Gravity and Strings

19.2 String-theory extended objects from effective-theory solutions 539
Using Eqs. (18.78), the Dp-brane tension formula, Eq. (19.11), and the value of G (10)
N ,
Eq. (19.26), we find
hDp = (2πℓs) 7−p g
, p < 7, hD7 =−
(7 − p)ω(8−p)
g
2π , hD8 =− g
. (19.65)
4πℓs
Several remarks are in order here.
1. The D-instanton (p =−1) solution is not included in this general case. It will be
dealt with in Section 19.2.7.
2. The D-string (p = 1) solution is related by IIB S duality (with S = η) tothe F1B
solution. More general S-duality transformations generate solutions that represent
bound states of q F1Bs and p D1s called pq-strings [822]. The same can be said
about the D5 and the S5B, which can be combined into pq 5-branes [670]. There are
also pq 7-brane solutions, but they have a more complicated interpretation. We will
study these solutions in Section 19.4.3.
3. The metric and dilaton of the p = 3solution are those of the self-dual D3-brane
solution, but the RR potential is different. The correct field strength is just the selfdual
part of the field strength of the generic solution and its components are
ˆG (5) mty1y3 −2
y3 =∓e−ˆϕ0
HD3 2 ∂m HD3,
ˆG (5) m 1 ···m 5 =± e−ˆϕ0
2 ɛm 1 ···m 5 m 6 ∂m 6 HD3.
(19.66)
4. The solutions for p < 7 are well defined for all values of |x9−p| > 0(HDp > 0). The
D7 and D8 solutions are well defined only in certain regions of the transverse space
for which HDp > 0 due to the negative signs of hD7 and hD8.Toobtain solutions that
are well defined everywhere in the transverse space, one has to consider configurations
with several branes and compact transverse spaces. The only singularities are
then at the positions of the branes. 9 We are going to study the simplest of these combinations
of D7-branes in Section 19.2.8. The simplest combination of D8-branes
which leads to a regular metric is the orbifold construction discussed on page 519.
5. The Hodge dual of the 10-form field strength associated with the D8-brane solution
is ⋆ ˆG (10) =±hD8g −1 =∓1/(4πℓs). This must, then, be the value of the mass parameter
m = ˆG (0) of the Romans massive N = 2A, d = 10 supergravity that describes
the effective string theory in the presence of one D8-brane [118, 782]. The parameter
m which was completely arbitrary from the supergravity point of view must be
quantized from the string-theory point of view.
6. The p = 9 solution is just flat spacetime.
9 The positions of the branes can be identified in general with the poles of the harmonic functions, although
we know that, in many cases, these poles are just coordinate singularities and correspond to regular event
horizons.