Maximilian Kreuzer

PALP: A Package for Analysing Lattice Polytopes with applications to toric geometry

We describe our package PALP of C programs for calculations with lattice polytopes and applications to toric geometry, which is freely available on the internet. It contains routines for vertex and facet enumeration, computation of incidences and symmetries, as well as completion of the set of lattice points in the convex hull of a given set of points. In addition, there are procedures specialized to reflexive polytopes such as the enumeration of reflexive subpolytopes, and applications to toric geometry and string theory, like the computation of Hodge data and fibration structures for toric Calabi–Yau varieties. The package is well tested and optimized in speed as it was used for time consuming tasks such as the classification of reflexive polyhedra in 4 dimensions and the creation and manipulation of very large lists of 5-dimensional polyhedra. While originally intended for low-dimensional applications, the algorithms work in any dimension and our key routine for vertex and facet enumeration compares well with existing packages.

Topological string amplitudes, complete intersection Calabi-Yau spaces and threshold corrections

We present the most complete list of mirror pairs of Calabi-Yau complete intersections in toric ambient varieties and develop the methods to solve the topological string and to calculate higher genus amplitudes on these compact Calabi-Yau spaces. These symplectic invariants are used to remove redundancies in examples. The construction of the B-model propagators leads to compatibility conditions, which constrain multi-parameter mirror maps. For K3 fibered Calabi-Yau spaces without reducible fibers we find closed formulas for all genus contributions in the fiber direction from the geometry of the fibration. If the heterotic dual to this geometry is known, the higher genus invariants can be identified with the degeneracies of BPS states contributing to gravitational threshold corrections and all genus checks on string duality in the perturbative regime are accomplished. We find, however, that the BPS degeneracies do not uniquely fix the non-perturbative completion of the heterotic string. For these geometries we can write the topological partition function in terms of the Donaldson-Thomas invariants and we perform a non-trivial check of S-duality in topological strings. We further investigate transitions via collapsing D5 del Pezzo surfaces and the occurrence of free 2 quotients that lead to a new class of heterotic duals.

Global SO(10) F-theory GUTs

Making use of toric geometry we construct a class of global F-theory GUT models. The base manifolds are blowups of Fano threefolds and the Calabi-Yau fourfold is a complete intersection of two hypersurfaces. We identify possible GUT divisors and construct SO(10) models on them using the spectral cover construction. We use a split spectral cover to generate chiral matter on the 10 curves in order to get more degrees of freedom in phenomenology. We use abelian flux to break SO(10) to SU(5) ×U(1) which is interpreted as a flipped SU(5) model. With the GUT Higgses in the SU(5) × U(1) model it is possible to further break the gauge symmetry to the Standard Model. We present several phenomenologically attractive examples in detail.

Completeness and Nontriviality of the Solutions of the Consistency Conditions

Abstract For the case of a compact gauge group we determine all solutions to the consistency conditions. In particular, our results imply that the known list of anomalies is complete also for nonrenormalizable models.

All consistent Yang-Mills anomalies

Abstract For the case of a compact gauge group we list all solutions to the consistency equations which have to be satisfied by anomalies. We describe the main algebraic tools and theorems required for this complete classification. Our results answer the question whether in nonrenormalizable gauge theories there exist additional up-to-now unknown anomalies to the negative.

On the Classification of Reflexive Polyhedra

Abstract: Reflexive polyhedra encode the combinatorial data for mirror pairs of Calabi–Yau hypersurfaces in toric varieties. We investigate the geometrical structures of circumscribed polytopes with a minimal number of facets and of inscribed polytopes with a minimal number of vertices. These objects, which constrain reflexive pairs of polyhedra from the interior and the exterior, can be described in terms of certain non-negative integral matrices. A major tool in the classification of these matrices is the existence of a pair of weight systems, indicating a relation to weighted projective spaces. This is the cornerstone for an algorithm for the construction of all dual pairs of reflexive polyhedra that we expect to be efficient enough for an enumerative classification in up to 4 dimensions, which is the relevant case for Calabi–Yau compactifications in string theory.

Searching for K3 fibrations

Abstract We present two methods for studying fibrations of Calabi-Yau manifolds embedded in toric varieties described by single weight systems. We analyze 184 026 such spaces and identify among them the 124 701 which are K 3 fibrations. As some of the weights give rise to two or three distinct types of fibrations, the total number we find is 167 406. With our methods one can also study elliptic fibrations of 3-folds and K 3 surfaces. We also calculate the Hodge numbers of the 3-folds obtaining more than three times as many as were previously known.

Toric construction of global F-theory GUTs

We systematically construct a large number of compact Calabi-Yau fourfolds which are suitable for F-theory model building. These elliptically fibered Calabi-Yaus are complete intersections of two hypersurfaces in a six dimensional ambient space. We first construct three-dimensional base manifolds that are hypersurfaces in a toric ambient space. We search for divisors which can support an F-theory GUT. The fourfolds are obtained as elliptic fibrations over these base manifolds. We find that elementary conditions which are motivated by F-theory GUTs lead to strong constraints on the geometry, which significantly reduce the number of suitable models. The complete database of models is available at [1]. We work out several examples in more detail.

Reflexive polyhedra, weights and toric Calabi-Yau fibrations

During the last years we have generated a large number of data related to Calabi-Yau hypersurfaces in toric varieties which can be described by reflexive polyhedra. We classified all reflexive polyhedra in three dimensions leading to K3 hypersurfaces and have also completed the four-dimensional case relevant to Calabi-Yau threefolds. In addition, we have analysed for many of the resulting spaces whether they allow fibration structures of the types that are relevant in the context of superstring dualities. In this survey we want to give background information both on how we obtained these data, which can be found at our web site, and on how they may be used. We give a complete exposition of our classification algorithm at a mathematical (rather than algorithmic) level. We also describe how fibration structures manifest themselves in terms of toric diagrams and how we managed to find the respective data. Both for our classification scheme and for simple descriptions of fibration structures the concept of weight systems plays an important role.

Calabi-Yau 4-folds and toric fibrations

Abstract We present a general scheme for identifying fibrations in the framework of toric geometry and provide a large list of weights for Calabi-Yau 4-folds. We find 914 164 weights with degree d ≤ 150 whose maximal Newton polyhedra are reflexive and 525 572 weights with degree d ≤ 4000 that give rise to weighted projective spaces such that the polynomial defining a hypersurface of trivial canonical class is transversal. We compute all Hodge numbers, using Batyrevs formulas (derived by toric methods) for the first and Vafas formulas (obtained by counting of Ramond ground states in N = 2 LG models) for the latter class, checking their consistency for the 109 308 weights in the overlap. Fibrations of k -folds, including the elliptic case, manifest themselves in the N lattice in the following simple way: The polyhedron corresponding to the fiber is a subpolyhedron of that corresponding to the k -fold, whereas the fan determining the base is a linear projection of the fan corresponding to the k -fold.