Tristan Hübsch

Rolling among Calabi-Yau vacua

Abstract For a very large number of Calabi-Yau manifolds of many different numerical invariants and hence distinct homotopy types, the relevant moduli spaces can be assembled into a connected web. Here we study the geometry of these moduli spaces, especially near the interfacing regions which correspond to conifolds, certain rather mild singularizations of the manifolds in question. In the natural metric, which we show to coincide also with the point field limit of the Zamolodchikov metric, all the distances in this web are finite.

Calabi‐Yau Manifolds: A Bestiary for Physicists

Calabi-Yau spaces are complex spaces with a vanishing first Chern class, or equivalently, with trivial canonical bundle (canonical class). They are used to construct possibly realistic (super)string models and are thus being studied vigorously in the recent physics literature.In the main part of the Book, collected and reviewed are relevant results on (1) several major techniques of constructing such spaces and (2) computation of physically relevant quantities such as massless field spectra and their Yukawa interactions. Issues of (3) stringy corrections and (4) moduli space and its geometry are still in the stage of rapid and continuing development, whence there is more emphasis on open problems here. Also is included a preliminary discussion of the conjectured universal moduli space and related open problems. Finally, several detailed models and sample computations are included throughout the Book to exemplify the techniques and the general discussion.The Book also contains a Lexicon (28 pages) of 150 assorted terms, key-words and main results and theorems, well suited for a handy reference. Although cross-referenced with the main part of the Book, the Lexicon can also be used independently.The level of mathematics is guided and developed between that of the popular Physics Reports of Eguchi, Gilkey and Hanson and the book Superstrings (Vol. 2) by Green, Schwarz and Witten on one end and Principles of Algebraic Geometry of Griffiths and Harris on the other.This is the first systematic exposition in book form of the material on Calabi-Yau spaces, related mathematics and the physics application, otherwise scattered through research articles in journals and conference proceedings.

Connecting Moduli Spaces of Calabi-yau Threefolds

We demonstrate that many families of Calabi-Yau threefolds consist generically of small resolutions of nodal forms in other families and, in fact, that a large class of families is connected by this relation. Our result resonates with a conjecture of Reid that Calabi-Yau threefolds may have a universal moduli space even though they are of different homotopy types. Such ideas tie quite naturally to alluring prospects of unifying (super)string models.

Periods for Calabi-Yau and Landau-Ginzburg vacua☆

Abstract The complete structure of the moduli space of Calabi-Yau manifolds and the associated Landau-Ginzburg theories, and hence also of the corresponding low-energy effective theory that results from (2, 2) superstring compactification, may be determined in terms of certain holomorphic functions called periods. These periods are shown to be readily calculable for a great many such models. We illustrate this by computing the periods explicitly for a number of classes of Calabi-Yau manifolds. We also point out that it is possible to read off from the periods certain important information relating to the mirror manifolds.

Calabi-Yau Manifolds as Complete Intersections in Products of Complex Projective Spaces

We consider constructions of manifolds withSU(3) holonomy as embedded in products of complex projective spaces by imposing certain homogeneous holomorphic constraints. We prove that every such construction leads to one deformation class of manifolds withSU(3) holonomy. For a subset of these manifolds we prove simple connectedness, address the problem of calculating the second Betti number and explicitly calculate it for a class of constructions. This establishes a very wide class of manifolds withSU(3) holonomy, that can give rise to yet many more constructions via dividing out the action of suitably chosen discrete groups. Some of the examples studied may yield phenomenologically acceptable models.

Calabi-yau Manifolds: Motivations and Constructions

The possible ways of compacitification of theE8 ⊗E8 Superstring theory to four dimensions are reviewed. The phenomenological need forN=1 supersymmetry is argued (on quite general grounds) to favour the choice of a Calabi-Yau manifold for the compact internal manifold. The massless spectrum after compactification is derived in full detail revealing, beside the usual particles, others that may have great phenomenological impact. The technical aspects of the construction of such manifolds are examined and the methods of calculation of the relevant topological properties are given. A big family of such constructions, giving rise to many new Calabi-Yau manifolds, is presented and its relevance to the search of a phenomenologically acceptable solution is discussed.

All the Hodge numbers for all Calabi-Yau complete intersections

The authors compute explicitly all the Hodge numbers for all Calabi-Yau manifolds realised as complete intersections of hypersurfaces in products of complex projective spaces. This determines the essential part of the matter superfield spectrum for the heterotic superstring compactified on any of these manifolds. They use various techniques presented in the recent literature to obtain 265 distinct Hodge diamonds; they exemplify these techniques, giving all necessary details of their computations.

Polynomial deformations and cohomology of Calabi-Yau manifolds

We study the method of polynomial deformations that is used in the physics literature to determine the Hodge numbers of Calabi-Yau manifolds as well as the related Yukawa couplings. We show that the argument generally presented in the literature in support of these computations is seriously misleading, give a correct proof which applies to all the cases we found in the literature, and present examples which show that the method is not universally valid. We present a general analysis which applies to all Calabi-Yau manifolds embedded as complete intersections in products of complex projective spaces, yields sufficient conditions for the validity of the polynomial deformation method, and provides an alternative computation of all the Hodge numbers in many cases in which the polynomial method fails.

Codes and supersymmetry in one dimension

Adinkras are diagrams that describe many useful supermultiplets in D=1 dimensions. We show that the topology of the Adinkra is uniquely determined by a doubly even code. Conversely, every doubly even code produces a possible topology of an Adinkra. A computation of doubly even codes results in an enumeration of these Adinkra topologies up to N=28, and for minimal supermultiplets, up to N=32.

On the instanton contributions to the masses and couplings of E6 singlets

Abstract We consider the gauge neutral matter in the low-energy effective action for string theory compactification on a Calabi-Yau manifold with (2,2) world-sheet supersymmetry. At the classical level these states (the 1s of E6) correspond to the cohomology group H1( M , End T). We examine the first order contribution of instantons to the mass matrix of these particles. In principle, these corrections depend on the Kahler parameters ti through factors of the form e2πiti and also depend on the complex structure parameters. For simplicity we consider in greatest detail the quintic threefold P4[5]. It follows on general grounds that the total mass is often, and perhaps always, zero. The contribution of individual instantons is however non-zero and the contribution of a given instanton may develop poles associated with instantons coalescing for certain values of the complex structure. This can happen when the underlying Calabi-Yau manifold is smooth. Hence these poles must cancel between the coalescing instantons in order that the superpotential be finite. We examine also the Yukawa couplings involving neutral matter 13 and neutral and charged fields 27 · 27 · 1 , which have been little investigated even though they are of phenomenological interest. We study the general conditions under which these couplings vanish classically. We also calculate the first-order world-sheet instanton correction to these couplings and argue that these also vanish.