Gravity and Strings

19.2 String-theory extended objects from effective-theory solutions 543
T
e2 πi/3 eπi/3
-1/2 +1/2
Fig. 19.2. The fundamental domain of the modular group.
We can now interpret the D7-brane solution in the light of the preceding discussion. First,
it is useful to rewrite it using ω = x 1 + ix 2 in the form
where
d ˜ ˆs 2 E = dt2 − d y 2
7
i
S
τ
− Im(H) dωd ¯ω, τ = H, (19.74)
H = ie −ˆϕ0 hD7 ln ω, or H = ie −ˆϕ0 hD7 ln ¯ω, (19.75)
for D7- and anti-D7-branes (positive and negative charge w.r.t. Ĉ (0) ), respectively, where we
have eliminated the constant 1 in HD7 since the solution is not asymptotically flat anyway.
If we go around the origin ω = 0atwhich the (anti-)D7 is placed, then, according to the
source calculation,
ω → e 2πi ω, ⇒ τ → τ ± 1 = T ±1 (τ). (19.76)
The D7-brane solution has, as we expected from our general discussion, non-trivial monodromy.
Furthermore, the monodromy around a D7-brane with charge n is T n .For(d − 3)branes
monodromy plays the role of charge (they are equivalent, when standard charge can
be defined), which can be represented by a monodromy matrix.
The D7-brane solution is, however, defined only in the disk |ω| < 1 due to the negative
sign of hD7 and we may be interested in different transverse spaces. The corresponding
solutions may be found thanks to the following observation: the Ansatz Eqs. (19.74) is
a solution for any H that is a holomorphic or antiholomorphic function of ω [474]. D7branes
will be placed at points around which the monodromy of τ is T n . The restriction
to (anti)holomorphicity has to do with the impossibility of having objects with opposite
charges in equilibrium.
Observe that what appears in the metric is Im(H), not Im(τ), even ifthey coincide in
this form of the solution. Then gω ¯ω does not transform under G transformations of τ, but
the relation gω ¯ω = Im(τ) breaks down. In fact, the general 7-brane solution can be written