Moduli-space problem

The previous chapter described Calabi“Yau compacti¬cation for a product

manifold M4 — M . When the ten-dimensional heterotic string is compacti-

¬ed on such a manifold the resulting low-energy e¬ective action has N = 1

supersymmetry, which makes it phenomenologically attractive in a num-

ber of respects. Certain speci¬c Calabi“Yau compacti¬cations even lead to

three-generation models.

An unrealistic feature of these models is that they contain massless scalars

with undetermined vacuum expectation values (vevs). Therefore, they do

not make speci¬c predictions for many physical quantities such as coupling

constants. These scalar ¬elds are called moduli ¬elds, since their vevs are

moduli for which there is no potential in the low-energy four-dimensional

e¬ective action. This moduli-space problem or moduli-stabilization problem

has been recognized, but not emphasized, in the traditional string theory

literature. This situation changed with the discovery of string dualities and

recognition of the key role that branes play in string theory.

As discussed in Chapter 8, the moduli-space problem already arises for

simple circle compacti¬cation of D = 11 supergravity, where the size of the

circle is a modulus, dual to the vev of the type IIA dilaton, which is un-

determined. A similar problem, in a more complicated setting, appears for

the volume of the compact space in conventional Calabi“Yau compacti¬ca-

tions of any superstring theory. In this case the size of the internal manifold

cannot be determined.

Warped compacti¬cations

Recently, string theorists have understood how to generate a potential that

can stabilize the moduli ¬elds. This requires compactifying string theory

456

Flux compacti¬cations 457

on a new type of background geometry, a warped geometry.1 Warped com-

pacti¬cations also provide interesting models for superstring and M-theory

cosmology. Furthermore, they are relevant to the duality between string

theory and gauge theory discussed in Chapter 12.

In a warped geometry, background values for certain tensor ¬elds are

nonvanishing, so that associated ¬‚uxes thread cycles of the internal manifold.

An n-form potential A with an (n + 1)-form ¬eld strength F = dA gives a

magnetic ¬‚ux of the form2

F, (10.1)

γn+1

that depends only on the homology of the cycle γn+1 . Similarly, in D di-

mensions the same ¬eld gives an electric ¬‚ux

F, (10.2)

γD’n’1

where the star indicates the Hodge dual in D dimensions. This ¬‚ux depends

only on the homology of the cycle γD’n’1 .

Flux quantization

This chapter explores the implications of ¬‚ux compacti¬cations for the

moduli-space problem, and it presents recent developments in this active

area of research. The ¬‚uxes involved are strongly constrained. This is im-

portant if one hopes to make predictions for physical parameters such as the

masses of quarks and leptons. The form of the n-form tensor ¬elds that solve

the equations of motion is derived, and the important question of which of

these preserve supersymmetry and which do not is explored.

In addition to the equations of motion, a second type of constraint arises

from ¬‚ux-quantization conditions. Section 10.5 shows that when branes are

the source of the ¬‚uxes, the quantization is simple to understand: the ¬‚ux

(suitably normalized) through a cycle surrounding the branes is the number

of enclosed branes, which is an integer. For manifolds of nontrivial homology,

there can be integrally quantized ¬‚uxes through nontrivial cycles even when

there are no brane sources, as is explained in Section 10.1. In such cases, the

quantization is a consequence of the generalized Dirac quantization condition

explained in Chapter 6. In special cases, there can be an o¬set by some

fraction in the ¬‚ux quantization rule due to e¬ects induced by curvature.

1 Warped geometries have been known for a long time, but their role in the moduli-stabilization

problem was only recognized in the 1990s.

2 It is a matter of convention which ¬‚ux is called magnetic and which ¬‚ux is called electric.

458 Flux compacti¬cations

This happens in M-theory, for example, due to higher-order quantum gravity

corrections to the D = 11 supergravity action, as is explained in Section 10.5.

Flux compacti¬cations

Let us begin by considering compacti¬cations of M-theory on manifolds that

are conformally Calabi“Yau four-folds. For these compacti¬cations, the met-

ric di¬ers from a Calabi“Yau metric by a conformal factor. Even though

these models are phenomenologically unrealistic, since they lead to three-

dimensional Minkowski space-time, in some cases they are related to N = 1

theories in four dimensions. This relatively simple class of models illustrates

many of the main features of ¬‚ux compacti¬cations. More complicated ex-

amples, such as type IIB and heterotic ¬‚ux compacti¬cations, are discussed

next. In the latter case nonzero ¬‚uxes require that the internal compacti¬ca-

tion manifolds are non-K¨hler but still complex. It is convenient to describe

a

them using a connection with torsion.

The dilaton and the radial modulus

Two examples of moduli are the dilaton, whose value determines the string

coupling constant, and the radial modulus, whose value determines the size

of the internal manifold. Classical analysis that neglects string loop and in-

stanton corrections is justi¬ed when the coupling constant is small enough.

Similarly, a supergravity approximation to string theory is justi¬ed when

the size of the internal manifold is large compared to the string scale. When

there is no potential that ¬xes these two moduli, as is the case in the ab-

sence of ¬‚uxes, these moduli can be tuned so that these approximations are

arbitrarily good. Therefore, even though compacti¬cations without ¬‚uxes

are unrealistic, at least one can be con¬dent that the formulas have a regime

of validity. This is less obvious for ¬‚ux compacti¬cations with a stabilized

dilaton and radial modulus, but it will be shown that the supergravity ap-

proximation has a regime of validity for ¬‚ux compacti¬cations of M-theory

on manifolds that are conformally Calabi“Yau four-folds.

More generally, moduli ¬elds are stabilized dynamically in ¬‚ux compacti¬-

cations. While this is certainly what one wants, it also raises new challenges.

How can one be sure that a classical supergravity approximation has any

validity at all, once the value of the radial modulus and the dilaton are sta-

bilized? There is generally a trade-o¬ between the number of moduli that

are stabilized and the amount of mathematical control that one has. This

poses a challenge, since in a realistic model all moduli should be stabilized.

Flux compacti¬cations 459

Some models are known in which all moduli are ¬xed, and a supergravity

approximation still can be justi¬ed. In these models the ¬‚uxes take integer

values N , which can be arbitrarily large in such a way that the supergravity

description is valid in the large N limit.

The string theory landscape

Even though ¬‚ux compacti¬cations can stabilize the moduli ¬elds appear-

ing in string theory compacti¬cations, there is another troubling issue. Flux

compacti¬cations typically give very many possible vacua, since the ¬‚uxes

can take many di¬erent discrete values, and there is no known criterion for

choosing among them. These vacua can be regarded as extrema of some po-

tential, which describes the string theory landscape. Section 10.6 discusses

one approach to addressing this problem, which is to accept the large de-

generacy and to characterize certain general features of typical vacua using

a statistical approach.

Fluxes and dual gauge theories

Chapter 12 shows that superstring theories in certain backgrounds, which

typically involve nonzero ¬‚uxes, have a dual gauge-theory description. The

simplest examples involve conformally invariant gauge theories. However,

there are also models that provide dual supergravity descriptions of con¬ning

supersymmetric gauge theories. Section 10.2 describes a ¬‚ux model that is

dual to a con¬ning gauge theory in the context of the type IIB theory, the

Klebanov“Strassler (KS) model. The dual gauge theory aspects of this model

are discussed in Chapter 12.