Classification of Reflexive Polyhedra in Three Dimensions

We present the last missing details of our algorithm for the classification of reflexive polyhedra in arbitrary dimensions. We also present the results of an application of this algorithm to the case of three dimensional reflexive polyhedra. We get 4319 such polyhedra that give rise to K3 surfaces embedded in toric varieties. 16 of these contain all others as subpolyhedra. The 4319 polyhedra form a single connected web if we define two polyhedra to be connected if one of them contains the other.Comment: 19 pages, LaTeX2

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-H"ubsch 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.Comment: 12 pages, LaTe

Efficiently Computing Minimal Sets of Critical Pairs

In the computation of a Gr"obner basis using Buchberger's algorithm, a key issue for improving the efficiency is to produce techniques for avoiding as many unnecessary critical pairs as possible. A good solution would be to avoid _all_ non-minimal critical pairs, and hence to process only a_minimal_ set of generators of the module generated by the critical syzygies. In this paper we show how to obtain that desired solution in the homogeneous case while retaining the same efficiency as with the classical implementation. As a consequence, we get a new Optimized Buchberger Algorithm.Comment: LaTeX using elsart.cls, 27 page

Near Flat Space limit of strings on AdS_4 x CP^3

The non-linear nature of string theory on non-trivial backgrounds related to the AdS/CFT correspondence suggests to look for simplifications. Two such simplifications proved to be useful in studying string theory. These are the pp-wave limit which describes point-like strings and the so called "near flat space" limit which connects two different sectors of string theory -- pp-waves and "giant magnons". Recently another example of AdS/CFT duality emerged - $AdS_4/CFT_3$, which suggests duality between $\mathcal N=6$ CS theory and superstring theory on AdS_4\times \cp. In this paper we study the "near flat space" limit of strings on the AdS_4\times \cp background and discuss possible applications of the reduced theory.Comment: 20 pages, typos corrected, references adde

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

Toric Geometry and Calabi–Yau Compactifications

These notes contain a brief introduction to the construction of toric Calabi–Yau hypersurfaces and complete intersections with a focus on issues relevant for string duality calculations. The last two sections can be read independently and report on recent results and work in progress, including torsion in cohomology, classification issues, and topological transitions.Цi нотатки мiстять короткий вступ до побудови торiчних Калабi–Яу гiперповерхней та повних перетинiв з акцентом на розрахунках, що стосуються дуальностi струн. Останнi два роздiли можуть бути прочитанi незалежно вiд iнших i присвяченi недавнiм результатам та роботам, якi ще не закiнчено, включаючи кручення в когомологiї, питання класифiкацiї та топологiчних переходiв

(0,2) string compactifications

Using the simple current method we study a class of $(0,2)$ SCFTs which we conjecture to be equivalent to (0,2) sigma models constructed in the framework of gauged linear sigma models.Comment: Talk at the International Symposium on the Theory of Elementary Particles Buckow, August 27-31, 1996; LaTeX, fleqn.sty, espcrc2.sty; 6 page

Simple currents versus orbifolds with discrete torsion -- a complete classification

We give a complete classification of all simple current modular invariants, extending previous results for (\Zbf_p)^k to arbitrary centers. We obtain a simple explicit formula for the most general case. Using orbifold techniques to this end, we find a one-to-one correspondence between simple current invariants and subgroups of the center with discrete torsions. As a by-product, we prove the conjectured monodromy independence of the total number of such invariants. The orbifold approach works in a straightforward way for symmetries of odd order, but some modifications are required to deal with symmetries of even order. With these modifications the orbifold construction with discrete torsion is complete within the class of simple current invariants. Surprisingly, there are cases where discrete torsion is a necessity rather than a possibility.Comment: 28 page