String Theory and M-Theory

6.5 World-volume actions for D-branes 235
description of the result. The answer takes the form
dΩp+1 = d ¯Θ A T AB
p dΘ B , (6.117)
where T AB
p is a 2×2 matrix of p-form valued Dirac matrices and A, B = 1, 2
is summed. Comparing to the result for the D0-brane given in Chapter 5,
gives in that case
which implies that
Ω1 = −m ¯ ΘΓ11dΘ = m( ¯ Θ 1 dΘ 2 − ¯ Θ 2 dΘ 1 ), (6.118)
0 1
T0 = m
−1 0
The formula for D-brane tensions gives the identification
. (6.119)
m = TD0 = 1
√ . (6.120)
α ′
Now let us present the general result for S2. It turns out to be simpler to
give all the results at once rather than to enumerate them one by one. In
other words, the expression for
T AB =
∞
p=0
gs
T AB
p
(6.121)
can be written relatively compactly. 9 In the type IIA case the sum is over
even values of p, and in the type IIB case the sum is over odd values of p.
Given T , which is a sum of differential forms of various orders, one simply
extracts the p-form part to obtain Tp and construct the Chern–Simons term
S2 of the Dp-brane action. Forms of order higher than 9 are not relevant.
The expression for T turns out to have the form
T AB = m e 2πα′ F f AB (ψ), (6.122)
where F is given in Eq. (6.109), and ψ is a matrix-valued one-form given by
In the type IIA case
ψ =
f(ψ) =
1
√ 2πα ′ Γµ Π µ α dσ α . (6.123)
0 cos ψ
− cosh ψ 0
(6.124)
9 Recall that sums of differential forms of various orders were encountered earlier in the anomaly
discussion of Chapter 5.