String Theory and M-Theory

356 String geometry
At sufficiently high energy, supersymmetry in ten or 11 dimensions
should be manifest. The issue being considered here is whether at energies
that are low compared to the compactification scale, where there is an
effective four-dimensional theory, there should be N = 1 supersymmetry.
One intriguing piece of evidence for this is that supersymmetry ensures
that the three gauge couplings of the standard model unify at about 10 16
GeV suggesting supersymmetric grand unification at this energy.
A technical advantage of supersymmetry, which appeared in the discussion
of dualities in Chapter 8, and is utilized in Chapter 11 in the
context of black hole physics, is that supersymmetry often makes it possible
to extrapolate results from weak coupling to strong coupling, thereby
providing information about strongly coupled theories. Supersymmetric
theories are easier to solve than their nonsupersymmetric counterparts.
The constraints imposed by supersymmetry lead to first-order equations,
which are easier to solve than the second-order equations of motion. For
the type of backgrounds considered here a solution to the supersymmetry
constraints that satisfies the Bianchi identity for the three-form field
strength is always a solution to the equations of motion, though the converse
is not true.
If the ten-dimensional heterotic string is compactified on an internal manifold
M, one wants to know when this gives N = 1 supersymmetry in four
dimensions. Given a certain set of assumptions, it is proved in Section 9.3
that the internal manifold must be a Calabi–Yau three-fold.
A first glance at Calabi–Yau manifolds
Calabi–Yau manifolds are complex manifolds, and they exist in any even
dimension. More precisely, a Calabi–Yau n-fold is a Kähler manifold in n
complex dimensions with SU(n) holonomy. The only examples in two (real)
dimensions are the complex plane £ and the two-torus T 2 . Any Riemann
surface, other than a torus, is not Calabi–Yau. In four dimensions there are
two compact examples, the K3 manifold and the four-torus T 4 , as well as
noncompact examples such as £ 2 and £ ×T 2 . The cases of greatest interest
are Calabi–Yau three-folds, which have six real (or three complex) dimensions.
In contrast to the lower-dimensional cases there are many thousands
of Calabi–Yau three-folds, and it is an open question whether this number
is even finite. Compactification on a Calabi–Yau three-fold breaks 3/4 of
the original supersymmetry. Thus, Calabi–Yau compactification of the het-