String Theory and M-Theory

474 Flux compactifications
Superpotential for complex-structure moduli
The complex-structure moduli T I appear in chiral multiplets, and the interactions
responsible for stabilizing them are encoded in the superpotential
W 3,1 (T ) = 1
Ω ∧ F, (10.62)
2π
where Ω is the holomorphic four-form of the Calabi–Yau four-fold, and we
have set κ11 = 1. There are several different methods to derive Eq. (10.62).
The simplest method, which is the one used here, is to verify that this superpotential
leads to the supersymmetry constraints Eq. (10.34). An alternative
derivation is presented in Section 10.3, where it is shown that Eq. (10.62)
arises from Kaluza–Klein compactification of M-theory on a manifold that
is conformally Calabi–Yau four-fold.
In space-times with a vanishing cosmological constant, the conditions for
unbroken supersymmetry are the vanishing of the superpotential and the
vanishing of the Kähler covariant derivative of the superpotential, that is,
M
W 3,1 = DIW 3,1 = 0 with I = 1, . . . , h 3,1 , (10.63)
where DIW 3,1 = ∂IW 3,1 − W 3,1∂IK3,1 , and K3,1 is the Kähler potential on
the complex-structure moduli space introduced in Section 9.6, namely
K 3,1
= − log Ω ∧ Ω . (10.64)
The Kähler potential is now formulated in terms of the holomorphic fourform
instead of the three-form used in Chapter 9. The condition W 3,1 = 0
implies
M
F 4,0 = F 0,4 = 0. (10.65)
As in the three-fold case of Section 9.6, ∂IΩ generates the (3, 1) cohomology
so that the second condition in Eq. (10.63) imposes the constraint
F 1,3 = F 3,1 = 0. (10.66)
The form of the superpotential in Eq. (10.62) holds to all orders in perturbation
theory, because of the standard nonrenormalization theorem for
the superpotentials. This theorem, which is most familiar for N = 1 theories
in D = 4, also holds for N = 2 theories in D = 3. 7 Supersymmetry
7 The basic argument is that since the superpotential is a holomorphic function, the size of
the internal manifold could only appear in the superpotential paired up with a corresponding
axion. However, the superpotential cannot depend on this axion, as otherwise the axion shift
symmetry would be violated. Correspondingly, the superpotential does not depend on the size
of the internal manifold, and its form is not corrected in perturbation theory. Nonperturbative
corrections are nevertheless allowed, as they violate the axion shift symmetry. For more details
see GSW, Vol. II.