String Theory and M-Theory

SOLUTION
10.1 Flux compactifications and Calabi–Yau four-folds 479
In analogy to the three-fold case discussed in Chapter 9, the following formulas
hold for four-folds:
and
∂IΩ = KIΩ + χI, I = 1, ..., h 3,1
J = K A eA, A = 1, ..., h 1,1 ,
where χI and eA describe bases of harmonic (3, 1)-forms and (1, 1)-forms,
respectively. Since Ω is a (4, 0)-form one obtains from Eq. (10.63)
M
Ω ∧ F 0,4 = 0 and
M
χI ∧ F 1,3 = 0.
Since h0,4 = 1, the first constraint leads to F 0,4 = 0. Since χI describes a
basis of harmonic (3, 1)-forms, ⋆F 3,1 = h3,1 I=1 AIχI, which leads to
⋆F 3,1 ∧ F 1,3
= ⋆(F 1,3 ) ∗ ∧ F 1,3
= |F 1,3 | 2√ g d 8 x = 0,
M
as F is real. This leads to the flux constraint
M
M
F 1,3 = F 3,1 = F 0,4 = F 4,0 = 0.
Using ∂AW 1,1 = 0 and Eq. (10.68), one gets
eA ∧ J ∧ F 2,2 = 0.
Since ⋆(J ∧ F 2,2 ) is a harmonic (1, 1)-form, we have
⋆(J ∧ F 2,2
) = U A eA.
So the above constraint results in
⋆(J ∧ F 2,2 ) ∧ (J ∧ F 2,2
) =
M
h 1,1
A=1
M
|J ∧ F 2,2 | 2√ g d 8 x = 0,
which leads to the primitivity condition Eq. (10.67). Notice that the condition
W 1,1 = 0 is then satisfied, too. ✷