Toric geometry and dualities of string theory

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.