String Theory Demystified

CHAPTER 15 The Holographic Principle 259
• But, gravity can propagate into the bulk. In the bulk, which is the interior
volume of the AdS sphere, gravity is the only interaction. Inside the ball of the
AdS geometry, the theory is supergravity. We won’t get into supergravity in
this book but you can look it up on the arXiv if interested in learning about it.
The conformal theory that describes particles and their interactions is
supersymmetric and is called super Yang-Mills theory or SYM for short. The
gauge group for SYM is SU(N). So the AdS/CFT correspondence can be framed
as follows:
• There is a super Yang-Mills theory with SU(N) on the surface of the ball.
• There is bulk supergravity in the interior of the ball.
In string theory, the number of degrees of freedom for the SYM is constrained by
three factors:
• The fundamental string length
• The string coupling
• The curvature of AdS space
The number of degrees of freedom for SYM is ∼ N 2 since the gauge group is
SU(N) and it has a gauge coupling g YM . The constraint on N is quantifi ed in the
following relationship:
R= ( g N)
s s
/ 14
The gauge-coupling is related to the string-coupling constant as:
2
g = g YM s
Now we would like to introduce a cutoff in the bulk. We divide up the sphere into
little cells such that the total number of cells in the sphere is ∼ δ −3
for some cell d.
That is,
• We cut off the information storage capacity by replacing the continuum of
space by cells of size d.
• There is a single degree of freedom in each cell.
With the total number of degrees of freedom for the SYM theory proportional to
N 2 , we fi nd that the total number of degrees of freedom with the cutoff is
N
dof
N
A N
2
= = 3
δ
R
2
3