It seems that all string theories and D = 11 supergravity are different lim
its of one underlying theory. These 'different' string theories are related
by dualities. One of these leads to the following identifications:
Het[K3 x T-2, V-Y] = IIB[Y]
Het[Z(X), V-X] = F[X]
Here Y and Z are Calabi-Yau threefolds, X is a Calabi-Yau fourfold and the
V's on the right hand side remind us that heterotic compactifications depen
d, in general, on a background gauge field and hence on a vector bundle. Th
e above identifications provide insight into the important class of (0,2) v
acua in addition to providing a highly non trivial test of string duality.
The class of (0,2) vacua, although important is much less well understood t
han the more familiar class of (2,2) vacua. The (0,2) vacua require an unde
rstanding of vector bundles on Calabi-Yau manifolds and these are much less
well understood than the Calabi-Yau manifolds themselves. The point of vie
w adopted here is that the methods of Toric Geometry afford a certain syste
matization - many Calabi-Yau manifolds may be understood in terms of reflex
ive polyhedra. The dualities above relate vector bundles on K3 surfaces to
Calabi-Yau threefolds and vector bundles on Calabi-Yau threefolds to Calabi
-Yau fourfolds. The right hand side of these identities can, in many cases,
be related to reflexive polyhedra so are would expect the left hand side t
o have also a natural interpretation in these terms. The subject of this te
legraphic review is that this is in fact the case.