String Theory and M-Theory

6.5 World-volume actions for D-branes 229
at least if we assume g9µ = ˜g9µ = 0. Equation (6.66) can be understood as
the vacuum expectation value of this relation.
R–R sector fields
The massless spectrum of each of the superstring theories also contains
bosonic fields in the R–R sector. There is an obstruction to describing
their coupling to the string world sheet in the RNS formulation, a fact that
is a fundamental limitation of this approach. They can be coupled to the
world sheet in the GS formulation, in which case they have couplings of the
form ¯ ΘΓ µ1···µn ΘFµ1···µn.
A possible approach to understanding the behavior of R–R background
fields under T-duality is to go back to the construction of these fields as
bilinears in fermionic fields in the GS formulation of the superstring and
use the fact that under T-duality the right-moving fermions are multiplied
by a Dirac matrix (see Exercise 6.6). Alternatively, since they couple to
D-branes, one can use the T-duality properties of D-branes to deduce the
transformation rules. Either method leads to the same conclusion. In total,
the effect of T-duality on the R–R tensor fields of the type IIA theory is to
give the following type IIB R–R fields:
C9 = C, Cµ = Cµ9, Cµν9 = Cµν, Cµνλ = Cµνλ9. (6.99)
As a result, the odd-form potentials of the type IIA theory are mapped to
the even-form potentials of the type IIB theory. These formulas can be read
backwards to describe the transformations in the other direction, that is,
from type IIB to type IIA. These formulas are only valid for trivial NS–NS
backgrounds (Bµν = 0, gµν = ηµν and constant Φ). Otherwise, they need to
be generalized.
6.5 World-volume actions for D-branes
Let us now turn to the construction of world-volume actions for D-branes.
The basic idea is that modes of the open strings that start and end on a
given D-brane can be described by fields that are restricted to the world
volume of the D-brane. In order to describe the dynamics of the D-brane at
energies that are low compared to the string scale, only the massless openstring
modes need to be considered, and one can construct a low-energy
effective action based entirely on them. Thus, associated with a Dp-brane,
there is a (p+1)-dimensional effective field theory of massless fields (scalars,