String Theory and M-Theory

9.6 Special geometry 393
In analogy with the torus example, we can define coordinates X I on the
moduli space by using the A periods of the holomorphic three-form
X I
= Ω with I = 0, . . . , h 2,1 . (9.107)
A I
The number of coordinates defined this way is one more than the number of
moduli fields. However, the coordinates X I are only defined up to a complex
rescaling, since the holomorphic three-form has this much nonuniqueness.
To take account of this factor consider the quotient 19
t α = Xα
X 0 with α = 1, . . . , h 2,1 , (9.108)
where the index α now excludes the value 0. This gives the right number
of coordinates to describe the complex-structure moduli. Since the X I give
the right number of coordinates to span the moduli space, the B periods
FI = Ω (9.109)
must be functions of the X, that is, FI = FI(X). It follows that
BI
Ω = X I αI − FI(X)β I . (9.110)
A simple consequence of Eq. (9.103) is
Ω ∧ ∂IΩ = 0, (9.111)
which implies
or, equivalently,
FI = X
FI = ∂F
∂X I
J ∂FJ 1
=
∂X I 2
∂
∂X I
J
X FJ , (9.112)
where F = 1
2 XI FI. (9.113)
As a result, all of the B periods are expressed as derivatives of a single
function F called the prepotential. Moreover, since
2F = X
I ∂F
, (9.114)
∂X I
F is homogeneous of degree two, which means that if we rescale the coordinates
by a factor λ
F (λX) = λ 2 F (X). (9.115)
19 As usual in this type of construction, these coordinates parametrize the open set X 0 = 0.