da = e’8u(x) db.

Setting

b

ρ = √ + ie4u ,

2

the resulting low-energy e¬ective action is

1 ‚µ „ ‚ µ „

¯ 3 ‚µ ρ‚ µ ρ

√ ¯

1 4

d x ’g R ’ ’

S= 2 .

2 (Im „ )2 2 (Im ρ)2

2κ4

Here the four-dimensional gravitational coupling constant is given by κ2 =

4

2 /V, where V is the volume of the Calabi“Yau three-fold computed using

κ10

the metric gmn . The kinetic terms for „ and ρ correspond to the ¬rst two

terms in the K¨hler potential K.

a 2

10.3 Moduli stabilization

The important fact about compacti¬cations with ¬‚ux is that there is a non-

trivial scalar potential for the moduli ¬elds.17 This should not be surprising,

since the background ¬‚ux modi¬es the equations that determine the geom-

etry. The complete scalar potential V for the moduli ¬elds can be obtained

from the superpotential and the K¨hler potential by a standard supergravity

a

formula, as was discussed earlier, or by a direct Kaluza“Klein compacti¬ca-

tion, as is done here.

Scalar potential for M-theory

In the following the scalar potential for ¬‚ux compacti¬cations of M-theory

on a Calabi“Yau four-fold is derived from the low-energy expansion of the

action Eq. (10.3) on the warped geometry described by Eq. (10.5). This

further illustrates that the constraints derived from W 3,1 in Eq. (10.62)

stabilize the complex-structure moduli, while the equations derived from

W 1,1 in Eq. (10.68) stabilize the K¨hler moduli.

a

17 Calling these ¬elds moduli in this setting is a bit of an oxymoron, since moduli are de¬ned to

have no potential. However, this has become standard usage.

500 Flux compacti¬cations

As you are asked to check in Problem 10.18, ¬‚uxes generate a scalar

potential for the moduli

1 1

F § F ’ TM2 χ ,

V (T, K) = (10.168)

4V 3 6

M

where we set κ11 = 1, as in Section 10.1. The terms that contribute to the

potential originate from the internal component of the ¬‚ux while the fm

term has been dropped, because it gives a subleading contribution in the

large-volume limit.

Since F is a four-form it lies in the middle-dimensional cohomology of the

Calabi“Yau four-fold. According to Eq. (10.44) the (2, 2)-component of the

four-form ¬‚ux has the Lefschetz decomposition

F 2,2 = Fo + J § Fo + J § JFo ,

2,2 1,1 0,0

(10.169)

where the subindex o indicates that the ¬‚ux is primitive. As was shown in

Eq. (10.67), only the primitive term, that is, the ¬rst term, is nonzero for

a supersymmetric solution. However, here all terms are included in order

to allow for the possibility of supersymmetry breaking. Since the ¬rst and

third terms are self-dual, and the second term is anti-self-dual,

F 2,2 = F 2,2 ’ 2J § Fo ,

1,1

(10.170)

where denotes the Hodge dual on the internal manifold. It follows from

Exercise 10.4 that

F 4,0 = F 4,0 F 3,1 = ’F 3,1 ,

and (10.171)

and similarly for the (0, 4) and (1, 3) components, since F is real. Taking

the previous two equations into account, the total four-form ¬‚ux satis¬es

F = F ’ 2F 3,1 ’ 2F 1,3 ’ 2J § Fo .

1,1

(10.172)

Therefore, after taking the wedge product with F , the kinetic term for the

¬‚ux appearing in Eq. (10.168) can be rewritten in the form

F 3,1 §F 1,3 ’2 1,1 1,1

F§ F = F §F ’4 J §Fo §J §Fo . (10.173)

M M M M

All other terms vanish by orthogonality relations given by the Hodge de-

composition and the Lefschetz decomposition. Inserting this into the scalar

potential Eq. (10.168), we realize that the ¬rst term on the right-hand side

of Eq. (10.173) cancels due to the tadpole-cancellation condition Eq. (10.60)

with N = 0. As a result, only the anti-self-dual part of F contributes to the

scalar potential.

10.3 Moduli stabilization 501

Supersymmetry-breaking solutions

The preceding results imply the existence of supersymmetry-breaking solu-

tions of the equations of motion. Indeed, any ¬‚ux satisfying

(2,2)

F ∈ Hprimitive

F= F and / (10.174)

solves the equations of motion and breaks supersymmetry. Fluxes of the

form

F ∼„¦ F ∼J §J

or (10.175)

provide examples. Moreover, since these ¬‚ux components do not appear in

the scalar potential they do not generate a cosmological constant.

The scalar potential

The second term on the right-hand side of Eq. (10.173) can be rewritten

according to

¯ 3,1

3,1

F 3,1 § F 1,3 = ’eK GI J DI W 3,1 DJ W , (10.176)

¯

M

and as a result yields a scalar potential for the complex-structure moduli.

This result is obtained by expanding F 3,1 in a basis of (3, 1)-forms. The

explicit calculation is rather similar to Exercise 10.5. Analogously, the last

term on the right-hand side of Eq. (10.173) generates a potential for the

K¨hler-structure moduli

a

J § Fo § J § Fo = ’V ’1 GAB DA W 1,1 DB W 1,1 ,

1,1 1,1

(10.177)

M

where18

1

DA = ‚A ’ ‚A K1,1 K1,1 = ’3 log V,

with (10.178)

2

and GAB is the inverse of the metric GAB

1

GAB = ’ ‚A ‚B logV. (10.179)

2

1

Here V = 24 J § J § J § J is the Calabi“Yau volume. In total, the scalar

potential becomes

1

¯ 3,1

V (T, K) = eK GI J DI W 3,1 DJ W + V ’4 GAB DA W 1,1 DB W 1,1 , (10.180)