The large N limit of Field Theories and Gravity

QCD, Strings and Black holes
The large N limit of Field Theories and
Gravity
Juan Maldacena

Field Theory

=

Gravity theory

Gauge Theories QCD
Plan
QCD, Strings, the large N limit Supersymmetric QCD
N large
Gravitational theory in 10 dimensions
Calculations Correlation functions Quark-antiquark potential
Black holes

Quantum Gravity String theory

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

This model explained “Regge trajectories”

Rotating String model
m2 ~ TJmax + const

From E. Klempt hep-ex/0101031

Strong Interactions from Quantum ChromoDynamics

Experiments at higher energies revealed quarks and gluons

3 colors (charges) They interact exchanging gluons

Electrodynamics

photon

g

g

electron
Gauge group

U(1)

Chromodynamics (QCD)

g
g gluon

SU(3)

3 x 3 matrices
Gluons carry color charge, so they interact among themselves

Coupling constant decreases at high energy

Gross, Politzer, Wilczek

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

Flux tubes of color field = glue

V = T L

At low energies we have something that looks like a string

Can we have an effective theory in terms of strings ?

Large N limit

Take N colors instead of 3, SU(N)

t’ Hooft ‘74

Large N and strings
Gluon: color and anti-color
Open strings Æ mesons Closed strings Æ glueballs Looks like a string theory, but…

1. Simplest action = Area
Not consistent in D=4 ( D=26 ? )
generate
At least one more dimension (thickness)

Polyakov

2. Strings theories always contain a state with m=0, spin =2: a Graviton.
For this reason strings are commonly used to study quantum gravity

Scherk-Schwarz Yoneya

We combine these two problems into a solution. We will look for a 5 dimensional theory that contains gravity. We have to find an appropriate 5 dimensional curved spacetime.

Most supersymmetric QCD

Supersymmetry Bosons

Fermions

Ramond Wess, Zumino

Gluon

Gluino

Many supersymmetries

B1

F1

B2

F2

Maximum 4 supersymmetries, N = 4 Super Yang Mills

Aμ Vector boson Ψα 4 fermions (gluinos) ΦI 6 scalars
All NxN matrices

spin = 1 spin = 1/2 spin = 0

SO(6) symmetry

Susy might be present in the real world but spontaneously broken at low energies.

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
The string should move in a space of the form
ds2 = R2 w2 (z) ( dx23+1 + dz2 )
redshift factor = warp factor ~ gravitational potential
Demanding that the metric is symmetric under scale transformations x Æ λ x , we find that w(z) = 1/z

ds2 = R2 (dx23+1 + dz2) z2

R4

Boundary

AdS5

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