It is assumed here that the orbifold group action acts only on spatial dimen-

sions. When the time direction is involved, new phenomena, such as closed

time-like curves, can result.

Some simple examples

Compact examples

A circle is obtained by identifying points on the real line according to x ∼

x + 2πR. The simplest example of an orbifold is the interval S1 / 2 resulting

after the identi¬cation of the circle coordinate x ’ ’x. This identi¬cation

transforms a circle into an interval as shown in Fig. 9.2. This orbifold plays

a crucial role in connection with the strong-coupling limit of the E8 — E8

heterotic string, as discussed in Chapter 8.

x

x˜-x

π

0

Fig. 9.2. The simplest example of an orbifold is the interval S1 /¥ 2.

360 String geometry

Noncompact examples

A simple noncompact example of an orbifold results from considering the

complex plane , described by a local coordinate z in the usual way, and

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the isometry given by the transformation

z ’ ’z. (9.1)

This operation squares to one, and therefore it generates the two-element

group 2 . The orbifold / 2 is de¬ned by identifying points that are in the

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same orbit of the group action, that is, by identifying z and ’z. Roughly

speaking, this operation divides the complex plane into two half-planes.

More precisely, the orbifold corresponds to taking the upper half-plane and

identifying the left and right halves of the boundary (the real axis) according

to the group action. As depicted in Fig. 9.3, the resulting space is a cone.

Fig. 9.3. To construct the orbifold /¥ 2 the complex plane is divided into two

¦

parts and identi¬ed along the real axis (z ∼ ’z). The resulting orbifold is a cone.

This orbifold is smooth except for a conical singularity at the point (0, 0),

because this is the ¬xed point of the group action. One consequence of

the conical singularity is that the circumference of a circle of radius R,

centered at the origin, is πR and the conical de¬cit angle is π. An obvious

generalization is the orbifold / N , where the group is generated by a

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rotation by 2π/N . In this case there is again a singularity at the origin

and the conical de¬cit angle is 2π(N ’ 1)/N . This type of singularity is

an AN singularity. It is included in the more general ADE classi¬cation of

singularities, which is discussed in Sections 9.11 and 9.12.

The example / 2 illustrates the following general statement: points

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that are invariant (or ¬xed) under some nontrivial group element map to

singular points of the quotient space. Because of the singularities, these

quotient spaces are not manifolds (which, by de¬nition, are smooth), and

9.1 Orbifolds 361

they are called orbifolds instead. Not every discrete group action has ¬xed

generated by a translation z ’ z + a

points. For example, the group

gives rise to the quotient space / , which is a cylinder. Since there are no

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¬xed points, the cylinder is a smooth manifold, and it would not be called

an orbifold. When there are two such periods, whose ratio is not real, the

quotient space /( — ) is a smooth torus.

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The spectrum of states

What kind of physical states occur in the spectrum of free strings that live

on an orbifold background geometry? In general, there are two types of

states.

• The most obvious class of states, called untwisted states, are those that

exist on X and are invariant under the group G. In other words, the

Hilbert space of string states on X can be projected onto the subspace

of G-invariant states. A string state Ψ on X corresponds to an orbifold

string state on X/G if