String Theory and M-Theory

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
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 defined by identifying points that are in the
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 identified 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 fixed 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 deficit angle is π. An obvious
generalization is the orbifold £ / N, where the group is generated by a
rotation by 2π/N. In this case there is again a singularity at the origin
and the conical deficit angle is 2π(N − 1)/N. This type of singularity is
an AN singularity. It is included in the more general ADE classification of
singularities, which is discussed in Sections 9.11 and 9.12.
The example £ / 2 illustrates the following general statement: points
that are invariant (or fixed) 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 definition, are smooth), and