## F-theory on all toric hypersurface fibrations and its Higgs branches

## Abstract

We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflexive 2D polyhedra. We present a base-independent analysis of the codimension one, two and three singularities of these fibrations. We use these geometric results to determine the gauge groups, matter representations, 6D matter multiplicities and 4D Yukawa couplings of the corresponding effective theories. All these theories have a non-trivial gauge group and matter content. We explore the network of Higgsings relating these theories. Such Higgsings geometrically correspond to extremal transitions induced by blow-ups in the 2D toric varieties. We recover the 6D effective theories of all 16 toric hypersurface fibrations by repeatedly Higgsing the theories that exhibit Mordell-Weil torsion. We find that the three Calabi-Yau manifolds without section, whose fibers are given by the toric hypersurfaces in $$\mathbb P^{2}$$, $$\mathbb P^{1}$$ × $$\mathbb P^{1}$$ and the recently studied $$\mathbb P^{2}$$ (1,1, 2) , yield F-theory realizations of SUGRA theories with discrete gauge groups $$\mathbb Z$$ _{3}, $$\mathbb Z$$ _{2} and $$\mathbb Z$$ _{4}.This opens up a whole new arena for model building with discrete global symmetries in F-theory. In these three manifolds, we also find codimension two I _{2}-fibers supporting matter charged only under these discrete gauge groups. Their 6D matter multiplicities are computed employing ideal techniques and the associated Jacobian fibrations. Here, we also show that the Jacobian of the biquadric fibration has one rational section, yielding one U(1)-gauge field in F-theory. Furthermore, the elliptically fibered Calabi-Yau manifold based on dP _{1} has a U(1)-gauge field induced by a non-toric rational section. In this model, we find the first F-theory realization of matter with U(1)-charge q = 3.

- Authors:

- Univ. of Pennsylvania, Philadelphia, PA (United States). Dept. of Physics and Astronomy; European Organization for Nuclear Research (CERN), Geneva (Switzerland). Dept. of Physics, Theory Group
- Univ. of Bonn (Germany). Bethe Center for Theoretical Physics, Inst. der Physikalisches
- Univ. of Pennsylvania, Philadelphia, PA (United States). Dept. of Physics and Astronomy

- Publication Date:

- Research Org.:
- Univ. of Pennsylvania, Philadelphia, PA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22); National Science foundation (NSF)

- OSTI Identifier:
- 1454515

- Grant/Contract Number:
- SC0007901; PHY05-51164

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of High Energy Physics (Online)

- Additional Journal Information:
- Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2015; Journal Issue: 1; Journal ID: ISSN 1029-8479

- Publisher:
- Springer Berlin

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; F-Theory; Differential and Algebraic Geometry; Discrete and Finite Symmetries; Supergravity Models

### Citation Formats

```
Klevers, Denis, Mayorga Pena, Damian Kaloni, Oehlmann, Paul-Konstantin, Piragua, Hernan, and Reuter, Jonas. F-theory on all toric hypersurface fibrations and its Higgs branches. United States: N. p., 2015.
Web. doi:10.1007/JHEP01(2015)142.
```

```
Klevers, Denis, Mayorga Pena, Damian Kaloni, Oehlmann, Paul-Konstantin, Piragua, Hernan, & Reuter, Jonas. F-theory on all toric hypersurface fibrations and its Higgs branches. United States. doi:10.1007/JHEP01(2015)142.
```

```
Klevers, Denis, Mayorga Pena, Damian Kaloni, Oehlmann, Paul-Konstantin, Piragua, Hernan, and Reuter, Jonas. Tue .
"F-theory on all toric hypersurface fibrations and its Higgs branches". United States. doi:10.1007/JHEP01(2015)142. https://www.osti.gov/servlets/purl/1454515.
```

```
@article{osti_1454515,
```

title = {F-theory on all toric hypersurface fibrations and its Higgs branches},

author = {Klevers, Denis and Mayorga Pena, Damian Kaloni and Oehlmann, Paul-Konstantin and Piragua, Hernan and Reuter, Jonas},

abstractNote = {We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflexive 2D polyhedra. We present a base-independent analysis of the codimension one, two and three singularities of these fibrations. We use these geometric results to determine the gauge groups, matter representations, 6D matter multiplicities and 4D Yukawa couplings of the corresponding effective theories. All these theories have a non-trivial gauge group and matter content. We explore the network of Higgsings relating these theories. Such Higgsings geometrically correspond to extremal transitions induced by blow-ups in the 2D toric varieties. We recover the 6D effective theories of all 16 toric hypersurface fibrations by repeatedly Higgsing the theories that exhibit Mordell-Weil torsion. We find that the three Calabi-Yau manifolds without section, whose fibers are given by the toric hypersurfaces in $\mathbb P^{2}$, $\mathbb P^{1}$ × $\mathbb P^{1}$ and the recently studied $\mathbb P^{2}$ (1,1, 2) , yield F-theory realizations of SUGRA theories with discrete gauge groups $\mathbb Z$3, $\mathbb Z$2 and $\mathbb Z$4.This opens up a whole new arena for model building with discrete global symmetries in F-theory. In these three manifolds, we also find codimension two I 2-fibers supporting matter charged only under these discrete gauge groups. Their 6D matter multiplicities are computed employing ideal techniques and the associated Jacobian fibrations. Here, we also show that the Jacobian of the biquadric fibration has one rational section, yielding one U(1)-gauge field in F-theory. Furthermore, the elliptically fibered Calabi-Yau manifold based on dP 1 has a U(1)-gauge field induced by a non-toric rational section. In this model, we find the first F-theory realization of matter with U(1)-charge q = 3.},

doi = {10.1007/JHEP01(2015)142},

journal = {Journal of High Energy Physics (Online)},

number = 1,

volume = 2015,

place = {United States},

year = {2015},

month = {1}

}

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