String Theory and M-Theory

9.10 Heterotic string theory on Calabi–Yau three-folds 415
As a result, W is a T 3 fibration over a base B. By definition, a Calabi–
Yau manifold is a T 3 fibration if it can be described by a three-dimensional
base space B, with a three-torus above each point of B assembled so as to
make a smooth Calabi–Yau manifold. A T 3 fibration is more general than a
T 3 fiber bundle in that isolated T 3 fibers are allowed to be singular, which
means that one or more of their cycles degenerate. Turning the argument
around, M must also be a T 3 fibration. Mirror symmetry is a fiber-wise
T-duality on all of the three directions of the T 3 . A simple example of a
fiber bundle is depicted in Fig. 9.9.
Fig. 9.9. A Moebius strip is an example of a nontrivial fiber bundle. It is a line
segment fibered over a circle S 1 . Calabi–Yau three-folds that have a mirror are
conjectured to be T 3 fibrations over a base B. In contrast to the simple example
of the Moebius strip, some of the T 3 fibers are allowed to be singular.
Since the number of T-dualities is odd, even forms and odd forms are interchanged.
As a result, the (1, 1) and (2, 1) cohomologies are interchanged,
as is expected from mirror symmetry. Moreover, there exists a holomorphic
three-form on W , which implies that W is Calabi–Yau. The three
T-dualities, of course, also interchange type IIA and type IIB.
The argument given above probably contains the essence of the proof of
mirror symmetry. A note of caution is required though. We already pointed
out that there are Calabi–Yau manifolds whose mirrors are not Calabi–Yau,
so a complete proof would need to account for that. The T-duality rules and
the condition that a supersymmetric three-cycle has to be special Lagrangian
are statements that hold to leading order in α ′ , while the full description of
the mirror W requires, in general, a whole series of α ′ corrections.
9.10 Heterotic string theory on Calabi–Yau three-folds
As was discussed earlier, the fact that dH is an exact four-form implies that
tr(R∧R) and tr(F ∧F ) = 1
30Tr(F ∧F ) must belong to the same cohomology
class. The curvature two-form R takes values in the Lie algebra of the