Gravity and Strings

550 The extended objects of string theory
For the Dp-brane solutions (p < 7) Eqs. (19.64) compactified on p circles we calculate
first the (10 − p)-dimensional dilaton, which is given by
Then, according to Eq. (19.90),
e −2(φ−φ0) = e −2( ˆφ− ˆφ0)
ˆg y 1 y 1 ··· ˆgy p y
p−6
2(8−p)
˜gE µν = HDp
gµν, ⇒ ˜gEtt = H
− p−7
p−8
Dp
p − 7
∼ 1 −
which, compared with Eq. (19.93), gives again the right value,
MDp =
(7 − p)hDpω(8−p)
16πG (10−p)
N
= (7 − p)(2π)p R9 ···R10−phDpω(8−p)
16πG (10)
N
19.3.2 Charges
p−6
4
p = HDp . (19.99)
p − 8 hDp
1
, (19.100)
|x 7−p
p−9|
= R9 ···R10−p
ℓ p+1
s g
.
(19.101)
With the normalization of the superstring effective actions Eqs. (16.38) and (17.4), the
(electric) charges associated with the KR 2-form and the RR (p + 1)-form potentials,
which are carried, respectively, by fundamental strings and Dp-branes, can be defined by
the integrals19 g2
g2
qF1 =
16πG (10)
N
S 7 ∞
e −2 ˆφ ⋆ ˆH, qDp =
16πG (10)
N
S 8−p
∞
⋆ ˆG (p+2) , (19.102)
whereas the charge associated with the NSNS 6-form potential, carried by the S5, is given
by
qS5 =
g2
e 2 ˆφ ⋆ ˆH (7) =
g2
ˆH. (19.103)
16πG (10)
N
S 4 ∞
16πG (10)
N
qF1 and qS5 are the electric–magnetic duals of each other, as are qDp and qD ˜p. With the
above normalization, the generalization of the Dirac quantization condition for extended
objects reads
qDpqD ˜p = 2πn 16πG(10)
N
ˆg 2
, qF1qS5 = 2πn 16πG(10)
N
ˆg 2
, n ∈ Z. (19.104)
It is easy to see that the values of the charges of the string/M-theory solutions coincide
with the values we gave in Section 19.1.1. It should be stressed that both the masses and
the charges of the extreme solutions are determined by the same h. This is due to the fact
that the masses (tensions) and charges of these objects saturate BPS bounds. The solutions
preserve half of the supersymmetries of the corresponding supergravity theory (see
Section 19.5.1).
19 These definitions are valid for field configurations in which only one potential is non-trivial. In general, the
charge is obtained by integrating the form F such that dF = 0isthe equation of motion. (This is sometimes
called the Page charge [687].) The presence of non-trivial Chern–Simons terms in the action implies that F
consists in various terms, as we will discuss in Section 19.6.1.
S 4 ∞