Affine Kac-Moody algebras, CHL strings and the classification of tops
Vincent Bouchard, Harald Skarke
TL;DR
The paper broadens Candelas–Font’s top concept to a comprehensive class of 3D lattice polytopes that encode elliptic-fibration degenerations, and it furnishes a systematic method to assign affine Kac–Moody algebras (untwisted or twisted) to each top via root-length and null-root data. It shows that tops organize into infinite families and that twisted cases arise only in special lattice pairings, corresponding to CHL-type rank reductions and specific string dualities. By embedding tops in toric geometry and exploiting Kodaira-type singularity resolutions, the authors connect geometric degenerations to Dynkin diagrams and folding patterns that yield both standard and twisted algebras such as $A_n^{(1)}$, $E_6^{(2)}$, and $D_4^{(3)}$. The results illuminate how toric degenerations govern non-simply-laced gauge groups in string compactifications and clarify the dualities between M-/F-theory and heterotic CHL strings.
Abstract
Candelas and Font introduced the notion of a `top' as half of a three dimensional reflexive polytope and noticed that Dynkin diagrams of enhanced gauge groups in string theory can be read off from them. We classify all tops satisfying a generalized definition 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 fibration. We give a prescription for assigning an affine, possibly twisted Kac-Moody algebra to any such top (and more generally to any elliptic fibration structure) in a precise way that involves the lengths of simple roots and the coefficients of null roots. Tops related to twisted Kac-Moody algebras can be used to construct string compactifications with reduced rank of the gauge group.