String Theory Demystified

CHAPTER 13 D-Branes 231
a set of D-branes with spatial dimensions pqr , , ,... in various orientations. However,
here we will stick to the simplest case, which is to consider two Dp-branes that are
a a
parallel but located at different coordinates x1 and x2.
We will describe this case in
a moment and see how the energy from stretching a string between the branes
changes the mass spectrum. However, before doing that we take a brief aside to
introduce Chan-Paton factors.
Chan-Paton factors were introduced into string theory because Yang-Mills
theories are necessary to describe the particle interactions of the standard model of
particle physics. Before D-branes were known about, the technique used was to
attach non-abelian degrees of freedom to the endpoints of open strings. These
degrees of freedom were denoted quark and antiquark, respectively. These names
came about by historical accident, string theory was originally proposed as a
description of the strong interaction, but it was later displaced from that role by
quantum chromodynamics(QCD).
There are i = 1,..., N possible states of a string endpoint. Since an open string
has two endpoints, it has two Chan-Paton indices ij. An open string state can be
written as:
a
The λij N
∑
a
pa ; pij ; λij
=
ij , = 1
are matrices that are called Chan-Paton factors. It turns out that amplitudes
obtained when including Chan-Paton factors are invariant under U (N) transformations,
which can be transformed into a local U (N) gauge symmetry in spacetime.
This is exactly what is required for Yang-Mills theories, so it provides a basis
for including the standard model in string theory.
After D-branes were discovered, the Chan-Paton indices were reinterpreted.
Now we suppose that there are multiple D-branes with integer labels, and string
endpoints can be located at D-brane i and j for example. It turns out that multiple
D-branes are what give rise to the standard model of particle physics in string
theory. In particular, coincident D-branes give rise to massless gauge fi elds in the
following way:
• If there are N coincident Dp-branes, there are N 2
massless gauge fi elds.
• This characterizes a U( N)
Yang-Mills theory on the world-volume of the N
coincident D-branes.
We have already seen that a single Dp-brane has a photon state. This is consistent
with the outline we are developing here. We have a single D-brane, and the gauge
group of the electromagnetic fi eld is U( 1 ) . If we add more D-branes in the right
way, we can get the number of gauge fi elds that we want.