How to Classify Reflexive Gorenstein Cones

Two of my collaborations with Max Kreuzer involved classification problems related to string vacua. In 1992 we found all 10,839 classes of polynomials that lead to Landau-Ginzburg models with c=9 (Klemm and Schimmrigk also did this); 7,555 of them are related to Calabi-Yau hypersurfaces. Later we found all 473,800,776 reflexive polytopes in four dimensions; these give rise to Calabi-Yau hypersurfaces in toric varieties. The missing piece - toric constructions that need not be hypersurfaces - are the reflexive Gorenstein cones introduced by Batyrev and Borisov. I explain what they are, how they define the data for Witten's gauged linear sigma model, and how one can modify our classification ideas to apply to them. I also present results on the first and possibly most interesting step, the classification of certain basic weights systems, and discuss limitations to a complete classification.Comment: 16 pages; contribution to the memorial volume `Strings, Gauge Fields, and the Geometry Behind - The Legacy of Maximilian Kreuzer

Towards finiteness without supersymmetry

Some aspects of finite quantum field theories in 3+1 dimensions are discussed. A model with non--supersymmetric particle content and vanishing one-- and two--loop beta functions for the gauge coupling and one--loop beta functions for Yukawa--couplings is presented.Comment: 14 pages, latex, ITP-UH-2/93, TUW-93-0

Non-perturbative Gauge Groups and Local Mirror Symmetry

We analyze D-brane states and their central charges on the resolution of C^2/Z_n by using local mirror symmetry. There is a point in the moduli space where all n(n-1)/2 branches of the principal component of the discriminant locus coincide. We argue that this is the point where compactifications of Type IIA theory on a K3 manifold containing such a local geometry acquire a non-perturbative gauge symmetry of the type A_{n-1}. This analysis, which involves an explicit solution of the GKZ system of the local geometry, explains how the quantum geometry exhibits all positive roots of A_{n-1} and not just the simple roots that manifest themselves as the exceptional curves of the classical geometry. We also make some remarks related to McKay correspondence.Comment: 14 pp, LaTex2

Cosmic Acceleration as an Optical Illusion

We consider light propagation in an inhomogeneous irrotational dust universe with vanishing cosmological constant, with initial conditions as in standard linear perturbation theory. A non-perturbative approach to the dynamics of such a universe is combined with a distance formula based on the Sachs optical equations. Then a numerical study implies a redshift-distance relation that roughly agrees with observations. Interpreted in the standard homogeneous setup, our results would appear to imply the currently accepted values for the Hubble rate and the deceleration parameter; furthermore there is consistency with density perturbations at last scattering. The determination of these three quantities relies only on a single parameter related to a cutoff scale. Discrepancies with the existing literature are related to subtleties of higher order perturbation theory which make both the reliability of the present approach and the magnitude of perturbative effects beyond second order hard to assess.Comment: 34 pages, 7 figures; v2: references added; v3: stronger modifications, particularly in the discussion section concerning the reliability of result

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 \IP^3's that allow transverse polynomials, whereas for $n=4$ there are 184026 weight systems, including the 7555 weight systems for weighted \IP^4's. It is proven (without computer) that the Newton polyhedra corresponding to all of these weight systems are reflexive.Comment: Latex, 14 page