Large N and stringsGluon: color and anti-color

Open strings mesonsClosed strings glueballs

Take N colors instead of 3, SU(N)

Large N limit

t’ Hooft ‘74

g2N = effective interaction strength when colors are correlated

General Idea

- Solve first the N=∞ theory.

- Then do an expansion in 1/N.

- Set 1/N =1/3 in that expansion.

The N=∞ case

- It is supposed to be a string theory

- Try to guess the correct string theory

- Two problems are encountered.

1. Simplest action = Area

Not consistent in D=4 ( D=26 ? )

At least one more dimension (thickness)Polyakovgenerate

2. Strings theories always contain a state with m=0, spin =2: a Graviton.

But: - In QCD there are no massless particles. - This particle has the interactions of gravity For this reason strings are commonly used to study quantum gravity. Forget about QCD and use strings as a theory of quantum gravity. Superstring theory, unification, etc.

But what kind of string theory should describe QCD ?

Scherk-SchwarzYoneya

Lovelace

We need to find the appropriate 5 dimensional geometry

It should solve the equations of string theory

They are a kind of extension of Einstein’s equations

Very difficult so solve

Consider a simpler case first. A case with more symmetry.

We consider a version of QCD with more symmetries.

Most supersymmetric QCD

Supersymmetry

Bosons Fermions

Gluon Gluino

Many supersymmetries

B1 F1B2 F2

Maximum 4 supersymmetries, N = 4 Super Yang Mills

Susy might be present in the real world but spontaneously broken at low energies.So it is interesting in its own right to understand supersymmetric theories.

We study this case because it is simpler.

RamondWess, Zumino

Similar in spirit to QCD

Difference: most SUSY QCD is scale invariant

Classical electromagnetism is scale invariant V = 1/rQCD is scale invariant classically but not quantum mechanically, g(E)

Most susy QCD is scale invariant even quantum mechanically

Symmetry group

Lorentz + translations + scale transformations + other

These symmetries constrain the shape of the five dimensional space.

ds2 = R2 w2 (z) ( dx23+1 + dz2 )

redshift factor = warp factor ~ gravitational potential

Demanding that the metric is symmetric under scale transformations x x , we find that w(z) = 1/z

This metric is called anti-de-sitter space. It has constant negative curvature, with a radius of curvature given by R.

Boundary

ds2 = R2 (dx23+1 + dz2)

z2

R4

AdS5

z = 0z

z = infinity

Gravitational potential w(z)

z(The gravitational potential does not have a minimum can have massless excitationsScale invariant theory no scale to set the mass )

Anti de Sitter space

Solution of Einstein’s equations with negative cosmological constant.

De Sitter solution with positive cosmological constant, accelerated expanding universe( )

Two dimensional negatively curved space

Spatial section of AdS = Hyperbolic space

R = radius of curvature

Light rays Massive particles

Time

The Field theory is defined on the boundary of AdS.

The space has a boundary.

It is infinitely far in spatial distance

A light ray can go to the boundary and back in finite time, as seen from an observer in the interior. The time it takes is proportional to R.

Building up the Dictionary

Graviton stress tensor

z)y)x) Field theory

= Probability amplitude that gravitons go between given points on the boundary

Gubser, Klebanov,Polyakov - Witten

Other operatorsOther fields (particles) propagating in AdS.

Mass of the particle scaling dimension of the operator

2)(42 mR

We expected to have string theory on AdS. Supersymmetry D=10 superstring theory on AdS x (something)5 5

S5

Type IIB superstrings on AdS x S

5

55

5-form field strength F = generalized magnetic field quantized

NFS

5

(J. Schwarz)

Most supersymmetric QCD

String Theory

Free strings

String Tension = T = 2

1

slsl = string length

Relativistic, so T = (mass)/(unit length)

Excitations along a stretched string travel at the speed of light

Closed strings

Can oscillate Normal modes Quantized energy levels

Mass of the object = total energy

,

sl

M=0 states include a graviton (a spin 2 particle)

First massive state has M ~ T 2

VenezianoScherkSchwarzGreen …..

String Interactions

Splitting and joining

g String theory Feynman diagram

Simplest case: Flat 10 dimensions and supersymmetric

Precise rules, finite results, constrained mathematical structure

At low energies, energies smaller than the mass of the first massive string state

Gravity theory

( Incorporates gauge interactions Unification )

Radius of curvature >> string length gravity is a good approximation

ls

R

Most supersymmetry QCDtheory

String theory on AdS x S 5

5=

Radius of curvature

(J.M.)

sYMAdSSlNgRR

4/12

55

Duality:

g2 N is small perturbation theory is easy – gravity is bad

g2 N is large gravity is good – perturbation theory is hard

Strings made with gluons become fundamental strings.

Particle theory = gravity theory

N colors N = magnetic flux through S5

Where Do the Extra Dimensions Come From?

3+1 AdS5 radial dimension

z Boundary

Interior Gluons live here

Strings live here

What about the S5 ?

• Related to the 6 scalars• S5 other manifolds = Most susy QCD less susy

QCD.• Large number of examples

Klebanov, Witten, Gauntlett, Martelli, Sparks,Hannany, Franco, Benvenutti, Tachikawa, Yau …..

string

Boundary

q

q

Quark anti quark potential

V = potential = proper length of the string in AdS

L

NgV

2

L

NgV

2

Weak coupling result:

Confining Theories

Add masses to scalars and fermions pure Yang Mills at low energies confining theory. There are many concrete examples. At strong coupling gravity solution is a good description.

w(z0) > 0

boundary

String at z0 has finite tension from the point of view of the boundary theory. Graviton in the interior massive spin=2 particle in the boundary theory = glueball.

Gravitational potential or warp factor

w(z)

zz0

Checking the conjecture• It is hard because either one side is strongly

coupled or the other. • Supersymmetry allows many checks. Quantities

that do not depend on the coupling. • More recently, ``integrability’’ allowed to check

the conjecture for quantities that have a non-trivial dependence on the coupling, g2N.

• One can vividly see how the gluons that live in four dimensions link up to produce strings that move in ten dimensions. …

Minahan, Zarembo, Beisert, Staudacher, Arutyunov, Frolov, Hernandez, Lopez, Eden

What can we learn about gravity from the field theory ?

• Useful for understanding quantum aspects of black holes

The relation connects a quantum field theory to gravity.

Black holesGravitational collapse leads to black holes

Classically nothing can escape once it crosses the event horizon

Quantum mechanics implies that black holes emit thermal radiation. (Hawking)

MGrT

Ns

11

M

MKT sun810

Black holes evaporate

Evaporation time 3

12universe 10

Kg

M

Temperature is related to entropy

dM = T dS S =Area of the horizon 4 LPlanck

2

What is the statistical interpretation of this entropy?

(Hawking-Bekenstein)

Black holes in AdS

Thermal configurations in AdS.

Entropy: SGRAVITY = Area of the horizon = SFIELD THEORY = Log[ Number of states]

Evolution: Unitary

Solve the information paradox raised by S. Hawking

Confining Theories and Black Holes

Confinement

Deconfinement= black hole (black brane)

Low temperatures

High temperatures

Gravitational potential

Extra dimension Extra dimension

Horizon

z=z0 z=0

Black hole Very low shear viscosity similar to what is observed at RHIC: “ the most perfect fluid”

Kovtun, Son, Starinets, Policastro

High energy collision produces a black hole = droplet of deconfined phase ~ quark gluon plasma .

z=z0 , z=0

Black holes in the Laboratory

QCD 5d string theory

Very rough model, we do not yet know the precise string theory

Emergent space time

Spacetime: like the fermi surface, only defined in the classical limit

Lin, Lunin, J.M.

A theory of some universe

• Suppose that we lived in anti-de-sitter space• Then the ultimate description of the

universe would be in terms of a 2+1 dimensional field theory living on the sphere at infinity. (With around 10120 fields to give a universe of the size of ours)

• Out universe is close to de-Sitter. Could we have a similar description in that case ?

Conclusions

- Gravity and particle physics are “unified” Usual: Quantum gravity particle physics.

New: Particle physics quantum gravity.

- Black holes and confinement are related- Emergent space-time. Started from a theory

without gravity got a theory in higher dimensions with gravity.

- Tool to do computations in gauge theories.- Tool to do computations in gravity.

Future

Field theory:

Gravity:

Theories closer to the theory of strong interactions

Solve large N QCD

Quantum gravity in other spacetimes

Understand cosmological singularities