Gravity and Strings

18.2 General p-brane solutions 517
ds2 = e +2aϕ 1
H −2 ˜p+1 Wdt2 − d y 2
˜p
− e +2aϕ H −2− 1
p+1 dz 2 q + W −1dρ2 + ρ2d 2
,
(˜δ−2)
e−2aϕ = H −2x , F(p+2)z1···zqψ1···ψ (˜δ−2) = (˜δ − 3)αh (˜δ−2)
ψ1···ψ (˜δ−2) ,
H = 1 + h
ω
, W = 1 + ,
˜δ−3 ˜δ−3 ρ ρ
ω = h 1 − a2
4x α2
, x = (a2 /2)c
1 + (a2 ,
/2)c
(p + 1) + ( ˜p + 1)
c =
(p + 1)( ˜p + 1) ,
(18.68)
where ˜δ = d − ( ˜p + q) and (n) is the volume form of an n-sphere.
Most of the single-p-brane solutions we are going to deal with are included in these
general solutions, except for the self-dual ones. It is easy to deal with them using solutions
describing two p-branes and therefore we will study them after we study intersecting pbrane
solutions in Section 19.6.
18.2.3 Sources for solutions of the p-brane a-model
Our experience tells us that we may find charged-p-brane sources for the extreme, and only
for the extreme (ω = 0), charged-p-brane solutions of the a-model that we have just found.
On finding these sources, we will be able to relate the integration constants h and α of the
solution to the brane tension T(p) and charge parameter µp.
We consider the following generic coupled system:
S = Sa + Sp, (18.69)
where Sa is the bulk a-model action (18.61) and Sp is the charged p-brane action:
Sp[X µ ,γij] =− T(p)
d
2
p+1 ξ |γ | e −2bϕ γ ij ∂i X µ ∂ j X ν gµν − (p − 1)
+ (−1)p+1
µp
d
(p + 1)!
p+1 ξɛ i1···i p+1
A(p+1)µ1···µp+1∂i1 X µ1 ···∂i p+1 X µp+1 .
(18.70)
The coupling of the scalar to the p-brane is, in principle, arbitrary. However, in all the
relevant cases the parameters a and b turn out to be related by
a =−(p + 1)b, (18.71)
and, actually, only then do we have a solution of the coupled system, as we are going to
see.