# Gauge groups and matter fields for F-theory without Section on Double
Covers of $\mathbb{P}^1\times\mathbb{P}^1$ and Complete Intersections in
$\mathbb{P}^1\times\mathbb{P}^3$

Research paper by **Yusuke Kimura**

Indexed on: **15 Jul '16**Published on: **15 Jul '16**Published in: **High Energy Physics - Theory**

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#### Abstract

In this paper, we investigate F-theory compactifications without a section to
the fibration. We construct genus-one fibered Calabi-Yau 4-folds without a
section, and compactify F-theory on these spaces. To build such models, we in
particular focus our attention on the direct product of K3's. We consider i)a
double cover of $\mathbb{P}^1\times\mathbb{P}^1$ ramified over a bidegree (4,4)
curve, and ii)the complete intersection of bidgree (1,2) and (1,2)
hypersurfaces in $\mathbb{P}^1\times\mathbb{P}^3$. A generic member of these
surfaces is a genus-one fibered K3 surface lacking a section to the fibration.
The direct product of such surface times a K3 gives a genus-one fibered
Calabi-Yau 4-fold without a section. Among double covers of
$\mathbb{P}^1\times\mathbb{P}^1$ ramified over a bidegree (4,4) curve, we
choose to consider members defined by special forms of equations, so that each
genus-one fiber has the complex multiplication of order 4. With this choice, a
detailed analysis of physics is possible from the geometry. With the above
settings, we analyse F-theory compactification on a genus-one fibration without
a section. We precisely determine the gauge groups on the 7-branes. We also
determine potential matter spectra. When singular fibers collide, gauge group
enhances. The exceptional gauge group $E_7$ arises for some enhancements.