String Dualities and Toric Geometry: An Introduction

TL;DR

Addresses how toric geometry encodes string and F-theory dualities via homogeneous coordinates, fans and reflexive polyhedra. It develops a practical toolkit to construct Calabi–Yau hypersurfaces, read off gauge symmetries from polyhedral data, and analyze fibrations (especially elliptic and K3) in a duality context. The framework connects Kodaira/ADE singularities to gauge enhancements and demonstrates dualities between F-theory, heterotic and type II on toric Calabi–Yau manifolds through tops and Dynkin diagrams. This toric approach provides a concrete, computational bridge between geometry and non-perturbative string phenomenology.

Abstract

This note is supposed to be an introduction to those concepts of toric geometry that are necessary to understand applications in the context of string and F-theory dualities. The presentation is based on the definition of a toric variety in terms of homogeneous coordinates, stressing the analogy with weighted projective spaces. We try to give both intuitive pictures and precise rules that should enable the reader to work with the concepts presented here.

Figures (5)

• Figure 1: The fans for ${\mathbb P}^2$ and ${\mathbb P}^{(2,3,1)}$
• Figure 2: The fan for the blow-up of ${\mathbb P}^{(2,3,1)}$
• Figure 3: The polyhedra corresponding to $P$ for ${\mathbb P}^2$ and ${\mathbb P}^{(2,3,1)}$
• Figure 4: The polyhedron $\Delta^*$ corresponding to a smooth elliptically fibered K3 surface
• Figure 5: The polyhedron $\Delta^*$ corresponding to the heterotic string with unbroken gauge symmetry