Toric Geometry, Enhanced non Simply laced Gauge Symmetries in Superstrings and F-theory Compactifications

TL;DR

This paper advances the geometric engineering of supersymmetric quantum field theories with non simply laced gauge groups using toric geometry, extending methods to new singularities and providing explicit F-theory and string compactification results.

Contribution

It develops new toric methods for non simply laced singularities, extends affine toric data for F-theory, and introduces polyvalent toric geometry for engineering gauge theories.

Findings

Extended toric methods to non simply laced singularities.

Explicit affine non simply laced toric data for F-theory.

Derived new solutions for affine D4 singularity.

Abstract

We study the geometric engineering of supersymmetric quantum field theories (QFT), with non simply laced gauge groups, obtained from superstring and F-theory compactifications on local Calabi-Yau manifolds. First we review the main lines of the toric method for ALE spaces with ADE singularities which we extend to non simply laced ordinary and affine singularities. Then, we develop two classes of solutions depending on the two possible realisations of the outer-automorphism group of the toric graph $\Delta$(ADE). In F-theory on elliptic Calabi-Yau manifolds, we give explicit results for the affine non simply laced toric data and the corresponding BCFG mirror geometries. The latters extend known results obtained in litterature for the affine ADE cases. We also study the geometric engineering of $ N=1$ supersymmetric gauge theory in eight dimensions. In type II superstring compactifications on local Calabi-Yau threefolds, we complete the analysis for ordinary ADE singularities by giving the explict derivation of the lacking non simply ones. Finally we develop the basis of polyvalent toric geometry. The latter extends bivalent and trivalent geometries, considered in the geometric engineering method, and use it to derive a new solution for the affine $\hat D_4$ singularity. Other features are also discussed.