Harald Skarke

No mirror symmetry in Landau-Ginzburg spectra!

We use a recent classification of non-degenerate quasihomogeneous polynomials to construct all Landau-Ginzburg (LG) potentials for N=2 superconformal field theories with c=9 and calculate the corresponding Hodge numbers. Surprisingly, the resulting spectra are less symmetric than the existing incomplete results. It turns out that models belonging to the large class for which an explicit construction of a mirror model as an orbifold is known show remarkable mirror symmetry. On the other hand, half of the remaining 15% of all models have no mirror partners. This lack of mirror symmetry may point beyond the class of LG-orbifolds.

ON THE CLASSIFICATION OF QUASIHOMOGENEOUS FUNCTIONS

We give a criterion for the existence of a non-degenerated quasihomogeneous polynomial in a configuration, i.e. in the space of polynomials with a fixed set of weights, and clarify the relation of this criterion to the necessary condition derived from the formula for the Poincaré polynomial. We further prove finiteness of the number of configurations for a given value of the singularity index. For the value 3 of this index, which is of particular interest in string theory, a constructive version of this proof implies an algorithm for the calculation of all non-degenerate configurations.

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.

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.

An Abundance of K3 Fibrations from Polyhedra with Interchangeable Parts

Even a cursory inspection of the Hodge plot associated with Calabi-Yau threefolds that are hypersurfaces in toric varieties reveals striking structures. These patterns correspond to webs of elliptic-K3 fibrations whose mirror images are also elliptic-K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along slices corresponding to the K3 fibers. Any two half-polytopes over a given slice can be combined into a reflexive polytope. This fact, together with a remarkable relation on the additivity of Hodge numbers, explains much of the structure of the observed patterns.

ALL ABELIAN SYMMETRIES OF LANDAU-GINZBURG POTENTIALS

We present an algorithm for determining all inequivalent abelian symmetries of nondegenerate quasi-homogeneous polynomials and apply it to the recently constructed complete set of Landau–Ginzburg potentials for N = 2 superconformal field theories with c = 9. A complete calculation of the resulting orbifolds without torsion increases the number of known spectra by about one third. The mirror symmetry of these spectra, however, remains at the same low level as for untwisted Landau–Ginzburg models. This happens in spite of the fact t.

Weight systems for toric Calabi-Yau varieties and reflexivity of Newton polyhedra

According to a recently proposed scheme for the classification of reflexive polyhedra, weight systems of a certain type play a prominent role. These weight systems are classified for the cases n = 3 and n = 4, corresponding to toric varieties with K3 and Calabi-Yau hypersurfaces, respectively. For n = 3 we find the well-known 95 weight systems corresponding to weighted ℙ3’s that allow transverse polynomials, whereas for n = 4 there are 184,026 weight systems, including the 7555 weight systems for weighted ℙ4’s. It is proven (without computer) that the Newton polyhedra corresponding to all these weight systems are reflexive.

LANDAU-GINZBURG ORBIFOLDS WITH DISCRETE TORSION

We complete the classification of (2, 2) vacua that can be constructed from Landau-Ginzburg models by Abelian twists with arbitrary discrete torsions. Compared to the case without torsion, the number of new spectra is surprisingly small. In contrast to a popular expectation mirror symmetry does not seem to be related to discrete torsion (at least not in the present compactification framework). The Berglund-Hubsch construction naturally extends to orbifolds with torsion; for more general potentials, on the other hand, the new spectra neither have nor provide mirror partners in our class of models.

ADE string vacua with discrete torsion

Abstract We complete the classification of (2,2) string vacua that can be constructed by diagonal twists of tensor products of minimal models with ADE invariants. Using the Landau-Ginzburg framework, we compute all spectra from inequivalent models of this type. The completeness of our results is only possible the systematically avoiding the huge redundancies coming from permutation symmetries of tensor products. We recover the results for (2,2) vacua of an extensive computation of simple current invariants by Schellekens and Yankielowicz, and find 4 additional mirror pairs of spectra. For the model (1) 9 we observe a relation between redundant spectra and groups that are related in a particular way.