Brane-world scenarios

An alternative to the usual Kaluza“Klein compacti¬cation method of hid-

ing extra dimensions, called the brane-world scenario, is described in Section

10.2. One of the goals of this approach is to solve the gauge hierarchy prob-

lem, that is, to explain why gravity is so much weaker than the other forces.

The basic idea is that the visible Universe is a 3-brane, on which the stan-

dard model ¬elds are con¬ned, embedded in a higher dimension space-time.

Extra dimensions have yet to be observed experimentally, of course, but in

this set-up it is not out of the question that this could be possible.3 While

3 The search for extra dimensions is one of the goals of the Large Hadron Collider (LHC) at

CERN, which is scheduled to start operating in 2007.

460 Flux compacti¬cations

the standard model ¬elds are con¬ned to the 3-brane, gravity propagates in

all 4 + n dimensions. Section 10.2 shows that the hierarchy problem can be

solved if the (4 + n)-dimensional background geometry is not factorizable,

that is, if it involves a warp factor, like those of string theory ¬‚ux compact-

i¬cations. In fact, ¬‚ux compacti¬cations of string theory give a warp factor

in the geometry, which could provide a solution to the hierarchy problem.

This is an alternative to the more usual approach to the hierarchy problem

based on supersymmetry broken at the weak scale.

Fluxes and superstring cosmology

The Standard Big Bang model of cosmology (SBB) is the currently accepted

theory that explains many features of the Universe such as the existence of

the cosmic microwave background (CMB). The CMB accounts for most of

the radiation in the Universe. This radiation is nearly isotropic and has

the form of a black-body spectrum. However, there are small irregularities

in this radiation that can only be explained if, before the period described

by the SBB, the Universe underwent a period of rapid expansion, called

in¬‚ation. This provides the initial conditions for the SBB theory. Di¬erent

models of in¬‚ation have been proposed, but in¬‚ation ultimately needs to be

derived from a fundamental theory, such as string theory. This is currently

a very active area of research in the context of ¬‚ux compacti¬cations, and it

is described towards the end of this chapter. Cosmology could provide one

of the most spectacular ways to verify string theory, since strings of cosmic

size, called cosmic strings, could potentially be produced.

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

In the traditional string theory literature, compacti¬cations to fewer than

four noncompact space-time dimensions were not considered to be of much

interest, since the real world has four dimensions. However, this situa-

tion changed with the discovery of the string dualities described in Chap-

ters 8 and 9. In particular, it was realized that M-theory compacti¬cations

on conformally Calabi“Yau four-folds, which are discussed in this section,

are closely related to certain F-theory compacti¬cations to four dimensions.

Since the three-dimensional theories have N = 2 supersymmetry, which

means that there are four conserved supercharges, they closely resemble

four-dimensional theories with N = 1 supersymmetry.

Recall that Exercise 9.4 argued that a supersymmetric solution to a the-

ory with global N = 1 supersymmetry is a zero-energy solution of the equa-

10.1 Flux compacti¬cations and Calabi“Yau four-folds 461

tions of motion. By solving the ¬rst-order supersymmetry constraints one

obtains solutions to the second-order equations of motion, and thus a con-

sistent string-theory background. One has to be careful when generalizing

this to theories with local supersymmetry, since solving the Killing spinor

equations does not automatically ensure that a solution to the full equations

of motion. This section shows that the supersymmetry constraints for ¬‚ux

compacti¬cations, together with the Bianchi identity, yield a solution to the

equations of motion, which can be derived from a potential for the moduli,

and that this potential describes the stabilization of these moduli.

M-theory on Calabi“Yau four-folds

The bosonic part of the action for 11-dimensional supergravity, presented in

Chapter 8, is

√ 1 1

2κ2 S = d11 x ’G R ’ |F4 |2 ’ A3 § F 4 § F 4 . (10.3)

11

2 6

The only fermionic ¬eld is the gravitino and a supersymmetric con¬guration

is a nontrivial solution to the Killing spinor equation

1 (4)

“M F(4) ’ 3FM µ = 0.

δΨM = Mµ + (10.4)

12

The notation is the same as in Section 8.1. This equation needs to be solved

for some nontrivial spinor µ and leads to constraints on the background

metric as well as the four-form ¬eld strength. In Chapter 9 a similar analysis

of the supersymmetry constraints for the heterotic string was presented.

However, there the three-form tensor ¬eld was set to zero for simplicity.

In general, it is inconsistent to set the ¬‚uxes to zero, unless additional sim-

plifying assumptions (or truncations) are made. This section shows that van-

ishing ¬‚uxes are inconsistent for most M-theory compacti¬cations on eight

manifolds due to the e¬ects of quantum corrections to the action Eq. (10.3).

Warped geometry

Let us now construct ¬‚ux compacti¬cations of M-theory to three-dimensional

Minkowski space-time preserving N = 2 supersymmetry.4 The most gen-

eral ansatz for the metric that is compatible with maximal symmetry and

Poincar´ invariance of the three-dimensional space-time is a warped metric.

e

This means that the space-time is not a direct product of an external space-

time with an internal manifold. Rather, a scalar function depending on the

4 A similar analysis can be performed to obtain models with N = 1 supersymmetry.

462 Flux compacti¬cations

coordinates of the internal dimensions ∆(y) is included. The explicit form

for the metric ansatz is

ds2 = ∆(y)’1 ·µν dxµ dxν +∆(y)1/2 gmn (y)dy m dy n , (10.5)

8D internal

3D ¬‚at

manifold

space-time

where xµ are the coordinates of the three-dimensional Minkowski space-time

M3 and y m are the coordinates of the internal Euclidean eight-manifold

M . In the following we consider the case in which the internal manifold

is a Calabi“Yau four-fold, which results in N = 2 supersymmetry in three

dimensions. The scalar function ∆(y) is called the warp factor. The powers

of the warp factor in Eq. (10.5) have been chosen for later convenience.

In general, a warp factor can have a dramatic in¬‚uence on the properties

of the geometry. Consider the example of a torus, which can be described

by the ¬‚at metric

ds2 = dθ2 + d•2 0 ¤ θ ¤ π, 0 ¤ • ¤ 2π.

with (10.6)

By including a suitable warp factor, the torus turns into a sphere

ds2 = dθ2 + sin2 θd•2 , (10.7)

leading to topology change. Moreover, once the warp factor is included,

it is no longer clear that the space-time splits into external and internal

components. However, this section shows (for ¬‚ux compacti¬cations of M-

theory on Calabi“Yau four-folds) that the e¬ects of the warp factor are

subleading in the regime in which the size of the four-fold is large. In this

regime, one can use the properties of Calabi“Yau manifolds discussed in

Chapter 9.

Decomposition of Dirac matrices

To work out the dimensional reduction of Eq. (10.4), the 11-dimensional

Dirac matrices need to be decomposed. The decomposition that is required

for the 11 = 3 + 8 split is

“µ = ∆’1/2 (γµ — γ9 ) “m = ∆1/4 (1 — γm ),

and (10.8)

where γµ are the 2 — 2 Dirac matrices of M3 . Concretely, they can be

represented by

γ0 = iσ1 , γ1 = σ2 and γ 2 = σ3 , (10.9)

10.1 Flux compacti¬cations and Calabi“Yau four-folds 463

where the σ™s are the Pauli matrices