String Theory and M-Theory

234 T-duality and D-branes
One reason D-branes are useful probes of string geometry is that a tension
proportional to 1/gs does allow for such a regime. Chapter 12 considers a
situation in which the number of D-branes N is increased at the same time
as gs → 0 with N ∼ 1/gs. The gravitational effects of the D-branes survive
in this limit.
The same type of reasoning used earlier to relate the type IIA and IIB
string coupling constants can be used to determine D-brane tensions. Tduality
exchanges a wrapped Dp-brane in the type IIA theory and an unwrapped
D(p − 1)-brane in the type IIB theory (and vice versa). Using this
fact, compactification of the D-brane action on a circle gives (for p even) the
relation 2πRTDp = T D(p−1), or
2πRcp
gs
= cp−1
. (6.113)
˜gs
Inserting the relation between the string coupling constants in Eq. (6.66)
gives
1
cp =
2π √ α ′ cp−1. (6.114)
√
If one sets TD0 = (gs α ′ ) −1 , a result that is derived in Chapter 8, then one
obtains the precise formula
TDp =
1
gs(2π) p (α ′ . (6.115)
) (p+1)/2
As before, it is understood that the type IIA string coupling constant is used
if p is even, and the type IIB coupling constant is used if p is odd.
The construction of S2
Supersymmetric D-brane actions require κ symmetry in order to have the
right number of fermionic degrees of freedom. As in the examples of Chapter
5, this requires the addition of a Chern–Simons term, which can be
written as the integral of a (p + 1)-form
S2 = Ωp+1. (6.116)
However, as in the case of the superstring, it is easier to construct the (p+2)form
dΩp+1. It is manifestly invariant under supersymmetry, whereas the
supersymmetry variation of Ωp+1 is a total derivative.
The analysis is rather lengthy, but it involves the same techniques that
were described for simpler examples in Chapter 5. Let us settle here for a