Gravity and Strings

19.6 Intersections 571
The coordinates yr = (y1 1 ,...,yr 1 ) (plus, of course, time) correspond to the common
directions of the two branes relative to the worldvolume of the intersection and the solution
is assumed to be independent of them. The coordinates y(p1−r) = (y2 1 ,...,y(p1−r) 2 ) and
y(p2−r) = (y3 1 ,...,y(p2−r) 3 ) are relative transverse coordinates (to the p1- and p2-branes,
respectively). For simplicity, we will also assume the solution to be independent of them
(i.e. it will be delocalized). The solution may depend only on the overall transverse coordinates
(the rest), but, as we did for single-brane solutions, we include, for completeness,
q additional isometries and the solution will not depend on yq. Finally, the parameters
z1, z2, y1, and y2 are determined by the single-brane solutions of the a-model, Eq. (18.66).
The dilaton is assumed to be a certain product of powers of H1 and H2. Finally, the Ansatz
for the potentials is the usual one and we need only take into account that the pi-brane lies
in the directions yr and y(pi −r):
−1
A(pi +1)11···r112···(pi −r)2 = αi(H i − 1), i = 1, 2. (19.149)
If this Ansatz is to work, then, by insisting on the independence of the two would-be
harmonic functions H1 and H2, weshould simply acquire constraints on ω,r, a1, and a2.
On plugging the Ansatz into the equations of motion, we find, after a long and boring
calculation, that it does indeed lead to solutions under a few conditions on those constants:
ds2 = e−2a1ϕ1 H −2
1
1
− e −2a1ϕ1 H −2
1
− e −2a1ϕ1 H −2
1
p 1 +1 e −2a2ϕ2 H −2
2
1
1
p 1 +1 e −2a2ϕ2 H −2
2
− 1
− 1
˜p 1 +1 e −2a2ϕ2 H −2
2
p 2 +1 Wdt 2 − d y 2
r
− 1
− e−2a1ϕ1 H −2 ˜p 1 +1
1
e−2a2ϕ2 H −2
2
2 d y q + W −1dρ2 + ρ2d 2
(δ−2) .
A(pi +1)11···r112···(pi −r)2
= α1(H −1
1
˜p 2 +1 d y 2
(p1−r)
1
p 2 +1 d y 2
(p2−r)
1
− ˜p 2 +1
− 1),
e−2ai ϕi 2xi ≡ Hi , e−2ai ϕ −2ai ϕi −2a = e (e j ϕ j x ) j , i = j,
Hi = 1 + hi
ω
, W = 1 + ,
ρδ−3 ρδ−3
ω = hi 1 − a2 i
α
4xi
2
i , li = (xi − 1) ci(r + 1) − p
j + 1
,
˜pi + 1
i = j,
xi = (a2 i /2)ci
1 + (a2 i /2)ci
, ci = (pi + 1) + ( ˜pi + 1)
(pi + 1)( ˜pi + 1) ,
a1a2 =−2(r − r0), r0 = (p1 + 1)(p2 + 1)
− 1.
d − 2
(19.150)
This solution generalizes to the non-extreme regime extreme intersecting solutions obtained
in [51] in another (a1–a2) model (see also [50, 925] and references therein). Some
l i