Affine Kac-Moody algebras, CHL strings and the classification of tops

Candelas and Font i n troduced the notion of a `top' as half of a three dimensional re exive polytope and noticed that Dynkin diagrams of enhanced gauge groups in string theory can be read o from them. We classify all tops satisfying a generalized de nition as a lattice polytope with one facet containing the origin and the other facets at distance one from the origin. These objects torically encode the local geometry of a degeneration of an elliptic bration. We give a prescription for assigning an a ne, possibly twisted Kac-Moody algebra to any such top (and more generally to any elliptic bration structure) in a precise way that involves the lengths of simple roots and the coe cients of null roots. Tops related to twisted Kac-Moody algebras can be used to construct string compacti cations with reduced rank of the gauge group.