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

TL;DR

This paper classifies certain lattice polytopes called tops, which encode local geometries of elliptic fibrations, and establishes a link to affine Kac-Moody algebras, aiding in string compactification models.

Contribution

It provides a complete classification of tops satisfying a generalized definition and a method to assign affine or twisted Kac-Moody algebras to them, connecting geometry with algebraic structures in string theory.

Findings

Classified all tops satisfying the generalized definition.

Established a prescription to assign affine Kac-Moody algebras to tops.

Showed how twisted Kac-Moody algebras relate to string compactifications with reduced gauge group rank.

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.