String Theory and M-Theory

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
configurations 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 supersymmetry
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
are also applicable to compact orbifolds of the form T 2n / N.
The orbifold £ n / N
Let us parametrize £ n by coordinates (z 1 , . . . , z n ), and define a generator g
of N by a simultaneous rotation of each of the planes
g : z a → e iφa
z a , a = 1, . . . , n, (9.4)
where the φ a are integer multiples of 2π/N, so that g N = 1. The example
of the cone corresponds to n = 1, N = 2 and φ 1 = π.
Unbroken supersymmetries are the components of the original supercharge
Qα 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 examine
how a spinor transforms under this rotation. The weights of spinor
representations of a rotation generator in 2n dimensions have the form
(± 1 1 1
2 , ± 2 , . . . , ± 2 ), a total of 2n states. This corresponds to dividing the
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 representation
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
g : Qα → exp i ε a αφ a
Qα, (9.5)
where εα is a spinor weight. Suppose, for example, that the φa are chosen
so that
n 1
φ
2π
a = 0 mod N. (9.6)
a=1
a=1