monic forms of T 2 — T 2 that are invariant under action 3 . The other one

comes from the ¬xed points.

The 3 -invariant harmonic forms are:

1, dz 1 § dz 2 , d¯1 § d¯2 , dz 1 § d¯1 , dz 2 § d¯2 , dz 1 § dz 2 § d¯1 § d¯2 .

z z z z z z

Each of the nine singularities has a P 1 — P 1 blow-up whose boundary

£ £

is S 3 / 3 . Each of these contributes two two-cycles or h1,1 = 2. The two

two-cycles intersect at one point. Thus, the nonvanishing Hodge numbers

of the orbifold are

h2,2 = h0,0 = h2,0 = h0,2 = 1, h1,1 = 2 + 9 — 2 = 20,

while the other Hodge numbers vanish. These numbers are the same as

those for K3. This orbifold is a singular limit of a smooth K3, like the 2

orbifold considered in the text. 4 and 6 orbifolds also give singular K3 s.

2

EXERCISE 9.3

Consider 2 /G, where G is the subgroup of SU (2) generated by (z 1 , z 2 ) ’

£

(ωz 1 , ω ’1 z 2 ) and (z 1 , z 2 ) ’ (’z 2 , z 1 ) with ω 2n = 1. Show that in terms of

variables invariant under the action of G the resulting (singular) space can

be described by9

xn+1 + xy 2 + z 2 = 0.

SOLUTION

The variables

i 1

x = (z 1 z 2 )2 ,

y = ((z 1 )2n + (z 2 )2n ), z = ((z 1 )2n ’ (z 2 )2n )z 1 z 2

2 2

are invariant under the action of G. Thus

xn+1 = (z 1 z 2 )2n+2 ,

1

xy 2 = ’ ((z 1 )4n + (z 2 )4n + 2(z 1 z 2 )2n )(z 1 z 2 )2 ,

4

1 1 4n

z2 = ((z ) + (z 2 )4n ’ 2(z 1 z 2 )2n )(z 1 z 2 )2 .

4

9 The singularity of this space is called a Dn+2 singularity, because the blown-up geometry has

intersection numbers encoded in the Dn+2 Dynkin diagram. Intersection number is de¬ned in

Section 9.6, and the Dynkin diagram is explained in Section 9.11.

374 String geometry

This leads to the desired equation

xn+1 + xy 2 + z 2 = 0.

2

9.4 Calabi“Yau compacti¬cations of the heterotic string

Calabi“Yau compacti¬cations of ten-dimensional heterotic string theories

give theories in four-dimensional space-time with N = 1 supersymmetry.10

In other words, 3/4 of the original 16 supersymmetries are broken. As

mentioned in the introduction, the motivation for this is the appealing,

though unproved, possibility that this much supersymmetry extends down

to the TeV scale in the real world. Another motivation for considering these

compacti¬cations is that it is rather easy to embed the standard-model gauge

group, or a grand-uni¬cation gauge group, inside one of the two E8 groups

of the E8 — E8 heterotic string theory.

Ansatz for the D = 10 space-time geometry

Let us assume that the ten-dimensional space-time M10 of the heterotic

string theory decomposes into a product of a noncompact four-dimensional

space-time M4 and a six-dimensional internal manifold M , which is small

and compact

M10 = M4 — M. (9.41)

Previously, ten-dimensional coordinates were labeled by a Greek index and

denoted xµ . Now, the symbol xM denotes coordinates of M10 , while xµ

denotes coordinates of M4 and y m denotes coordinates of the six-dimensional

space M . This index rule is summarized by M = (µ, m). Generalizations of

the ansatz in Eq. (9.41) are discussed in Chapter 10.

Maximally symmetric solutions

Let us consider solutions in which M4 is maximally symmetric, that is, a

homogeneous and isotropic four-dimensional space-time. Symmetries alone

imply that the Riemann tensor of M4 can be expressed in terms of its metric

according to

R

Rµνρ» = (gµρ gν» ’ gµ» gνρ ), (9.42)

12

10 This amount of supersymmetry is unbroken to every order in perturbation theory. In some

cases it is broken by nonperturbative e¬ects.

9.4 Calabi“Yau compacti¬cations of the heterotic string 375

where the scalar curvature R = g µρ g ν» Rµνρ» is a constant. It is proportional

to the four-dimensional cosmological constant. Maximal symmetry restricts

the space-time M4 to be either Minkowski (R = 0), AdS (R < 0) or dS

(R > 0). The assumption of maximal symmetry along M4 also requires

the following components of the NS“NS three-form ¬eld strength H and the

Yang“Mills ¬eld strength to vanish

Hµνρ = Hµνp = Hµnp = 0 and Fµν = Fµn = 0. (9.43)

In this chapter it is furthermore assumed that the internal three-form ¬eld

strength Hmnp vanishes and the dilaton ¦ is constant. These assumptions,

made for simplicity, give rise to the backgrounds described in this chapter.

Backgrounds with nonzero internal H-¬eld and a nonconstant dilaton are

discussed in Chapter 10.

Conditions for unbroken supersymmetry

The constraints that N = 1 supersymmetry imposes on the vacuum arise

in the following way. Each of the supersymmetry charges Q± generates an

in¬nitesimal transformation of all the ¬elds with an associated in¬nitesimal

parameter µ± . Unbroken supersymmetries leave a particular background

invariant. This is the classical version of the statement that the vacuum

state is annihilated by the charges. The invariance of the bosonic ¬elds

is trivial, because each term in the supersymmetry variation of a bosonic

¬eld contains at least one fermionic ¬eld, but fermionic ¬elds vanish in a

classical background. Therefore, the only nontrivial conditions come from

the fermionic variations

δµ (fermionic ¬elds) = 0. (9.44)