to be of order 10’16 , so as to ¬nd TeV scale physics on the SM brane by

starting with Planck-scale physics on the Planck brane. This is an interest-

ing proposal (due to Randall and Sundrum) for solving the gauge hierarchy

problem. This scenario has a number of remarkable implications. It be-

comes conceivable that phenomena that used to be relegated to ultra-high

energy scales may be accessible at accelerator energies. Thus, Kaluza“Klein

modes, fundamental strings, black holes, gravitational radiation could all be

observable. The LHC experiments are preparing to search for all of these

possibilities. Supersymmetry, which many view as more likely to be dis-

covered, seems quite mundane by comparison. Not surprisingly, these ideas

have attracted a lot of attention, and there is a large and rapidly growing

literature on the subject. In the following, we settle for a brief sketch of how

this scenario might be realized in string theory.

A large hierarchy on the deformed conifold

It is interesting that the above approach to solving the hierarchy problem

appears naturally in string theory.15 The branes that seem best suited

to this purpose are the D3-branes in a type IIB orientifold or F-theory

construction. One can imagine D3-branes placed at points on a compact

internal manifold. To get a large hierarchy two sets of D3-branes would

need to be separated by the distance r. This distance would then determine

the size of the hierarchy. However, r is a modulus in the four-dimensional

theory, since the D3-brane coordinates have no potential. In the following

we will see that one can obtain a warped background generating a large and

stable hierarchy by using the ¬‚ux backgrounds discussed at the beginning

of this section.

To be concrete, one can consider the deformed conifold geometry. Locally,

15 So does supersymmetry.

10.2 Flux compacti¬cations of the type IIB theory 497

near the tip of the cone, the ¬‚ux solution is similar to the one described in

the previous section. Globally, however, the background solution must be

changed, since we are interested in a compact solution. The conifold solution

presented in the previous section is noncompact with r unbounded. This

can be interpreted as a singular limit of a compact manifold in which one of

the cycles degenerates to in¬nite size.

Let us assume that there are M units of F3 ¬‚ux through an A-cycle and

’K units of H3 ¬‚ux through a B-cycle, that is,

1 1

H3 = ’2πK.

F3 = 2πM and (10.160)

2π± 2π±

A B

Using Poincar´ duality, the superpotential can then be written as

e

G3 § „¦ = (2π)2 ± „¦ ’ K„

W= M „¦, (10.161)

B A

The complex coordinate describing the cycle collapsing at the tip of the

conifold is

z= „¦. (10.162)

A

The discussion of special geometry in Section 9.6 explained that the dual

coordinates, that is, the coordinates de¬ning periods of the B-cycles, are

functions of the periods of the A-cycles. More concretely, since we are de-

scribing a conifold singularity, we can invoke the result derived in Section 9.8

that

z

„¦= log z + holomorphic. (10.163)

2πi

B

Using these results, the K¨hler covariant derivative of the superpotential

a

can be rewritten in the form16

M K

(2π)2 ±

Dz W log z ’ i + . . . (10.164)

2πi gs

in the limit in which K/gs is large. The equation Dz W = 0 is solved by

e’2πK/M gs .

z (10.165)

Thus, one obtains a large hierarchy of scales if, for example, M = 1 and

K/gs = 5. It is assumed that the dilaton is frozen in this solution.

The solution for the warp factor can be estimated in the following way. As

16 This assumes z 1, which is the case of interest.

498 Flux compacti¬cations

will be discussed in more detail in Chapter 12, close to a set of N D3-branes

the space-time metric takes the form

2

r R

2

2 2

dr2 + r2 d„¦2 R4 = 4πgs N (± )2 ,

| dx | +

ds = with

5

R r

(10.166)

where r is the distance from the D3-brane, which is located at r ≈ 0. We

would like to estimate the warp factor close to the D3-brane. Since the

background is the deformed conifold, r has a minimal value determined by

the deformation parameter z according to

2/3

e’2πK/3M gs ,

ρmin = z 1/3

rmin (10.167)

showing that the warp factor approaches a small and positive value close to

the D3-brane.

EXERCISES

EXERCISE 10.6

Show that in a Calabi“Yau three-fold compacti¬cation of type IIB super-

string theory the K¨hler potential for the radial modulus, the axion“dilaton

a

modulus and the complex-structure moduli is given by

¯

K = ’3 log [’i(ρ ’ ρ)] ’ log[’i(„ ’ „ )] ’ log i „¦§„¦ .

¯ ¯

M

SOLUTION

The part of the K¨hler potential depending on the complex-structure moduli

a

(the last term) was derived in Chapter 9. The way to derive the contribution

from the radial modulus ρ and the axion“dilaton modulus „ is to consider

the action on a background of the form

ds2 = e’6u(x) gµν dxµ dxν +e2u(x) gmn dy m dy n .

CY3

4D

Here u(x) parametrizes the volume of the Calabi“Yau three-fold. The power

of eu(x) in the ¬rst term has been chosen to give a canonically normalized

Einstein term in four dimensions.

The supersymmetric partner of the radial modulus is another axion b,

10.3 Moduli stabilization 499

which descends from the four-form according to

Cµνpq = aµν Jpq ,

where J is the K¨hler form. In four dimensions the two-form a can be

a