Gravity and Strings

348 The Kaluza–Klein black hole
11.6 Orbifold compactification
Sometimes it is possible to compactify on spaces that are not manifolds. The prototypes
of these spaces are orbifolds. These can be constructed as the quotients of manifolds by
discrete symmetries. The simplest case is the segment, which can be constructed as the
quotient S 1 /Z2. Todescribe the quotient we need to define the action of Z2, and for this it
is convenient to describe the circle itself as the quotient of the real line parametrized by z
by the group Z of discrete translations z → z + 2πnℓ. There are no fixed points of the real
line under this group and, therefore, we obtain a non-singular manifold.
Now, in terms of this coordinate z, Z2 acts by z →−z. The result is the segment of
line that goes from z = 0toz = 2π. There are two fixed points under this group z = 0,
for obvious reasons, and z = πℓ, since −πℓ∼−πℓ+ 2πℓ= πℓ, and they are the singular
endpoints of the segment, which is not a manifold. 33
The description of orbifolds as quotients is very convenient because in general the discrete
symmetries have a well-defined action on the fields of the theory. In standard KK
theory there are only tensor fields and their behavior under z reflections depends on the
number of z indices they have: they acquire a minus sign for each index z. Only the KK
vector has an odd number of z indices, Aµ =ˆgµz/ ˆgzz,and thus it reverses its sign while the
metric and KK scalar remain invariant.
The rule is that the spectrum of the KK theory on an orbifold can contain only fields
that are invariant under the discrete symmetry. The reason is that odd fields will be given
in solutions by odd functions of z on the circle and they would be double-valued (i.e. not
well defined) on the orbifold. Thus, in the standard KK theory the KK vector is projected
out of it. It is precisely this mechanism that was used by Hoˇrava and Witten in [543, 544]
to eliminate the RR 1-form Ĉ (1) in the reduction of 1one-dimensional supergravity (the
effective-field theory of “M theory” in some corner of moduli space) to obtain chiral N =
1, d = 10 supergravity (the effective-field theory of the heterotic string) instead of unchiral
N = 2A, d = 10 supergravity (see Section 16.4).
In supersymmetric KK theory one has to define the action of the Z2 group on fermions.
In odd dimensions one typically defines
ˆψ ′ =±ˆƔz ˆψ, (11.258)
where Ɣz is the gamma matrix associated with the direction z and is proportional to the
chirality matrix in one dimension fewer. Then, in the orbifold compactification only one
chiral half of the spinors survives the projection.
33 The corresponding spacetime, taking into account the metric would have a size of π Rz.