# Presentation: 673037

## Strings, Gravity and the Large N Limit of  Gauge Theories

- Juan Maldacena
- Institute for Advanced Study
- Princeton, New Jersey

## =

- Gauge Theories
- QCD

- Quantum Gravity
- String theory

- _Plan_

- QCD,  Strings,  the large  N limit
- Supersymmetric Yang Mills

- Gravitational theory in 10 dimensions

- N large

- Calculations      Correlation functions

- Quark-antiquark potential

-  Confining theories, Black holes

- Holography

## Strings and QCD

- In the sixties many new mesons and hadrons were discovered. It was suggested that these might not be new fundamental particles. Instead they could be viewed as different oscillation modes of a string.

## Gauge group 

- in terms of quarks and gluons.

- 3 colors  (charges)
- They interact exchanging gluons

- Electrodynamics

- Chromodynamics

- Gauge group

- U(1)

- SU(3)

- 3 x 3 matrices

- Gluons carry
- color charge, so
- they interact among
- themselves

## 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

- At low energies we have something that looks like a string

- V  =  T  L

- Can we have an effective theory in terms of strings?

-  t’ Hooft Large N limit

-  t’ Hooft ‘74

- Take N colors instead of 3, SU(N)

## ‘t Hooft Limit 

- Gluon propagator

- i

- j

- Interactions

- i

- j

- k

- Corrections

- i

- j

- i

- j

- k

- 1

- +

- +

-  g2N

- Planar corrections give factors of (g2N)

- power

## Sum of all planar diagrams will give us a general function  f(g2N)

- Sum of all planar diagrams will give us a general function  f(g2N)

_**Limit:**_

-  Keep g2N fixed and take N to infinity

- Non-planar diagrams are suppressed by powers of  1/N

- Thinking of the planar diagrams as discretizing the worldsheet of a string,
- we see that if  g2N becomes of order one we recover a continuum string
- worldsheet.

- The string coupling constant is of order  1/N

- This might be a good approximation to QCD at low energies, when the coupling is large.

## Closed strings would be glueballs.

- Open strings would be the mesons.

- _What is the precise  form of the continuum worldsheet action for this string?_

- _Problems:_

- 1) Strings do not make sense in 4 (flat) dimensions
- 2) Strings always include a graviton, ie., a particle with m=0,  s=2

-  For this reason strings are normally studied as a model for quantum
-  gravity.

## Most Supersymmetric Yang Mills Theory

- Supersymmetry

- Bosons                  Fermions
- Gluon                    Gluino

- Many supersymmetries

- B1              F1
- B2              F2

- Maximum      4 supersymmetries

-  Vector boson                    spin = 1
-  4 fermions (gluinos)         spin = 1/2
-  6 scalars                            spin = 0

- Susy might be present in the real world but spontaneously broken at low energies.
- We study this case because it is simpler.

- All NxN matrices

## Symmetry group 

- Difference:  ***N******  ***=4  Yang Mills  is scale invariant

- Classical electromagnetism is scale invariant
-                         V = 1/r
- QCD is scale invariant classically but not quantum mechanically, g(E)
***N ***=4    Yang Mills is scale invariant even quantum mechanically

- _Symmetry group_

- Lorentz + translations  + scale transformations + other

- The string should move in a space of the form
-  ds2  =  R2   w2 (z)   ( dx23+1  +   dz2 )

-  redshift factor  =  warp factor

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

- 

## AdS5 x  S5  

- curvature, with a curvature scale given by R.

- This Yang Mills theory has a large amount of supersymmetry, the same as ten dimensional superstring theory on flat space.

- We add  an  S5 so that we have a ten dimensional space.

**AdS****5**** x  S****5****  **

- Boundary

**ds****2****  =  R****2****    (dx****2****3+1****  +  dz****2****)**
**	****	****	****	****            ****z****2**

- R4

- AdS5

- z = 0

- z

- z = infinity

## Anti de Sitter Space 

- Solution of Einstein’s equations with negative cosmological constant.

- De Sitter      solution with positive cosmological constant (group SO(1,5) )

- Spatial cross section of AdS

## Spatial section of AdS = Hyperbolic space

## =

- theory

- String theory on

- Radius of curvature

- (J.M.)

- Duality:

- Strings made with gluons become fundamental strings.

- AdS5 x S5

## String Theory

- _Free strings_

- String

- Tension =  T  =

- = 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  M2 ~ T

## String Interactions

- Splitting and joining

- g

- String theory Feynman diagram

-  Simplest case:  Flat 10 dimensions and supersymmetric

- Precise rules for computing the amplitudes  that yield finite results

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

- Gravity theory

- Very constrained  mathematical structure

## Non-perturbative Aspects

- In field theories  we can have solitons

-  e.g. magnetic monopoles  (monopoles of GUT theories)

- Collective excitations that are stable  (topologically)

-  g = coupling constant

- In string theory

- we have D-p-branes

- Can have different dimensionalities

-  p=0             D-0-brane     D-particle

-    p=1             D-1-brane     D-string

-  p=2             D-2-brane     membrane

-   etc.

## A

- Their excitations are described by open strings ending on the brane.
- At low energies these lead to fields living on the brane. These include gauge fields.
- N coincident branes give rise to U(N) gauge symmetry.

- i

- j

-  ij

## =

- theory

- String theory on

- Radius of curvature

- (J.M.)

- Duality:

- Strings made with gluons become fundamental strings.

- AdS5 x S5

## How Do We Use This?

- We would like to do computations of the Yang Mills theory at strong coupling,
- then we just do computations in the gravity theory

- Example:
- Correlation functions of operators in the Yang Mills theory, eg. stress tensor
- correlator

- Gubser, Klebanov,
- Polyakov  - Witten

- Other operators

- Other fields  (particles) propagating in AdS.

- Mass of the particle             scaling dimension of the operator

- x

- y

- z

## Quark Anti-Quark Potential

- V = potential = proper  length of the  string
-                           in AdS

- Baryons                 D-branes

- This is the correct answer for a conformal theory, the theory is not
- confining.

- The reason that we get a small potential at large distances is that the
- string is goes into a region with very small redshift factor.

## Where Do the Extra Dimensions Come From?

- 3+1          AdS           radial dimension

- 5

- z

- Boundary

-  1/z ~  energy scale

- Renormalization group flow:  Motion in the radial direction

-  z=0

-  z=infinity

-  ultraviolet,
-  high energies

-  infrared,
-  low energies

- Five-sphere: is related to the scalars and fermions in the supersymmetric
- Yang-Mills theory.  For other theories the sphere is replaced by other
- manifolds, or it might even not be there.

## Quark Anti-Quark Potential

- V = potential = proper  length of the  string
-                           in AdS

- Baryons                 D-branes

- This is the correct answer for a conformal theory, the theory is not
- confining.

- The reason that we get a small potential at large distances is that the
- string is goes into a region with very small redshift factor.

## Where Do the Extra Dimensions Come From?

- 3+1          AdS           radial dimension

- 5

- z

- Boundary

-  1/z ~  energy scale

- Renormalization group flow:  Motion in the radial direction

-  z=0

-  z=infinity

-  ultraviolet,
-  high energies

-  infrared,
-  low energies

- Five-sphere: is related to the scalars and fermions in the supersymmetric
- Yang-Mills theory.  For other theories the sphere is replaced by other
- manifolds, or it might even not be there.

## Confining Theories

- We can add  masses to the scalars and fermions so that at low energies
- we get a pure Yang-Mills theory. At strong coupling it is possible to
- find the corresponding gravity solution.

- There are various examples of theories with one supersymmetry that are
- confining.

-  The geometry ends in such a way that the
-  warp factor is finite. We can think of this as
-  an end of the world brane. There are various ways
-  in which this can happen.

- Now the string cannot decrease its tension by going to a region with very small redshift factor. Similarly the spectrum of gravity excitations has a mass gap.

## Deconfinement and Black Holes

- For these confining theories we can raise the temperature.  Then we will
- find two phases:
- At low temperatures we just have a gas of gravitons (strings) in the
- geometry we had for T=0.
- At high temperatures a black hole (a black brane) horizon forms.

-  z=0

- boundary

- boundary

- Horizon.  Here the redshift factor g00=0.

- S ~  1

-  _Area of horizon_

- 4 GN

- S=

-   ~  N2

## Randall-Sundrum Models

-  z=z0

-  z=0

- boundary

-  z=z1

- We only consider a portion of the space  with  z1 < z < z0.
- Cutting off the region with z<z1 is equivalent to introducing
- a UV cutoff in the field theory, if we keep the metric on the
- surface z=z1 fixed.  Letting this metric fluctuate we are coupling
- four dimensional gravity.
- The RS models are equivalent to 4D gravity coupled to a conformal
- (or conformal over some energy range) field theory.

## =

- Gauge Theories
- QCD

- Quantum Gravity
- String theory

- _Plan_

- QCD,  Strings,  the large  N limit
- Supersymmetric Yang Mills

- Gravitational theory in 10 dimensions

- N large

- Calculations      Correlation functions

- Quark-antiquark potential

-  Confining theories, Black holes

- Holography

## Deconfinement and Black Holes

- For these confining theories we can raise the temperature.  Then we will
- find two phases:
- At low temperatures we just have a gas of gravitons (strings) in the
- geometry we had for T=0.
- At high temperatures a black hole (a black brane) horizon forms.

-  z=0

- boundary

- boundary

- Horizon.  Here the redshift factor g00=0.

- S ~  1

-  _Area of horizon_

- 4 GN

- S=

-   ~  N2

## Holography

- It has been suggested that all quantum theories of gravity should
- be *holographic.  *
* *
- This means that we should be able to describe all physics within some region
- in terms of a theory living on the boundary of the region, and this theory on the
- boundary should have less than one degree of freedom per Planck area.

-  t’ Hooft
-  Susskind

- Non local mapping

- The AdS/CFT  conjecture is a concrete realization of this holographic principle

- The AdS/CFT conjecture gives a non-perturbative definition of
- quantum gravity in AdS spaces.

## Speculations About Pure Yang Mills 

- In the large N limit of pure Yang Mills (no susy) we expect to find a string theory on five dimensional geometry as follows:

- R(z)

-  Near the boundary the AdS radius goes to zero logarithmically (asymptotic freedom). When R(z) is comparable to the string length  the geometry ends.

- Adding quarks corresponds to adding D-branes extended along all five
- dimensions. The open strings living on these D-branes are the mesons.

- A D0 brane in the interior corresponds to a baryon.

- _Challenges_:  1) Find the precise geometry
-                      2) Solve string theory on it

- R~ ls

## Future

- More examples and various aspects of this correspondence.
- Other large N theories that are closer to the theory of strong interactions (QCD).
- Understand quantum gravity in other spacetimes, especially time-dependent cosmological spacetimes.

## Slide 33