gΨ = Ψ, (9.2)

For a ¬nite group G, one can start with any state on X, Ψ0 , and construct

a G-invariant state Ψ by superposing all the images gΨ0 .

• There is a second class of physical string states on orbifolds whose exis-

tence depends on the fact that strings are extended objects. These states,

called twisted states, are obtained in the following way. In a theory of

closed strings, which is what is assumed here, strings must start and end

at the same point, that is, X µ (σ + 2π) = X µ (σ). A string that connects

a point of X to one of its G images would not be an allowed con¬gura-

tion on X, but it maps to an allowed closed-string con¬guration on X/G.

Mathematically, the condition is

X µ (σ + 2π) = gX µ (σ), (9.3)

for some g ∈ G. The untwisted states correspond to g = 1. Twisted states

are new closed-string states that appear after orbifolding. In general, there

are various twisted sectors, labeled by the group element used to make

the identi¬cation of the ends. More precisely, it is the conjugacy classes

of G that give distinct twisted sectors. This distinction only matters if G

is nonabelian.

In the example / 2 it is clear that the twisted string states enclose

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the singular point of the orbifold. This is a generic feature of orbifolds.

362 String geometry

In the quantum spectrum, the individual twisted-sector quantum states

of the string are localized at the orbifold singularities that the classical

con¬gurations enclose. This is clear for low-lying excitations, at least,

since the strings shrink to small size.

Orbifolds and supersymmetry breaking

String theories on an orbifold X/G generically have less unbroken supersym-

metry than on X, which makes them phenomenologically more attractive.

Let us examine how this works for a certain class of noncompact orbifolds

that are a generalization of the example described above, namely orbifolds

of the form n / N . The conclusions concerning supersymmetry breaking

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are also applicable to compact orbifolds of the form T 2n / N .

n/

The orbifold

N

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Let us parametrize n by coordinates (z 1 , . . . , z n ), and de¬ne a generator g

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of N by a simultaneous rotation of each of the planes

a

g : z a ’ eiφ z a , a = 1, . . . , n, (9.4)

where the φa are integer multiples of 2π/N , so that g N = 1. The example

of the cone corresponds to n = 1, N = 2 and φ1 = π.

Unbroken supersymmetries are the components of the original supercharge

Q± that are invariant under the group action. Since the group action in this

example is a rotation, and the supercharge is a spinor, we have to exam-

ine how a spinor transforms under this rotation. The weights of spinor

representations of a rotation generator in 2n dimensions have the form

(± 2 , ± 2 , . . . , ± 1 ), a total of 2n states. This corresponds to dividing the

1 1

2

exponents by two in Eq. (9.4), which accounts for the familiar fact that a

spinor reverses sign under a 2π rotation. An irreducible spinor representa-

tion of Spin(2n) has dimension 2n’1 . An even number of ’ weights gives

one spinor representation and an odd number gives the other one. Under

the same rotation considered above

n

µa φa

g : Q± ’ exp i Q± , (9.5)

±

a=1

where µ± is a spinor weight. Suppose, for example, that the φa are chosen

so that

n

1

φa = 0 mod N. (9.6)

2π

a=1

9.2 Calabi“Yau manifolds: mathematical properties 363

Then, in general, the only components of Q± that are invariant under g are

those whose weights µ± have the same sign for all n components, since then

µa φa = 0. In special cases, other components may also be invariant. For

±

each value of ± for which the supercharge is not invariant, the amount of

unbroken supersymmetry is cut in half. Thus, if there is invariance for only

one value of ±, the fraction of the supersymmetry that is unbroken is 21’n .

This chapter shows that the same fraction of supersymmetry is preserved

by compacti¬cation on a Calabi“Yau n-fold. In fact, some orbifolds of this

type are singular limits of smooth Calabi“Yau manifolds.

9.2 Calabi“Yau manifolds: mathematical properties

De¬nition of Calabi“Yau manifolds

By de¬nition, a Calabi“Yau n-fold is a K¨hler manifold having n complex

a

dimensions and vanishing ¬rst Chern class

1

c1 = [R] = 0. (9.7)

2π

A theorem, conjectured by Calabi and proved by Yau, states that any com-

pact K¨hler manifold with c1 = 0 admits a K¨hler metric of SU (n) holon-

a a

omy. As we will see below a manifold with SU (n) holonomy admits a spinor

¬eld which is covariantly constant and as a result is necessarily Ricci ¬‚at.

This theorem is only valid for compact manifolds. In order for it to be valid

in the noncompact case, additional boundary conditions at in¬nity need to

be imposed. As a result, metrics of SU (n) holonomy correspond precisely