String Theory and M-Theory

374 String geometry
This leads to the desired equation
x n+1 + xy 2 + z 2 = 0.
9.4 Calabi–Yau compactifications of the heterotic string
Calabi–Yau compactifications of ten-dimensional heterotic string theories
give theories in four-dimensional space-time with N = 1 supersymmetry. 10
In other words, 3/4 of the original 16 supersymmetries are broken. As
mentioned in the introduction, the motivation for this is the appealing,
though unproved, possibility that this much supersymmetry extends down
to the TeV scale in the real world. Another motivation for considering these
compactifications is that it is rather easy to embed the standard-model gauge
group, or a grand-unification gauge group, inside one of the two E8 groups
of the E8 × E8 heterotic string theory.
Ansatz for the D = 10 space-time geometry
Let us assume that the ten-dimensional space-time M10 of the heterotic
string theory decomposes into a product of a noncompact four-dimensional
space-time M4 and a six-dimensional internal manifold M, which is small
and compact
M10 = M4 × M. (9.41)
Previously, ten-dimensional coordinates were labeled by a Greek index and
denoted x µ . Now, the symbol x M denotes coordinates of M10, while x µ
denotes coordinates of M4 and y m denotes coordinates of the six-dimensional
space M. This index rule is summarized by M = (µ, m). Generalizations of
the ansatz in Eq. (9.41) are discussed in Chapter 10.
Maximally symmetric solutions
Let us consider solutions in which M4 is maximally symmetric, that is, a
homogeneous and isotropic four-dimensional space-time. Symmetries alone
imply that the Riemann tensor of M4 can be expressed in terms of its metric
according to
Rµνρλ = R
12 (gµρgνλ − gµλgνρ), (9.42)
10 This amount of supersymmetry is unbroken to every order in perturbation theory. In some
cases it is broken by nonperturbative effects.
✷