is somewhat misleading. Supersymmetry imposes the constraint

W 1,1 = ‚A W 1,1 = 0 A = 1, . . . , h1,1 ,

with (10.69)

which leads to the primitivity condition. Section 10.3 shows that W 1,1

appears in the scalar potential for the moduli ¬elds of M-theory compacti¬ed

on a Calabi“Yau four-fold.

F-theory on Calabi“Yau four-folds

The M-theory compacti¬cations on manifolds that are conformally Calabi“

Yau four-folds are dual to certain F-theory compacti¬cations on Calabi“

Yau four-folds, which lead to four-dimensional space-times with N = 1

supersymmetry. Thus, this dual formulation is more attractive from the

phenomenological point of view. The F-theory backgrounds one is interested

in are nonperturbative type IIB backgrounds in which the Calabi“Yau four-

fold is elliptically ¬bered, as was discussed in Chapter 9.

To be concrete, the Calabi“Yau four-fold one is interested in can be de-

scribed locally as a product of a Calabi“Yau three-fold times a torus.8 The

conditions on the four-form ¬‚uxes derived above correspond to conditions

on three-form ¬‚uxes in the type IIB theory. Concretely, the relation between

the F-theory four-form and type IIB three-form is

1

(G § dz ’ G3 § d¯) ,

F4 = z (10.70)

¯3

„ ’„

8 Locally, this is always possible, except at singular ¬bers.

476 Flux compacti¬cations

where

dz = dσ1 + „ dσ2 . (10.71)

σ1,2 are the coordinates parametrizing the torus, and „ is its complex struc-

ture, which in the type IIB theory is identi¬ed with the axion“dilaton ¬eld.

Moreover, G3 = F3 ’ „ H3 is a combination of the type IIB R“R and NS“NS

three-forms. In components, this implies that

1

F 1,3 = (G3 )0,3 § dz ’ (G3 )1,2 § d¯ ,

z (10.72)

„ ’„¯

1

F 0,4 = ’ (G3 )0,3 § d¯.

z (10.73)

„ ’„¯

Imposing the M-theory supersymmetry constraints F 0,4 = F 1,3 = 0 leads to

the supersymmetry constraints for the type IIB three-form

G3 ∈ H (2,1) , (10.74)

while the remaining components of G3 vanish. The next section shows that

any harmonic (2, 1)-form on a Calabi“Yau three-fold with h1,0 = 0 is primi-

tive. Therefore, primitivity is automatic if the background is a Calabi“Yau

three-fold with nonvanishing Euler characteristic. Otherwise, it is an addi-

tional constraint that has to be imposed.

Many examples of M-theory and F-theory compacti¬cations on Calabi“

Yau four-fold have been constructed in the literature. A simple example is

described by M-theory on K3 — K3, which leads to a theory with N = 4

supersymmetry in three dimensions. Other examples include orbifolds of

T 2 — T 2 — T 2 — T 2.

EXERCISES

EXERCISE 10.1

Explain the powers of ∆ in Eq. (10.20).

SOLUTION

The powers of ∆ in Eq. (10.20) are a straightforward consequence of the

powers of ∆ appearing in the gamma matrices in Eq. (10.8). 2

10.1 Flux compacti¬cations and Calabi“Yau four-folds 477

EXERCISE 10.2

Show that if the Killing spinor ξ has positive chirality, that is, if γ9 ξ = +ξ,

F is self-dual on the Calabi“Yau four-fold, as stated in the text. What

happens if we reverse the chirality of ξ?

SOLUTION

Using the gamma-matrix identities listed in the appendix of this chapter it

is possible to show that

1

Fm Fm ξ = ’2F2 ξ ’ Fmnpq (F mnpq F mnpq ) ξ,

12

where γ9 ξ = ±ξ. Since Fm ξ = Fξ = 0, it follows that

F )2 = 0.

(F

This quantity is positive and therefore

F =± F γ9 ξ = ±ξ.

for

Thus, positive-chirality spinors lead to a self-dual F . If the chirality is

reversed, self-duality is replaced by anti-self-duality. 2

EXERCISE 10.3

Show that a harmonic four-form on a Calabi“Yau four-fold satisfying Fm ξ =

(2,2)

0 belongs to Hprimitive .

SOLUTION

In complex coordinates the condition Fm ξ = 0 implies

¯ ¯

Fm¯¯c γ ab¯ξ + 3Fm¯¯ γ abc ξ = 0,

¯c ¯

ab¯ abc

where m can be a holomorphic or antiholomorphic index. Each of these

terms has to vanish separately:

¯

• Using Eq. (10.33), the condition Fm¯¯c γ ab¯ξ = 0 implies

¯c

ab¯

¯ ¯

Fm¯¯c γd , γ ab¯ ξ = 6Fmd¯c γ b¯ξ = 0.

¯c c

¯b¯

¯

ab¯

This in turn implies that

¯

Fmd¯c γe , γ b¯ ξ = 4Fmd¯c γ c ξ = 0,

c ¯

¯b¯ ¯e ¯

¯

which yields

Fmd¯c γf , γ c ξ = 2Fmd¯f ξ = 0.

¯

¯e¯ ¯e ¯

¯

478 Flux compacti¬cations

Since m can be holomorphic or antiholomorphic and F is real, this results

in

F 4,0 = F 3,1 = F 1,3 = F 0,4 = 0.

¯

• Applying the same reasoning as above, the condition Fm¯¯ γ abc ξ = 0 im-

¯

abc

plies that

¯

Fa¯ d g cd = 0.

bc ¯

Using the self-duality of F shown in Exercise 10.2 and the relation between

J and the metric, this equation can be re-expressed as