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

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 ${\Delta }^{\ast }$ 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.

26 pages, LaTeX2e, 17 figures, references added

Cited by in corpus (20)

• Geometric Engineering in Toric F-Theory and GUTs with U(1) Gauge Factors
• SU(5) Tops with Multiple U(1)s in F-theory
• D7-Brane Motion from M-Theory Cycles and Obstructions in the Weak Coupling Limit
• An Abundance of K3 Fibrations from Polyhedra with Interchangeable Parts
• The Vertical, the Horizontal and the Rest: anatomy of the middle cohomology of Calabi-Yau fourfolds and F-theory applications
• Tops as Building Blocks for G2 Manifolds
• Box Graphs and Resolutions II: From Coulomb Phases to Fiber Faces
• Towards Generalized Mirror Symmetry for Twisted Connected Sum $G_2$ Manifolds
• Comparing elliptic and toric hypersurface Calabi-Yau threefolds at large Hodge numbers
• String Dualities and Toric Geometry: An Introduction
• Statistics of Limit Root Bundles Relevant for Exact Matter Spectra of F-Theory MSSMs
• Calabi-Yau Duals for CHL Strings
• F-theory and Heterotic Duality, Weierstrass Models from Wilson lines
• Brill-Noether-general Limit Root Bundles: Absence of vector-like Exotics in F-theory Standard Models
• Results from an Algebraic Classification of Calabi-Yau Manifolds
• Weyl invariant Jacobi forms along Higgsing trees
• Improved statistics for F-theory standard models
• Puzzles on the Duality between Heterotic and Type IIA Strings
• Effective Superpotentials of Type II D-brane/F-theory on Compact Complete Intersection Calabi-Yau Threefolds
• Reflexive Numbers and Berger Graphs from Calabi-Yau Spaces