ORBIFOLDS WITH DISCRETE TORSION AND MIRROR SYMMETRY

For a large class of $N=2$ SCFTs, which includes minimal models and many \s models on Calabi-Yau manifolds, the mirror theory can be obtained as an orbifold. We show that in such a situation the construction of the mirror can be extended to the presence of discrete torsions. In the case of the \ZZ_2\ex\ZZ_2 torus orbifold, discrete torsion between the two generators directly provides the mirror model. Working at the Gepner point it is, however, possible to understand this mirror pair as a special case of the Berglund--H"ubsch construction. This seems to indicate that the \ZZ_2\ex\ZZ_2 example is a mere coincidence, due to special properties of \ZZ_2 twists, rather than a hint at a new mechanism for mirror symmetry.Comment: 11 pages, LaTe

Strings on Calabi--Yau spaces and Toric Geometry

After a brief introduction into the use of Calabi--Yau varieties in string dualities, and the role of toric geometry in that context, we review the classification of toric Calabi-Yau hypersurfaces and present some results on complete intersections. While no proof of the existence of a finite bound on the Hodge numbers is known, all new data stay inside the familiar range $h_{11}+h_{12}\le 502$.Comment: error in Hodge data of complete intersections corrected, published in Nucl.Phys. B Conf. Suppl. 102 (2001) 8

The Mirror Map for Invertible LG Models

Calculating the (a,c) ring of the maximal phase orbifold for `invertible' Landau--Ginzburg models, we show that the Berglund--H"ubsch construction works for all potentials of the relevant type. The map that sends a monomial in the original model to a twisted state in the orbifold representation of the mirror is constructed explicitly. Via this map, the OP selection rules of the chiral ring exactly correspond to the twist selection rules for the orbifold. This shows that we indeed arrive at the correct point in moduli space, and that the mirror map can be extended to arbitrary orbifolds, including non-abelian twists and discrete torsion, by modding out the appropriate quantum symmetries.Comment: 10 pages, latex, CERN-TH.7165/94 and TUW-94/0

Gorenstein toric Fano varieties

We investigate Gorenstein toric Fano varieties by combinatorial methods using the notion of a reflexive polytope which appeared in connection to mirror symmetry. The paper contains generalisations of tools and previously known results for nonsingular toric Fano varieties. As applications we obtain new classification results, bounds of invariants and formulate conjectures concerning combinatorial and geometrical properties of reflexive polytopes.Comment: AMS-LaTeX, 29 pages with 5 figure

Reliable and efficient a posteriori error estimates for finite element approximations of the parabolic p-Laplacian

We generalize the a posteriori techniques for the linear heat equation in [Ver03] to the case of the nonlinear parabolic p-Laplace problem thereby proving reliable and efficient a posteriori error estimates for a fully discrete implicite Euler Galerkin finite element scheme. The error is analyzed using the so-called quasi-norm and a related dual error expression. This leads to equivalence of the error and the residual, which is the key property for proving the error bounds