String Theory Demystified

CHAPTER 15 The Holographic Principle 261
Using T = RTand
R= ( g N)
SYM s s
/ 14 we obtain
( TR)
Now if we take S = A/4G then we fi nd
T
3
3
SYM
Sg
= 8
R
A
= 3
R
Now we regulate the SYM so that the maximum T SYM is 1/d. Then we fi nd the
maximum area to be
A
max =
Regulation of the super Yang-Mills theory on the boundary gives a holographic
description with one bit per Planck area.
An interesting result derived by Susskind and Witten is the IR-UV connection.
This relates IR divergences in the bulk to UV divergences on the boundary. Consider
a string in the bulk that ends on the boundary. The ends of the string correspond to
a point charge in the Yang-Mills theory. Now, just thinking back to the self-energy
of an electron, you will realize that a point charge in the Yang-Mills theory has a
divergent infi nite self-energy. This is an UV divergence. The divergence of the bulk
string is proportional to 1/d, while d plays the role of a short distance regulator for
UV divergence in SYM theory.
The energy of the string is linearly divergent at the boundaries. Since this
divergence is softer, we say that it is an IR divergence. The propagator for a particle
of mass m in the bulk is given by
∆=
R
2 8
s s
3
3
δ
δ m
m
1 2
X − X
where we have relgulated the area using A≈R 3 δ and δ 1. Super Yang-Mills
theory is a conformal fi eld theory. Remember Chap. 5? We learned how to calculate
operator product expansions there. For super Yang-Mills theory:
p
1 2 1 2
− p
Y( X ) Y( X ) = µ X −X
You can see that you can transform between these two expressions. What this
means is that a propagator for a particle of mass m in the bulk can be transformed
into a power law in the conformal fi eld theory on the boundary.