String Theory and M-Theory

598 Black holes in string theory
where G (4)
µν is the four-dimensional metric, λ is a scalar and Aµ is a U(1)
gauge field. The four-dimensional metric satisfies the two-center attractor
equations.
EXERCISES
EXERCISE 11.10
Deduce Eq. (11.118) by projecting both sides of Eq. (11.120) on e −iα+K/2 Ω.
SOLUTION
Equation (11.120) is equivalent to
d
dτ
−U
e e −iα+K/2 Ω − e iα+K/2 Ω
∼ −iΓ.
Taking the wedge product with e−iα+K/2Ω, only the second term on the left
contributes since Ω ∧ Ω = Ω ∧ d
dτ Ω = 0. Thus
− d −U
e
dτ
e K Ω ∧ Ω − e −U e −iα+K/2 Ω ∧ d
e
dτ
iα+K/2
Ω
∼ −ie −iα+K/2 Ω ∧ Γ.
The imaginary part of this equation is now integrated over the manifold. A
useful identity that implies that the integral of the second term is real is
e iα+K/2 Ω ∧ d
e
dτ
−iα+K/2
Ω = e −iα+K/2 Ω ∧ d
e
dτ
iα+K/2
Ω ,
which is derived by differentiating Eq. (11.110) written in the form
e −iα+K/2
Ω ∧ e iα+K/2
Ω = −i.
In this way, one obtains
i d −U
e
dτ
e K
Ω ∧ Ω = − 1
2 eK/2
e −iα iα
Ω + e Ω ∧ Γ.
Using Eq. (11.110) to simplify the left-hand side and Eq. (11.114) to simplify
the right-hand side, one obtains
d −U
e
dτ
= |Z|,