ten-dimensional superstring theories is compacti¬ed to four dimensions on a

six-torus, then the resulting theory is very far from being phenomenologically

acceptable, since no supersymmetry is broken. This means that there is N =

4 or N = 8 supersymmetry in four dimensions, depending on which ten-

dimensional theory is compacti¬ed. This chapter explores possibilities that

are phenomenologically much more attractive, such as orbifolds, Calabi“Yau

manifolds and exceptional-holonomy manifolds. Compacti¬cation on these

spaces leads to vacua with less supersymmetry in four dimensions.

In order to make contact with particle phenomenology, there are various

properties of the D = 4 theory that one would like:

• The Yang“Mills gauge group SU (3) — SU (2) — U (1), which is the gauge

group of the standard model.

• An interesting class of D = 4 supersymmetric extensions of the standard

model have N = 1 supersymmetry at high energy. This supersymmetry

must be broken at some scale, which could be as low as a TeV, to make

contact with the physics observed at low energies. N = 1 supersymmetry

imposes restrictions on the theory that make calculations easier. Yet these

restrictions are not so strong as to make the theory unrealistic, as happens

in models with N ≥ 2.

356 String geometry

At su¬ciently high energy, supersymmetry in ten or 11 dimensions

should be manifest. The issue being considered here is whether at energies

that are low compared to the compacti¬cation scale, where there is an

e¬ective four-dimensional theory, there should be N = 1 supersymmetry.

One intriguing piece of evidence for this is that supersymmetry ensures

that the three gauge couplings of the standard model unify at about 1016

GeV suggesting supersymmetric grand uni¬cation at this energy.

A technical advantage of supersymmetry, which appeared in the dis-

cussion of dualities in Chapter 8, and is utilized in Chapter 11 in the

context of black hole physics, is that supersymmetry often makes it possi-

ble to extrapolate results from weak coupling to strong coupling, thereby

providing information about strongly coupled theories. Supersymmetric

theories are easier to solve than their nonsupersymmetric counterparts.

The constraints imposed by supersymmetry lead to ¬rst-order equations,

which are easier to solve than the second-order equations of motion. For

the type of backgrounds considered here a solution to the supersymme-

try constraints that satis¬es the Bianchi identity for the three-form ¬eld

strength is always a solution to the equations of motion, though the con-

verse is not true.

If the ten-dimensional heterotic string is compacti¬ed on an internal man-

ifold M , one wants to know when this gives N = 1 supersymmetry in four

dimensions. Given a certain set of assumptions, it is proved in Section 9.3

that the internal manifold must be a Calabi“Yau three-fold.

A ¬rst glance at Calabi“Yau manifolds

Calabi“Yau manifolds are complex manifolds, and they exist in any even

dimension. More precisely, a Calabi“Yau n-fold is a K¨hler manifold in n

a

complex dimensions with SU (n) holonomy. The only examples in two (real)

and the two-torus T 2 . Any Riemann

dimensions are the complex plane £

surface, other than a torus, is not Calabi“Yau. In four dimensions there are

two compact examples, the K3 manifold and the four-torus T 4 , as well as

noncompact examples such as 2 and — T 2 . The cases of greatest interest

£ £

are Calabi“Yau three-folds, which have six real (or three complex) dimen-

sions. In contrast to the lower-dimensional cases there are many thousands

of Calabi“Yau three-folds, and it is an open question whether this number

is even ¬nite. Compacti¬cation on a Calabi“Yau three-fold breaks 3/4 of

the original supersymmetry. Thus, Calabi“Yau compacti¬cation of the het-

String geometry 357

erotic string results in N = 1 supersymmetry in four dimensions, while for

the type II superstring theories it gives N = 2.

Conifold transitions and supersymmetric cycles

Nonperturbative e¬ects in the string coupling constant need to be included

for the four-dimensional low-energy theory resulting from Calabi“Yau com-

pacti¬cations to be consistent. For example, massless states coming from

branes wrapping supersymmetric cycles need to be included in the low-

energy e¬ective action, as otherwise the metric is singular and the action is

inconsistent. This is discussed in Section 9.8.

Mirror symmetry

Compacti¬cations on Calabi“Yau manifolds have an interesting property

that is related to T-duality, which is a characteristic feature of the toroidal

compacti¬cations described in Chapters 6 and 7. This chapter shows that

certain toroidal compacti¬cations also have another remarkable property,

namely invariance under interchange of the shape and size of the torus. This

is the simplest example of a symmetry known as mirror symmetry, which is a

property of more general Calabi“Yau manifolds. This property, discussed in

Section 9.9, implies that two distinct Calabi“Yau manifolds, which typically

have di¬erent topologies, can be physically equivalent. More precisely, type

IIA superstring theory compacti¬ed on a Calabi“Yau manifold M is equiv-

alent to type IIB superstring theory compacti¬ed on the mirror Calabi“Yau

manifold W .2 Evidence for mirror symmetry is given in Fig. 9.1. Some

progress towards a proof of mirror symmetry is discussed in Section 9.9.

Exceptional-holonomy manifolds

Calabi“Yau manifolds have been discussed a great deal since 1985. More

recently, other consistent backgrounds of string theory have been investi-

gated, partly motivated by the string dualities discussed in Chapter 8. The

most important examples, discussed in Section 9.12, are manifolds of G2

and Spin(7) holonomy. G2 manifolds are seven-dimensional and break 7/8

of the supersymmetry, while Spin(7) manifolds are eight-dimensional and

break 15/16 of the supersymmetry. Calabi“Yau four-folds, which are also

eight-dimensional, break 7/8 of the supersymmetry. They are discussed in

the context of ¬‚ux compacti¬cations in Chapter 10.

2 Even though it is called a symmetry, mirror symmetry is really a duality that relates pairs of

Calabi“Yau manifolds.

358 String geometry

Fig. 9.1. This ¬gure shows a plot of the sum h1,1 + h2,1 against the Euler number

χ = 2(h1,1 ’ h2,1 ) for a certain class of Calabi“Yau manifolds. The near-perfect

symmetry of the diagram illustrates mirror symmetry, which is discussed in Sec-

tion 9.7.

9.1 Orbifolds

Before discussing Calabi“Yau manifolds, let us consider a mathematically

simpler class of compacti¬cation spaces called orbifolds. Sometimes it is

convenient to know the explicit form of the metric of the internal space,

which for almost all Calabi“Yau manifolds is not known,3 not even for the

3 Exceptions include tori and the complex plane.

9.1 Orbifolds 359

four-dimensional manifold K3. Orbifolds include certain singular limits of

Calabi“Yau manifolds for which the metric is known explicitly.

Suppose that X is a smooth manifold with a discrete isometry group

G. One can then form the quotient space X/G. A point in the quotient

space corresponds to an orbit of points in X consisting of a point and all

of its images under the action of the isometry group. If nontrivial group

elements leave points of X invariant, the quotient space has singularities.

General relativity is ill-de¬ned on singular spaces. However, it turns out

that strings propagate consistently on spaces with orbifold singularities, pro-

vided so-called twisted sectors are taken into account. (Twisted sectors will

be de¬ned below). At nonsingular points, the orbifold X/G is locally indis-

tinguishable from the original manifold X. Therefore, it is natural to induce