Toric complete intersections and weighted projective space

It has been shown by Batyrev and Borisov that nef partitions of reflexive polyhedra can be used to construct mirror pairs of complete intersection Calabi--Yau manifolds in toric ambient spaces. We construct a number of such spaces and compute their cohomological data. We also discuss the relation of our results to complete intersections in weighted projective spaces and try to recover them as special cases of the toric construction. As compared to hypersurfaces, codimension two more than doubles the number of spectra with $h^{11}=1$. Alltogether we find 87 new (mirror pairs of) Hodge data, mainly with $h^{11}\le4$.Comment: 16 pages, LaTeX2e, error in Hodge data correcte

Searching for K3 Fibrations

We present two methods for studying fibrations of Calabi-Yau manifolds embedded in toric varieties described by single weight systems. We analyse 184,026 such spaces and identify among them 124,701 which are K3 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 K3 surfaces. We also calculate the Hodge numbers of the 3-folds obtaining more than three times as many as were previously known.Comment: 21 pages, LaTeX2e, 4 eps figures, uses packages amssymb,latexsym,cite,epi

On nondegeneracy of curves

A curve is called nondegenerate if it can be modeled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We show that up to genus 4, every curve is nondegenerate. We also prove that the locus of nondegenerate curves inside the moduli space of curves of fixed genus g > 1 is min(2g+1,3g-3)-dimensional, except in case g=7 where it is 16-dimensional

Investigating the transformations of polyoxoanions using mass spectrometry and molecular dynamics

The reactions of [γ-SiW10O36]8– represent one of the most important synthetic gateways into a vast array of polyoxotungstate chemistry. Herein, we set about exploring the transformation of the lacunary polyoxoanion [β2-SiW11O39]8– into [γ-SiW10O36]8– using high-resolution electrospray mass spectrometry, density functional theory, and molecular dynamics. We show that the reaction proceeds through an unexpected {SiW9} precursor capable of undertaking a direct β → γ isomerization via a rotational transformation. The remarkably low-energy transition state of this transformation could be identified through theoretical calculations. Moreover, we explore the significant role of the countercations for the first time in such studies. This combination of experimental and the theoretical studies can now be used to understand the complex chemical transformations of oxoanions, leading to the design of reactivity by structural control

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 Z2 quotients that lead to a new class of heterotic duals.Comment: 117 pages, 1 Postscript figur