String Theory and M-Theory

6.5 World-volume actions for D-branes 239
special cases it may be concentrated on a lower-dimensional hypersurface,
for example a brane within a brane.
In the presence of space-time curvature the Chern–Simons term contains
an additional factor involving differential forms constructed from the curvature
tensor. We won’t describe this factor, since it would require a rather
long digression. It reduces to 1 in a flat space-time, which is the case considered
here.
The nonabelian case
When N Dp-branes coincide, the world-volume theory is a U(N) gauge theory.
Almost all studies of nonabelian D-brane actions use the static gauge
from the outset, since otherwise it is unclear how to implement diffeomorphism
invariance and κ symmetry. In the static gauge the world-volume
fields are just those of a maximally supersymmetric vector supermultiplet:
gauge fields, scalars and spinors, all in the adjoint representation of U(N).
If one only wants to describe the leading nontrivial terms in a weak-field expansion,
the result is exactly super Yang–Mills theory. This approximation
is sufficient for many purposes including the important examples of Matrix
theory, based on D0-branes, and AdS/CFT duality, based on D3-branes,
which are discussed in Chapter 12.
When one tries to include higher powers of fields to give formulas that correctly
describe nonabelian D-brane physics for strong fields, the subject can
become mathematically challenging and physically confusing. The reason it
can be confusing concerns the domain of validity of DBI-type actions. They
are meant to capture the physics in the regime of approximation in which
the background fields and the world-volume gauge fields are allowed to be
arbitrarily large, but whose variation is small over distances of order the
string scale. The requirement of slow variation is meant to justify dropping
terms involving derivatives of the world-volume fields. The tricky issue in
the nonabelian case is that one should use covariant derivatives to maintain
gauge invariance, but there are relations of the form
[Dα, Dβ] ∼ Fαβ. (6.135)
This makes it somewhat ambiguous whether a term is derivative or not, and
so it is not obvious how to suppress rapid variation while allowing strong
fields. Nonetheless, some success has been achieved, which will now be
described.
Henceforth all fermion fields are set to zero and only bosonic actions are
considered. In addition to the background fields g, B, Φ and C, the desired
actions contain adjoint gauge fields A and 9 − p adjoint scalars Φ i , both of