String Theory and M-Theory

6.5 World-volume actions for D-branes 237
manner. However, a quicker method is to note that they can be obtained
by dimensional reduction of the gauge-fixed D9-brane action in Eq. (6.127).
Dimensional reduction simply means dropping the dependence of the worldvolume
fields on 9 − p of the coordinates. This works for both even and odd
values of p. For example, dimensional reduction of Eq. (6.127) to four dimensions
gives an exactly supersymmetric nonlinear extension of N = 4 super
Maxwell theory. The supersymmetry transformations are complicated, because
the gauge-fixing procedure contributes induced κ transformations to
the original ε transformations of the fields.
Bosonic D-brane actions with background fields
The D-brane actions obtained in the previous section are of interest as they
describe D-branes in flat space. However, one frequently needs a generalization
that describes the D-brane in a more general background in which
the various bosonic massless supergravity fields are allowed to take arbitrary
values. These actions exhibit interesting features, that we shall now address.
The abelian case
The background fields in the NS–NS sector are the space-time metric gµν,
the two-form Bµν and the dilaton Φ. These can be pulled back to the world
volume
P [g + B]αβ = (gµν + Bµν)∂αX µ ∂βX ν . (6.128)
Henceforth, for ease of writing, pullbacks are implicit, and this is denoted
gαβ + Bαβ. Note that this gαβ is the bosonic restriction of the quantity
that was called Gαβ previously. With this definition, the DBI term in static
gauge takes the form
SDp = −TDp
d p+1 σe −Φ0
− det (gαβ + Bαβ + k 2 ∂αΦ i ∂βΦ i + kFαβ).
(6.129)
Since the string coupling constant gs is already included in the tension TDp,
the dilaton field is shifted by a constant so that it has vanishing expectation
value (Φ = log gs + Φ0). This is the significance of the subscript. Note that
invariance under a two-form gauge transformation
δB = dΛ (6.130)
requires a compensating shift of the gauge field A.
The possibility of R–R background fields should also be considered. They
do not contribute to the DBI action, but they play an important role in