Enhanced Gauge Symmetry in Type II and F-Theory Compactifications: Dynkin Diagrams from Polyhedra

TL;DR

This paper demonstrates how Dynkin diagrams of gauge groups in type II and F-theory compactifications can be derived from toric polyhedra, linking geometric degenerations to algebraic classifications.

Contribution

It provides a method to read off gauge group Dynkin diagrams from toric polyhedra, connecting geometric degenerations with ADE and Kodaira classifications in string compactifications.

Findings

Dynkin diagrams are encoded in toric polyhedra structures.

Intersection patterns of divisors follow ADE and Kodaira classifications.

Method applies to elliptic K3 surfaces, threefolds, and fourfolds.

Abstract

We explain the observation by Candelas and Font that the Dynkin diagrams of nonabelian gauge groups occurring in type IIA and F-theory can be read off from the polyhedron $\Delta^*$ that provides the toric description of the Calabi-Yau manifold used for compacification. We show how the intersection pattern of toric divisors corresponding to the degeneration of elliptic fibers follows the ADE classification of singularities and the Kodaira classification of degenerations. We treat in detail the cases of elliptic K3 surfaces and K3 fibered threefolds where the fiber is again elliptic. We also explain how even the occurrence of monodromy and non-simply laced groups in the latter case is visible in the toric picture. These methods also work in the fourfold case.