where χ(X) is the Euler characteristic of the four-fold, and ND3 is the D3-

brane charge present in the compacti¬cation.12 The left-hand side of this

equation can be interpreted as the negative of the D3-brane charge induced

by curvature of the D7-branes. Thus, the equation is the condition for the

total D3-brane charge from all sources to cancel.

Conditions on the ¬‚uxes

What conditions does the background satisfy? To answer this question there

are several ways to proceed. One way is to solve the equations of motion pre-

viously described but now taking brane sources into account. Schematically,

this is done by inserting the F5 ¬‚ux of Eq. (10.81) into the Bianchi identity

Eq. (10.92) and subtracting the result from the contracted Einstein equa-

tion Eq. (10.86), taking the energy“momentum tensor contribution from the

brane sources into account. The resulting constraint is

1

| iG3 ’ G3 |2 +e’4A | ‚(e4A ’ ±) |2

∆ e4A ’ ± = 8A

6 Im„ e

(10.95)

+2κ2 e2A Jloc ’ T3 ρloc .

3

Most localized sources satisfy the BPS-like bound

Jloc ≥ T3 ρloc . (10.96)

3

As a result, for the kinds of sources that are considered here, the solutions

to the equations are characterized by the following conditions:

• The three-form ¬eld strength G3 is imaginary self-dual,

G3 = iG3 , (10.97)

where the denotes the Hodge dual in six dimensions. A solution to the

imaginary self-dual condition is a harmonic form of type (2, 1) + (0, 3).

It is shown below that only the primitive part of the (2, 1) component is

allowed in supersymmetric solutions.

• There is a relation between the warp factor and the four-form potential

e4A = ±. (10.98)

12 This includes D3-branes and instantons on D7-branes.

10.2 Flux compacti¬cations of the type IIB theory 485

• The sources saturate the BPS bound, that is,

Jloc = T3 ρloc . (10.99)

3

This equation is satis¬ed by D3-branes, for example. Indeed, using the

relevant terms in the world-volume action for the D3-brane in Eq. (10.90)

shows

T0 0 = T1 1 = T2 2 = T3 3 = ’T3 ρ3 Tm m = 0.

and (10.100)

This implies that the BPS inequality is not only satis¬ed but also satu-

rated. On the other hand, anti-D3-branes satisfy the inequality but do

not saturate it, since the left-hand side of Eq. (10.99) is still positive but

the right-hand side has the opposite sign. A di¬erent way to saturate the

bound is to use D7-branes wrapped on four-cycles and O3-planes. D5-

branes wrapped on collapsing two-cycles satisfy, but do not saturate, the

BPS bound.

The superpotential

The constraint Eq. (10.97) can be derived from a superpotential for the

complex-structure moduli ¬elds

„¦ § G3 ,

W= (10.101)

M

where „¦ denotes the holomorphic three-form of the Calabi“Yau three-fold.

Let us derive the conditions for unbroken supersymmetry using the super-

potential Eq. (10.101). For concreteness, consider the case of a Calabi“Yau

manifold with a single K¨hler modulus, which characterizes the size of the

a

Calabi“Yau. Before turning on ¬‚uxes, there are massless ¬elds describing

the complex-structure moduli z ± (± = 1, . . . , h2,1 ), the axion“dilaton „ and

the super¬eld ρ containing the K¨hler modulus.

a

As is explained in Exercise 10.6, the K¨hler potential can be computed

a

from the dimensional reduction of the ten-dimensional type IIB action by

taking the Calabi“Yau manifold to be large. The result for the radial mod-

ulus ρ is

K(ρ) = ’3 log[’i(ρ ’ ρ)].

¯ (10.102)

This should be added to the results for the axion“dilaton and complex-

structure moduli, which are

¯

K(z ± ) = ’ log i

K(„ ) = ’ log[’i(„ ’ „ )] „¦§„¦ .

¯ and

M

(10.103)

486 Flux compacti¬cations

The total K¨hler potential is given by

a

K = K(ρ) + K(„ ) + K(z ± ). (10.104)

Conditions for unbroken supersymmetry

Supersymmetry is unbroken if

Da W = ‚a W + ‚a KW = 0, (10.105)

where a = ρ, „, ± labels all the moduli super¬elds. In order to evaluate this

condition, ¬rst note that the superpotential in Eq. (10.101) is independent

of the radial modulus. As a result,

3

Dρ W = ‚ρ KW = ’ W = 0, (10.106)

ρ’ρ¯

which implies that supersymmetric solutions obey

W = 0. (10.107)

So the (0, 3) component of G3 has to vanish. The equation

1

D„ W = „¦ § G3 = 0 (10.108)

„ ’„¯ M

implies that the (3, 0) component of G3 has to vanish as well. The remaining

conditions are

D± W = χ± § G3 = 0, (10.109)

M

where χ± is a basis of harmonic (2, 1)-forms introduced in Chapter 9. Since

this condition holds for all harmonic (2, 1)-forms, one concludes that super-

symmetry is unbroken if

G3 ∈ H (2,1) (M ). (10.110)

Remark on primitivity

Compact Calabi“Yau three-folds with a vanishing Euler characteristic satisfy

h1,0 = 0. In this case any harmonic (2, 1)-form is primitive. To see this,

let us apply the Lefschetz decomposition to the present case. A harmonic

(2, 1)-form

1

χ = χab¯dz a § dz b § d¯c

z¯ (10.111)

c

2

can be decomposed into a part parallel to J and an orthogonal part according

to

χ = v § J + (χ ’ v § J) = χ + χ⊥ , (10.112)

10.2 Flux compacti¬cations of the type IIB theory 487

where

3

χap¯J p¯dz a ,

q

v= (10.113)

q

2

which has been chosen so that

χ⊥ § J = 0. (10.114)

On the other hand, if such a one-form v exists, it is harmonic, which implies

h1,0 = 0. As a result, χ = χ⊥ , and any harmonic (2, 1)-form is primitive.