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

#### 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.