Transcript QCD, Strings and Black holes

Remarks on the
gauge/gravity duality
Juan Maldacena
Field Theory
Gauge Theories
QCD
=
Gravity theory
Quantum Gravity
String 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
A ij
i j
Polchinski 95
Horowitz
Strominger 91
A ij
i j
String theory
on AdS5 x S5
N=4 SU(N) Yang Mills
theory in 3+1 dimensions
³
g2 N =
R
ls
´4
JM 97
Conformal
symmetry
AdS/CFT
Duality:
³
g2 N =
R
ls
´4
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…
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 mechanics
and 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
Is there a dS/CFT ?
Asymptotic future
Euclidean conformal
field theory
=?
Witten
Strominger
JM
De-Sitter
Initial singularity
Wave function of
the universe
Ψ[g] = Z[g]
Partition function of a
Euclidean field theory
No explicit example is known!
Objections: - dS decays
- dS is thermal
How do we get “emergent” time ?
Description of the string landscape,
and eternal inflation?
What are the future
boundaries?
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.
Gravitational potential
in the extra dimension
Graviton is
localized in
the extra dimension
4 dimensions
Extra dimension
Massive spin 2
particle in 4 dimensions
Jet Physics at strong coupling
Diego Hofman and J. M., to appear
Jet physics at strong coupling
q
eγ
e+
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.
Strong couling
Weak coupling
?
How do we describe the produced
state
• Not convenient to talk about partons
• Inclusive observable
Basham, Brown, Ellis, Love 78
• Energy correlation functions.
θ1
h²(µ1 )i
h²(µ1 )²(µ2 )i
h²(µ1 ) ¢¢¢²(µn )i
θ2
Integrated flux
R of energy at infinity
²(µ) =
dtT 0i n i
h²(µ)i = h0jj y
R
T j j0i
Three point function
Symmetries determine the three point function
up to two constants
h²(µ)i = a + b(cos2 µ ¡
1)
3
weak coupling in QCD:
(unpolarized e-beam)
²(µ) » 1 + cos2 µ
θ is angle to beam
In any theory with a gravity dual:
²(µ) » 1
In N=4 SYM at weak and strong coupling:
²(µ) » 1
N=1 superconformal field theory, j = R-current:
3a¡ c
1
2
²(µ) » 1 +
(cos µ ¡ )
2 c
3
Can we get a non-zero b from AdS ?
Bulk couplings between an on shell graviton
and two on shell gluons
S5¡
d
»
R
aF 2 + bR¹ º ±¾F ¹ º F ±¾
- 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)
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
h²(µ1 )²(µ2 )i » g2 N
1
µ2
;
µ12 ¿ 1
12
µ12
Strong coupling computation
Since we integrate the stress
x tensor  wavefunction localized on
an H3 subspace (SO(1,3) symmetry)
x
x
x
g(x,¢ )
h²(µ1 )²(µ2 )i =
R
H3
ª ¤ @2 ª f (µ1 ; x)f (µ2 ; x)
¡
General description
1/Δ
x
H3
²(µ; x) = f (µ; x)
In the gravity approximation the distribution of
energy depends just on three random variables:
a point on H3
Small angle singularity
h²(µ1 )²(µ2 )i » g2 N
1
µ4 ¡
2¢
12
The small angle singularity is governed by the
twist of the operators contributing to the light
cone OPE of the stress tensor
R
dx ¡ T¡
¡
(~
y)
R
Twist = Δ - S
dx ¡ T¡
¡
(0) » y¡
Balitsky Braun 88
4+ ¿n
R
dx ¡ O¡n ¡
¡
Non-local operator of spin 3
Single trace and double trace operators.
Double
trace
operators
of
the
form
O 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
¿3
=
=
2 + ¸ + ¢¢¢
21=2 ¸ 1=4 ;
for ¸ ¿ 1
for ¸ À 1
Polchinski Strassler 02
Related to deep inelastic scattering
It is related to a particular moment of the deep
inelastic amplitude
R of gravitons.
h²(¡ q)²(q)i » 1 dsA D I S (s; q2 )
¡ 1
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
m 2 ~ TJ max  const
From E. Klempt hep-ex/0101031
Strong Interactions from Quantum ChromoDynamics
3 colors (charges)
Experiments at higher energies
revealed quarks and gluons
They interact exchanging gluons
Chromodynamics (QCD)
Electrodynamics
photon
g
g
g
gluon
g
electron
3 x 3 matrices
Gauge group
U(1)
SU(3)
Gluons carry
color charge, so
they interact among
themselves
Gross, Politzer, Wilczek
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
q
V = T L
Flux 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.
How do strings emerge from QCD
Can we have an effective low energy theory in terms of strings ?
Large N and strings
Gluon: color and anti-color
Take N colors instead of 3, SU(N)
Large N limit
g2N = effective interaction strength
when colors are correlated
Open strings  mesons
Closed strings  glueballs
t’ Hooft ‘74
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 ? )
generate
At least one more dimension (thickness)
Lovelace
Polyakov
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
Scherk-Schwarz
Yoneya
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 ?
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
Ramond
Wess, Zumino
Many supersymmetries
B1
B2
Maximum
F1
F2
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.
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) ( dx23+1 + dz2 )
redshift factor = warp factor ~ gravitational potential
Demanding that the metric is symmetric under scale transformations
x  l x , we find that w(z) = 1/z
ds2 = R2 (dx23+1 + 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.
w(z)
Gravitational potential
z
(The gravitational potential does not have a minimum  can have massless excitations
Scale 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
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.
Building up the Dictionary
Graviton
Tmn
stress tensor
Gubser, Klebanov,
Polyakov - Witten
Tmn
Tmn
<
Tmn(x) Tmn(y)
Tmn(z) >
Field theory
= Probability amplitude that gravitons
go between given points on the boundary
Other operators
Other fields (particles) propagating in AdS.
Mass of the particle
scaling dimension of the operator
  2  4  (mR) 2
Most supersymmetric QCD
We expected to have string theory on AdS.
5
Supersymmetry
D=10 superstring theory on AdS 5x (something)
S5
Type IIB superstrings on AdS 5x S 5
(J. Schwarz)
5-form field strength F = generalized magnetic field quantized

S5
F N
Veneziano
Scherk
Schwarz
Green
…..
String Theory
Free strings
String
1
Tension = T = 2
ls
,
ls
= string length
Relativistic, so T = (mass)/(unit length)
Excitations along a stretched string travel at the speed of light
Closed strings
Can oscillate
ls
Normal modes
Mass of the object = total energy
M=0 states include a graviton (a spin 2 particle)
First massive state has M 2 ~ T
Quantized energy levels
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
Gravity theory
ls
R
Radius of curvature >> string length  gravity is a good approximation
( Incorporates gauge interactions  Unification )
Particle theory = gravity theory
Most supersymmetry QCD
theory
N colors
Radius of curvature
=
String theory on
AdS5 x S 5
(J.M.)
N = magnetic flux through S5
(
RS 5  RAdS5  g
2
YM

1/ 4
N
ls
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.
Where Do the Extra Dimensions Come From?
3+1  AdS5  radial dimension
Strings live here
Interior
z
Gluons live here
Boundary
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 …..
Quark anti quark potential
string
q
Boundary
q
Weak coupling result:
V = potential = proper length of the string
in AdS
g2N
V 
L
g2N
V 
L
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(z)
Gravitational potential
or warp factor
boundary
w(z0) > 0
z0
z
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.
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
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
Black holes
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)
T
1
1

rs GN M
M 
T  108 K  sun 
 M 
Black holes evaporate
Evaporation time
 M 
   universe 12 
 10 Kg 
3
Temperature is related to entropy
Area of the horizon
dM = T dS
S=
4 L2Planck
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
Low temperatures
Confinement
High temperatures
Deconfinement=
black hole (black brane)
Gravitational
potential
Horizon
Extra dimension
Extra dimension
Black holes in the Laboratory
QCD  5d string theory
z=0
z=z0 ,
High energy collision produces a black hole =
droplet of deconfined phase ~
quark gluon plasma .
Black hole Very low shear viscosity
similar to what is observed at RHIC:
“ the most perfect fluid”
z=0
z=z0
Kovtun, Son, Starinets, Policastro
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:
Theories closer to the theory of strong interactions
Solve large N QCD
Gravity:
Quantum gravity in other spacetimes
 Understand cosmological singularities