The integral of the ¬eld strength and its dual over any closed surface in space van-

ishes. Similarly, nontrivial ¬‚ux solutions exist in M-theory, even when no δ-function

sources, corresponding to M2-branes or M5-branes, are present.

this is small

6

p

∆ 1+O . (10.59)

v 3/4

In the approximation in which the size of the Calabi“Yau is very large, that

is, when p /v 1/8 ’ 0, the background metric becomes unwarped.

This analysis shows that nontrivial ¬‚ux solutions are possible even in the

absence of explicit delta function sources for M2-branes or M5-branes, which

would appear in the equation of motion and Bianchi identity for F4 . A rather

similar situation appears in ordinary Maxwell theory, where a magnetic ¬‚ux

is generated by an electric current running through a loop even though there

are no magnetic monopoles, as illustrated in Fig. 10.3.

According to Eq. (10.50), nonsingular solutions for the warp factor and

the background geometry are possible even in the absence of explicit brane

sources. In fact, a nonsingular background is necessary to justify rigorously

the validity of the supergravity approximation everywhere in space-time.

Nevertheless, the supergravity approximation is valid for singular solutions

provided that the delta-function singularities are treated carefully.

Inclusion of M2-brane sources

If M2-branes ¬lling the external Minkowski space are also present, an addi-

tional integer N (the number of M2-branes) appears on the left-hand side

of Eq. (10.56), resulting in

1 χ

F §F =

N+ . (10.60)

4κ2 TM2 24

M

11

10.1 Flux compacti¬cations and Calabi“Yau four-folds 473

Since F is self-dual, both terms on the left-hand side of this equation are

positive. So if χ > 0, there are supersymmetry preserving solutions with

nonvanishing ¬‚ux or M2-branes. The number of these solutions is ¬nite,

because of quantization constraints on the ¬‚uxes that are discussed in Sec-

tion 10.6. For χ < 0 there are no supersymmetric solutions.

Interactions of moduli ¬elds

As discussed in Chapter 9, a Calabi“Yau four-fold has three independent

Hodge numbers (h1,1 , h2,1 and h3,1 ), each of which gives the multiplicities

of scalar ¬elds in the lower-dimensional theory. The purpose here is to show

that many of these ¬elds can be stabilized by ¬‚uxes.

The D = 3 ¬eld content

The variations of the complex structure of a Calabi“Yau four-fold are parametrized

by h3,1 complex parameters T I , the complex-structure moduli ¬elds, which

belong to chiral supermultiplets. Deformations of the K¨hler structure give

a

1,1 real moduli K A . Thus, the K¨hler form is

rise to h a

h1,1

K A eA ,

J= (10.61)

A=1

where eA is a basis of harmonic (1, 1)-forms. Together with the h1,1 vec-

tors arising from the three-form A3 these give h1,1 three-dimensional vec-

tor supermultiplets. Moreover, h2,1 additional complex moduli N I , belong-

ing to chiral supermultiplets, arise from the three-form A3 . For simplic-

ity of the presentation, the scalars N I are ignored in the discussion that

follows. The conditions for unbroken N = 2 supersymmetry in three di-

mensions, described above, can be regarded as conditions that determine

some of the scalar ¬elds in terms of the ¬‚uxes. Let us therefore derive the

three-dimensional interactions that account for these conditions. A more

direct derivation, based on a Kaluza“Klein compacti¬cation, is given in Sec-

tion 10.3.

In the absence of ¬‚ux it is possible to make duality transformations that

replace the vector multiplets by chiral multiplets. In particular, the vectors

are replaced by scalars. Once this is done, the K¨hler moduli are complex-

a

i¬ed. When ¬‚uxes are present there are nontrivial Chern“Simons terms.

Nevertheless the duality transformation is still possible, but it becomes more

complicated. Thus, we prefer to work with the real K¨hler moduli.

a

474 Flux compacti¬cations

Superpotential for complex-structure moduli

The complex-structure moduli T I appear in chiral multiplets, and the inter-

actions responsible for stabilizing them are encoded in the superpotential

1

W 3,1 (T ) = „¦ § F, (10.62)

2π M

where „¦ is the holomorphic four-form of the Calabi“Yau four-fold, and we

have set κ11 = 1. There are several di¬erent methods to derive Eq. (10.62).

The simplest method, which is the one used here, is to verify that this super-

potential 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 compacti¬cation 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 K¨hler covariant derivative of the superpotential, that is,

a

W 3,1 = DI W 3,1 = 0 I = 1, . . . , h3,1 ,

with (10.63)

where DI W 3,1 = ‚I W 3,1 ’ W 3,1 ‚I K3,1 , and K3,1 is the K¨hler potential on

a

the complex-structure moduli space introduced in Section 9.6, namely

K3,1 = ’ log „¦§„¦ . (10.64)

M

The K¨hler potential is now formulated in terms of the holomorphic four-

a

form instead of the three-form used in Chapter 9. The condition W 3,1 = 0

implies

F 4,0 = F 0,4 = 0. (10.65)

As in the three-fold case of Section 9.6, ‚I „¦ 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 per-

turbation theory, because of the standard nonrenormalization theorem for

the superpotentials. This theorem, which is most familiar for N = 1 the-

ories 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.

10.1 Flux compacti¬cations and Calabi“Yau four-folds 475

implies that the superpotential Eq. (10.62) generates a scalar potential for

the complex-structure moduli ¬elds, so that these ¬elds are stabilized. This

potential is discussed in Section 10.3.

Interactions of the K¨hler moduli

a

The primitivity condition,

F 2,2 § J = 0, (10.67)

is the equation that stabilizes the K¨hler moduli. This condition can be

a

derived from the real potential

W 1,1 (K) = J § J § F, (10.68)

M

where J is the K¨hler form. This interaction is sometimes called a superpo-

a