Gravity and Strings

556 The extended objects of string theory
in the matrix ˆM0, and the charges qF1 and qD1 . The mass must be a function of those four
parameters (again, a saturated BPS bound). To find the values of the physical parameters of
this solution in terms of the constants a, b, and h pq1,wecanuseanSL(2, R) definition for
the charges:
qD1
qF1B
=q =
g 2
16πG (10)
N
S 7 ∞
6ω(7)
⋆ ˆM −1 ˆH = 6ω(7)h pq1 ˆg − 3 2
(2πℓs) 7 ℓs
ˆM −1
0 a, (19.110)
where ˆM0 =aa T + bb T . Using the property a T ˆM −1
0 a = 1, we find the relation between q
and h pq1:
h pq1 = (2πℓs) 7ℓs ˆg 3
2
q T ˆM
−1
0 q. (19.111)
We can now express the full solution in terms of the physical parameters M0 and q.
The object described by any of these solutions (usually called the pq-string) isap = 1
object (a string) that has both qF1B and qD1 charges in a IIB vacuum characterized by the
moduli M0 (e ˆϕ0 =ˆg and Ĉ (0)
0 = ˆθ/(2π)),and can be understood as the superposition of
D1s and F1Bs. The values of the charges are therefore quantized: they can only be multiples
of those of one (D, F) string: n/(2πℓ 2 s ).The tension of this object is proportional to h pq1,
and, therefore, for trivial moduli, to
q2 D1 + q2 F1B < |qD1|+|qF1B|.
The tension of two parallel (or coincident) strings of the same kind would be the sum
of the tensions of each of them (bound states at threshold), which means that there is
zero interaction energy and it costs zero energy to disintegrate the system. In this case, the
tension is in general smaller, which means that this solution represents a bound state of
F1Bs and D1s with non-zero binding energy (non-threshold bound states). However, the
solution is stable with respect to disintegration only if the numbers of D1s and F1As are
relatively prime: if they have a GCT different from 1, say N, the tension is N times that of
asingle pq-string with nD1/N and nF1/N strings, and it takes zero energy to disintegrate
it. The pq-strings with coprime numbers of strings are the basic states of the theory.
A solution describing analogous bound states of D5 and S5Bs (pq-5-branes) was constructed
in [670]:
d ˆs 2 E = H − 1 4
pq5 [dt2 − d y 2
5 ] − H 3 4 2
pq5d x 5 ,
ˆB ty1 ···y5 = ηb(H −1
pq5 − 1), ˆM
ˆg − 1 2
Hpq5 = 1 + h pq5
|x4| 2 , aT ηb = 1.
=aa T H 1 2
pq5 + bb T H − 1 2
pq5 ,
(19.112)
On T-dualizing these solutions, one obtains new bound states of F1s and Dps [671,
672]. These solutions describe intersecting branes with non-zero interaction energy. Other
intersecting solutions with non-zero interaction energy are, for instance, the systems of
Dp-branes and D(p + 2)-branes studied in [181, 256].