1. Remarks on the gauge/gravity duality
Juan Maldacena

2. = Field Theory Gravity theory Gauge Theories Quantum Gravity QCD
String theory

3. 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 = = + +

4. Look for a string theory in 4d  not consistent
At least 5 dimensions
Polyakov

5. 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
Polchinski 95
Horowitz
Strominger 91 i j A ij

6. A N=4 SU(N) Yang Mills String theory theory in 3+1 dimensionsj A ij
N=4 SU(N) Yang Mills
theory in 3+1 dimensions
String theory
on AdS5 x S5 g 2 N = R l s 4
JM 97

7. Conformal
symmetry
AdS/CFT
Duality: g 2 N = R l s 4

8. Development of the dictionary
Gubser, Klebanov,
Polyakov, Witten
Correlation functions
Wilson loops
Various deformations: masses, marginal deformations
Many new examples, both conformal and non-conformal
Continues today…

9. 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

10. Black holes Entropy = area = statistical entropy in the field theory
Unitary evolution:
Quantum mechanics
and gravity are
compatible

11. 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

12. 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’’

13. Emergent space time Spacetime: like the fermi surface,
only defined in the classical limit

14. = ? Ψ[g] = Z[g] Is there a dS/CFT ? De-Sitter
Witten
Strominger JM
Asymptotic future
Euclidean conformal
field theory
= ?
De-Sitter
Initial singularity
Wave function of
the universe
Partition function of a
Euclidean field theory
Ψ[g] = Z[g]
No explicit example is known!
How do we get “emergent” time ?
Objections: - dS decays
- dS is thermal

15. Description of the string landscape, and eternal inflation?
What are the future
boundaries?
Who measures that?

16. What are the field theory duals of the AdS4 vacua in the landscape (preserving N=1 susy) ?

17. 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 ?

18. 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.

19. 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.

20. 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.

21. Future More on the dictionary
Toy model for analyzing field theory dynamics.
Further exact results in N=4.

22. 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.

24. Gravitational potential
in the extra dimension
Graviton is
localized in
the extra dimension
Massive spin 2
particle in 4 dimensions
Extra dimension
4 dimensions

25. Jet Physics at strong coupling
Diego Hofman and J. M., to appear

26. Jet physics at strong couplingq e- γ e+ q J ( p ) j i

27. 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 ?

28. How do we describe the produced state
Not convenient to talk about partons
Inclusive observable
Energy correlation functions.
Basham, Brown, Ellis, Love 78 h ( 1 ) i θ1 θ2 h ( 1 ) 2 i h ( 1 ) n i

29. h ² ( µ ) i = j R T ² ( µ ) = R d t T n
Integrated flux of energy at infinity ( ) = R d t T i n h ( ) i = j y R T
Three point function
Symmetries determine the three point function
up to two constants h ( ) i = a + b c o s 2 1 3

30. In any theory with a gravity dual:
weak coupling in QCD:
In any theory with a gravity dual:
In N=4 SYM at weak and strong coupling:
(unpolarized e-beam)
θ is angle to beam ( ) 1 + c o s 2 ( ) 1 ( ) 1
N=1 superconformal field theory, j = R-current: ( ) 1 + 3 2 a c o s

31. S » R a F + b - a is fixed by the two point function of the currents
Can we get a non-zero b from AdS ?
Bulk couplings between an on shell graviton
and two on shell gluons S 5 d R a F 2 + b
- 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)

32. Two point function Consider a state produced by a scalar operator.
Two point function is a somewhat complicated
function of the angle between the two detectors
At very weak coupling is goes like h ( 1 ) 2 i g N ; 1 2

33. Strong coupling computation
Since we integrate the stress
tensor  wavefunction localized on
an H3 subspace (SO(1,3) symmetry) x x x x g ( x , ) h ( 1 ) 2 i = R H 3 @ f ; x

34. In the gravity approximation the distribution of
General description
1/Δ x H3 ( ; x ) = f
In the gravity approximation the distribution of
energy depends just on three random variables:
a point on H3

35. Small angle singularityh ( 1 ) 2 i g N 4
The small angle singularity is governed by the
twist of the operators contributing to the light
cone OPE of the stress tensor
Twist = Δ - S R d x T ( ~ y ) 4 + n O
Non-local operator of spin 3
Balitsky Braun 88

36. O Single trace and double trace operators.
Double trace operators of the form O
are the dominant contribution at strong coupling
At weak coupling single trace operators
have dimension one but at strong coupling
they get large dimension 3 = 2 + f o r 1 ( ) 4 ; À

37. Related to deep inelastic scattering
Polchinski Strassler 02
Related to deep inelastic scattering
It is related to a particular moment of the deep
inelastic amplitude of gravitons. h ( q ) i R 1 d s A D I S ; 2

38. 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.

42. 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
From E. Klempt hep-ex/

43. Strong Interactions from Quantum ChromoDynamics
3 colors (charges)
They interact exchanging gluons
Experiments at higher energies
revealed quarks and gluons
Chromodynamics (QCD)
Electrodynamics
electron
photon
gluon g
Gauge group
3 x 3 matrices
Gluons carry
color charge, so
they interact among
themselves
U(1)
SU(3)

44. g at high energies V = T L q
Coupling constant decreases at high energy
Gross, Politzer, Wilczek g
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
Flux tubes of color field = glue
V = T L
At low energies we have something that looks like a string. There are
approximate phenomenological models in terms of strings.
How do strings emerge from QCD
Can we have an effective low energy theory in terms of strings ?

45. Large N and strings Gluon: color and anti-color Open strings  mesons
Take N colors instead of 3, SU(N)
t’ Hooft ‘74
Large N limit
g2N = effective interaction strength
when colors are correlated
Open strings  mesons
Closed strings  glueballs

46. General Idea - Solve first the N=∞ theory.
- Then do an expansion in 1/N.
- Set 1/N =1/3 in that expansion.

47. The N=∞ case - It is supposed to be a string theory
- Try to guess the correct string theory
- Two problems are encountered.

48. Not consistent in D=4 ( D=26 ? )
1. Simplest action = Area
Not consistent in D= ( D=26 ? )
Lovelace
generate
Polyakov
At least one more dimension (thickness)
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-Schwarz
Yoneya

49. 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.

50. Most supersymmetric QCD
Supersymmetry
Ramond
Wess, Zumino
Bosons Fermions
Gluon Gluino
Many supersymmetries
B F1
B F2
Maximum 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.

51. Similar in spirit to QCD
Difference: most SUSY QCD is scale invariant
Classical electromagnetism is scale invariant
V = 1/r
QCD 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) ( dx 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 l

52. This metric is called anti-de-sitter space. It has constant negative
ds2 = R2 (dx dz2) z2 R4
AdS5
Boundary z
z = 0
z = infinity
This metric is called anti-de-sitter space. It has constant negative
curvature, with a radius of curvature given by R.
Gravitational potential
w(z) z
(The gravitational potential does not have a minimum  can have massless excitations
Scale invariant theory  no scale to set the mass )

53. 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

54. Spatial section of AdS = Hyperbolic space

55. The Field theory is defined on the boundary of AdS.
R = radius of curvature
Time
Light rays
Massive particles
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.
The Field theory is defined on the boundary of AdS.

56. < > Building up the Dictionary
Graviton stress tensor mn T
Gubser, Klebanov,
Polyakov - Witten mn T mn T
<
> mn
T (x) mn
T (y) mn
T (z)
= Probability amplitude that gravitons
go between given points on the boundary
Field theory
Other operators
Other fields (particles) propagating in AdS.
Mass of the particle scaling dimension of the operator

57. Most supersymmetric QCD
We expected to have string theory on AdS.
Supersymmetry D=10 superstring theory on AdS x (something) 5 5 5 S 5 5 5
Type IIB superstrings on AdS x S
(J. Schwarz)
5-form field strength F = generalized magnetic field quantized

58. String Theory , First massive state has M ~ T Free strings
Veneziano
Scherk
Schwarz
Green
…..
Free strings
Tension = T = ,
String
= 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
M=0 states include a graviton (a spin 2 particle)
First massive state has M ~ T 2

59. ( Incorporates gauge interactions  Unification )
String Interactions
Splitting and joining
String theory Feynman diagram g
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 ls
Gravity theory R
Radius of curvature >> string length  gravity is a good approximation
( Incorporates gauge interactions  Unification )

60. = Particle theory = gravity theory Most supersymmetry QCD theory
String theory on
(J.M.)
AdS x S 5
N = magnetic flux through S5
N colors
Radius of curvature
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.

61. z Where Do the Extra Dimensions Come From?
3+1  AdS5  radial dimension
Strings live here
Interior
Gluons live here z
Boundary

62. 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 …..

63. Quark anti quark potential
string
Boundary q
V = potential = proper length of the string
in AdS
Weak coupling result:

64. 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.
Gravitational potential
or warp factor
w(z) z z0
boundary
w(z0) > 0
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.

65. 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

66. What can we learn about gravity from the field theory ?
The relation connects a quantum field theory to gravity.
What can we learn about gravity from the field theory ?
Useful for understanding quantum aspects of black holes

67. Black holes Black holes evaporate S =
Gravitational 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)
Black holes evaporate
Evaporation time
Temperature is related to entropy
Area of the horizon
4 L
dM = T dS
S =
(Hawking-Bekenstein) 2
Planck
What is the statistical interpretation of this entropy?

68. 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

69. Confining Theories and Black Holes
Low temperatures
High temperatures
Deconfinement=
black hole (black brane)
Confinement
Gravitational
potential
Horizon
Extra dimension
Extra dimension

70. Black holes in the Laboratory
z=z0 ,
z=0
QCD  5d string theory
High energy collision produces a black hole =
droplet of deconfined phase ~
quark gluon plasma .
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
Very rough model, we do not yet know the precise string theory

71. Emergent space time Spacetime: like the fermi surface,
only defined in the classical limit
Lin, Lunin, J.M.

72. 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 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 ?

73. 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.

74. Future
Field theory:
Theories closer to the theory of strong interactions
Solve large N QCD
Gravity:
Quantum gravity in other spacetimes
Understand cosmological singularities