Toric Geometry and Dualities of String Theory

Abstract

It seems that all string theories and D = 11 supergravity are different limits of one underlying theory. These ‘different’ string theories are related by dualities. One of these leads to the following identifcations: Het[K3 × T2, VY] = IIB[Y] Het[ZX, VX] = 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 depend, in general, on a background gauge field and hence on a vector bundle. The above identifications provide insight into the important class of (0,2) vacua in addition to providing a highly nontrivial test of string duality. The class of (0,2) vacua, although important is much less well understood than the more familiar class of (2,2) vacua. The (0,2) vacua require an understanding of vector bundles on Calabi–Yau manifolds and these are much less well understood than the Calabi–Yau manifolds themselves. The point of view adopted here is that the methods of Toric Geometry afford a certain systematization — many Calabi–Yau manifolds may be understood in terms of re exive 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 re exive polyhedra so we would expect the left hand side to have also a natural interpretation in these terms. The subject of this telegraphic review is that this is in fact the case.