String Theory and M-Theory

396 String geometry
A change in the normalization of the right-hand side would correspond to
shifting the Kähler potential by an inconsequential constant. These equations
show that the space spanned by wα is a Kähler manifold and the
Kähler potential is given by the logarithm of the volume of the Calabi–Yau.
We also define the intersection numbers
καβγ = κ(eα, eβ, eγ) = eα ∧ eβ ∧ eγ
(9.130)
and use them to form
G(w) = 1 καβγw
6
αwβw γ
w0 = 1
6w0
J ∧ J ∧ J , (9.131)
which is analogous to the prepotential for the complex-structure moduli
space. Here we have introduced one additional coordinate, namely w 0 , in
order to make G(w) a homogeneous function of degree two. Then we find
e −K1,1
h
= i
1,1
w A ∂ ¯ G ∂G
− ¯wA
∂ ¯w A ∂wA
, (9.132)
A=0
where now the new coordinate w 0 is included in the sum. In Eq. (9.132) it
is understood that the right-hand side is evaluated at w 0 = 1. A homework
problem asks you to verify that Eq. (9.132) agrees with Eq. (9.129).
The form of the prepotential
To leading order the prepotential is given by Eq. (9.131). However, note that
the size of the Calabi–Yau belongs to M 1,1 (M) and as a result α ′ corrections
are possible. So Eq. (9.131) is only a leading-order result. However, the
corrections are not completely arbitrary, because they are constrained by
the symmetry. First note that the real part of w α is determined by B,
which has a gauge transformation. This leads to a Peccei–Quinn symmetry
given by shifts of the fields by constants ε α
δw α = ε α . (9.133)
Together with the fact that G(w) is homogeneous of degree two, this implies
that perturbative corrections take the form
G(w) = κABCw A w B w C
w 0 + iY(w 0 ) 2 , (9.134)
where Y is a constant. Note that the coefficient of (w 0 ) 2 is taken to be
purely imaginary. Any real contribution is trivial since it does not affect the