Magic square and half-hypermultiplets in F-theory

TL;DR

This paper explores the structure of half-hypermultiplets in six-dimensional F-theory models, focusing on a specific non-split $I_6$ type elliptic fibration with gauge group $Sp(3)$, revealing differences from previous magic square related models.

Contribution

It investigates a novel F-theory model with non-split $I_6$ fibers, identifying the origin of half-hypermultiplets at $E_6$ points and analyzing matter generation near $D_6$ points, expanding understanding of magic square related F-theory constructions.

Findings

Half-hypermultiplets arise at $E_6$ points in the non-split $I_6$ model.

Significant qualitative differences from previous magic square models.

Conifold transition relates split and non-split fiber models.

Abstract

In six-dimensional F-theory/heterotic string theory, half-hypermultiplets arise only when they correspond to particular quaternionic K\"ahler symmetric spaces, which are mostly associated with the Freudenthal-Tits magic square. Motivated by the intriguing singularity structure previously found in such F-theory models with a gauge group $SU(6)$,$SO(12)$ or $E_7$, we investigate, as the final magical example, an F-theory on an elliptic fibration over a Hirzebruch surface of the non-split $I_6$ type, in which the unbroken gauge symmetry is supposed to be $Sp(3)$. We find significant qualitative differences between the previous F-theory models associated with the magic square and the present case. We argue that the relevant half-hypermultiplets arise at the $E_6$ points, where half-hypermultiplets ${\bf 20}$ of $SU(6)$ would have appeared in the split model. We also consider the problem on the non-local matter generation near the $D_6$ point. After stating what the problem is, we explain why this is so by using the recent result that a split/non-split transition can be regarded as a conifold transition.