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We discuss recent constructions of global F-theory GUT models and explain how to make use of toric geometry to do calculations within this framework. After introducing the basic properties of global F-theory GUTs, we give a self-contained review of toric geometry and introduce all the tools that are necessary to construct and analyze global F-theory models. We will explain how to systematically obtain a large class of compact Calabi-Yau fourfolds which can support F-theory GUTs by using the software package PALP.

Even though it has been around for quite a while [

The construction of [

This paper discusses selected topics in toric geometry and F-theory GUTs. The paper is organized as follows: in Section

In this section, we introduce the basic concepts and notions used in global F-theory models. In the remainder of this paper we will explain the techniques which are necessary to do calculations within this framework. For more details on how the quantities introduced below come about, we refer to the original papers or the recent review article [

In [

The information about the F-theory model is encoded in the Tate equation (

In F-theory GUT models, chiral matter localizes on curves on

So far, we have summarized the basic structure of a global F-theory GUT. In the present section, we will discuss which properties of the GUT model are encoded in the geometries of the base manifold

Since the GUT brane

Having found a suitable base manifold, the next step is to identify divisors inside

We can identify candidates for del Pezzo divisors inside

A necessary condition for a divisor

In local F-theory GUTs, the del Pezzo property is sufficient to ensure the existence of a decoupling limit. For global models, further checks are in order. Gravity decouples from the gauge degrees of freedom if the mass ratio

Having found a suitable base manifold, we can also study matter curves and Yukawa couplings. The curve classes

Given a base manifold

Recently, there has been active discussion in the literature how to globally define fluxes in F-theory models [

In the previous section, we have introduced quantities which encode important information about F-theory GUT models in the geometry of the base manifold and the Calabi-Yau fourfold. In this section, we will provide the tools to calculate them. The input data needed for these calculations can be obtained by using toric geometry. After giving the basic definitions, we will discuss how to describe hypersurfaces and complete intersections of hypersurfaces in toric ambient spaces. Then, we explain how to obtain the intersection ring and the Kähler cone, or dually, the Mori cone. Finally, we will discuss how to use the computer program PALP [

We start by defining a toric variety

The crucial fact about toric geometry is that the geometric data of the toric variety can be described in terms of combinatorics of cones and polytopes in dual pairs of integer lattices. The information about the toric variety is encoded in a fan

The

In order to make contact with the definition (

There are two important properties of the fan

Finally, let us emphasize the significance of the homogeneous weights

Having defined a toric variety, we go on to discuss hypersurfaces and complete intersections of hypersurfaces in toric varieties. The hypersurface equations are sections of non-trivial line bundles. The information of these bundles can be recovered from their transition functions. In this context, we introduce the notions of Cartier divisors and Weil divisors. A Cartier divisor is given, by definition, by rational equations

Equations for hypersurfaces or complete intersections are sections of line bundles

In our discussion of F-theory model building, we also encounter complete intersection Calabi-Yaus. The concept of polar pairs of reflexive lattice polytopes can be generalized as follows:

In many string theory applications, and in particular also in F-theory, the fibration structure of a Calabi-Yau manifold is of great interest. For Calabi-Yaus which can be described in terms of toric geometry, the fibration structure can be deduced from the geometry of the lattice polytopes. If we are looking for toric fibrations where the fibers are Calabi-Yaus of lower dimensions, we have to search for reflexive subpolytopes of

Two further pieces of data that are necessary in many string theory calculations are the intersection numbers of the toric divisors and the Mori cone, which is the dual of the Kähler cone. Inside the Kähler cone, the volumes such as (

Let us start with discussing the intersection ring. For a compact toric variety

So far, we have only discussed the intersection ring of the toric variety

In order to be able to calculate all the quantities defined in Section

In string theory and F-theory, we deal with compactifications on Calabi-Yau threefolds and fourfolds. In F-theory model building, the base manifold

Let us now discuss some features and applications of PALP [

The program

Apart from recent applications in F-theory model building, which we will discuss in the next section, PALP has been used in many other contexts. A data base of Calabi-Yau threefolds has been generated by listing all 473 800 776 reflexive polyhedra in four dimensions [

In this section, we make the connection to F-theory model building and discuss how the calculations discussed in Section

The next step in the calculation is to construct the Calabi-Yau fourfold

Having found a reflexive fourfold polytope with at least one nef partition is not enough to have a good global F-theory model. If we further demand that the base

Having found a good Calabi-Yau fourfold, we can construct a GUT model on every (del Pezzo) divisor. A toric description on how to impose a specific GUT group on a Tate model has been given in [

In this paper we have discussed how toric geometry can be used to construct a large number of geometries that can support global F-theory GUTs. Using this technology, we could show that elementary consistency constraints greatly reduce the number of possible models. However, due to computational constraints, we did not quite succeed in systematically listing all possible F-theory models within a class of geometries. Such an endeavor would require substantial changes in the computer programs we are using. It is actually quite remarkable that we could make use of PALP for Calabi-Yau fourfolds and non-Calabi-Yau threefolds, since this goes beyond what it was originally designed for.

Let us present a list of suggestions to extend PALP in order to improve the applicability to the current problems in mathematics and physics and to make it more accessible for users. The original purpose of PALP was to solve a classification problem for polytopes. Over the years, it has been adjusted and extended in order to be applied to specific problems. Many of the basic routines that were implemented to tackle some special questions could be used in much more general contexts but cannot be easily accessed. Therefore, a better modularization of the software is necessary in order to have flexible access to these basic routines. Another problem of PALP is that one has to specify several parameters and bounds such as the number of points in a polytope in a given dimension at the compilation of the program. It would be practical to have fully dynamical dimensions in order to work with a precision tailored to the problem at hand without recompiling.

A fundamental change would be to step away from the description of polytopes and instead use the ray representation which has the full data of the cones. This is necessary if one wants to deal with non-reflexive polytopes. A further extension which has already been partially implemented is to include triangulations, intersection rings, and even the calculation of Picard-Fuchs operators needed for mirror symmetry calculations into PALP. The ultimate goal is to have an efficient and versatile program which can be used for toric calculations of all kinds without having to rely on commercial software. Finally, a detailed documentation of all the features of PALP would be helpful [

As for the search for F-theory models, an extended version of PALP would hopefully help to overcome the problems of nonreflexivity and overflows we have encountered in [

In October 2010, M. Kreuzer asked J. Knapp to be the coauthor of this paper. Sadly, he passed away on November 26, 2010, when this work was still in the early stages. J. Knapp is grateful for many years of collaboration with him, as well as for his constant support and encouragement. The author would like to thank her collaborators Ching-Ming Chen, Christoph Mayrhofer, and Nils-Ole Walliser for a pleasant and fruitful collaboration on the projects [

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_{6}GUTs