ansatz of the form

9

GM N dxM dxN = e2A(y) ·µν dxµ dxν +e’2A(y) gmn (y)dy m dy n ,

ds2 =

10

M,N =0

6D

4D

(10.80)

xµ

where denote the coordinates of four-dimensional Minkowski space-time,

and y m are local coordinates on M . Poincar´ invariance implies that the

e

warp factor A(y) is allowed to depend on the coordinates of the internal

manifold only.

Poincar´ invariance and the Bianchi identities restrict the allowed com-

e

ponents of the ¬‚ux. The three-form ¬‚ux G3 is allowed to have components

along M only, while the self-dual ¬ve-form ¬‚ux F5 should take the form

§ dx0 § dx1 § dx2 § dx3 ,

F5 = (1 + 10 )d± (10.81)

where ±(y) is a function of the internal coordinates, which will turn out to

be related to the warp factor A(y).

The no-go theorem is derived by using the equations of motion following

from the action Eq. (10.75). The ten-dimensional Einstein equation can be

written in the form

1

RM N = κ2 TM N ’ GM N T , (10.82)

8

where

2 δS

TM N = ’ √ (10.83)

’G δGM N

482 Flux compacti¬cations

is the energy“momentum tensor, and T is its trace. This equation has an

external piece (µν) and an internal piece (mn), but the mixed piece vanishes

trivially. The external piece takes the form10

1 1

|G3 |2 + e’8A |‚±|2

RM N = ’ GM N M, N = 0, 1, 2, 3.

4 2 Im „

(10.84)

Transforming to the metric ·µν gives an equation determining the warp

factor in terms of the ¬‚uxes

e4A 1

|G3 |2 + e’8A |‚±|2 ,

∆A = (10.85)

8 Im „ 4

or, equivalently

e8A

|G3 |2 + e’4A |‚±|2 + |‚e4A |2 .

4A

∆e = (10.86)

2 Im „

The no-go theorem is a simple consequence of this equation. If both sides

are integrated over the internal manifold M , the left-hand side vanishes,

because it is a total derivative. The right-hand side is a sum of positive-

de¬nite terms, which only vanishes if the individual terms vanish. As a

result, one is left with constant A, ± and vanishing G3 . The assumption of

maximal symmetry would, in principle, allow an external space-time with a

cosmological constant Λ, which for Λ < 0 results in AdS space-times while

for Λ > 0 gives dS space-times. It turns out that the above no-go theorem

can be generalized to include this cosmological constant. As you are asked

to show in Problem 10.8, Λ appears with a positive coe¬cient on the right-

hand side of Eq. (10.86). Using the same reasoning as above, one obtains

another no-go theorem which excludes dS solutions in the absence of sources

and/or singularities in the background geometry.

Flux-induced superpotentials

It turns out that brane sources can and do invalidate the no-go theorem.

There is an energy“momentum tensor associated with these sources, which

contributes to the right-hand side of Eq. (10.86) in the form

2κ2 e2A Jloc , (10.87)

where

9 3

1 M

TM M )loc ,

Jloc ’

=( TM (10.88)

4

M =5 M =0

10 Indices M, N are used (rather than µ, ν) to emphasize that this curvature is constructed using

the metric GM N .

10.2 Flux compacti¬cations of the type IIB theory 483

and T loc denotes the energy“momentum tensor associated with the local

sources given by

2 δSloc

loc

TM N = ’ √ . (10.89)

’G δGM N

Here Sloc is the action describing the sources. For a Dp-brane wrapping a

(p ’ 3)-cycle Σ in M the relevant interactions are

√

dp+1 ξTp ’g + µp

Sloc = ’ Cp+1 . (10.90)

4 —Σ 4 —Σ

§ §

This is the action to leading order in ± and for the case of vanishing ¬‚uxes

on the brane. This action was described in detail in Section 6.5. In order

to describe D7-branes wrapped on four-cycles it is necessary to include the

¬rst ± correction given by the Chern“Simons term on the D7-brane

p1 (R)

’µ3 C4 § . (10.91)

48

4 —Σ

§

It turns out that Eq. (10.87) can contribute negative terms on the right-hand

side of Eq. (10.86).

These sources also contribute to the Bianchi identity11 for F5

dF5 = H3 § F3 + 2κ2 T3 ρ3 . (10.92)

Here ρ3 is the D3 charge density from the localized sources and, as usual, it

contains a delta function factor localized along the source.

Tadpole-cancellation condition

Integrating Eq. (10.92) over the internal manifold M leads to the type IIB

tadpole-cancellation condition

1

H3 § F3 + Q3 = 0, (10.93)

2κ2 T3 M

where Q3 is the total charge associated with ρ3 . As a result, nonvanishing

Q3 charges induce three-form expectation values. It is shown below that G3

is imaginary self-dual. Therefore, three-form ¬‚uxes are only induced if Q3

is negative. Problem 10.12 asks you to check that the D7-branes generate

a negative contribution to the right-hand side of Eq. (10.86) by solving the

equations of motion in the presence of branes.

A useful way of describing the type IIB solution is by lifting it to F-theory

compacti¬ed on an elliptically ¬bered Calabi“Yau four-fold X. As explained

in Section 9.3, the base of the ¬bration encodes the type IIB geometry while

11 Because of self-duality, this is the same as the equation of motion.

484 Flux compacti¬cations

the ¬ber describes the behavior of the type IIB axion“dilaton „ . In this

description, the tadpole-cancellation condition takes a form similar to that

found for M-theory on a four-fold

χ(X) 1

H3 § F 3 ,

= ND3 + 2 (10.94)

24 2κ T3 M