Academic literature on the topic 'Strict Calabi-Yau manifolds'

Cette thèse est consacrée à l'étude des cycles algébriques dans les variétés hyper-Kähleriennes projectives et les variétés de Calabi-Yau strictes. Elle contribue à la compréhension des conjectures de Beauville et de Voisin sur les anneaux de Chow des variétés hyper-kählériennes projectives et des variétés de Calabi-Yau strictes. Elle étudie également certains invariants birationnels des variétés hyper-kählériennes projectives.La première partie de la thèse, parue dans Mathematische Zeitschrift [C. Bai, On Abel-Jacobi maps of Lagrangian families, Math. Z. 304, 34 (2023)] et présentée dans le chapitre 2, étudie si les sous-variétés lagrangiennes dans une variété hyper-kählérienne partageant la même classe cohomologique ont également la même classe de Chow. Nous étudions la notion de familles lagrangiennes et ses applications aux applications d'Abel-Jacobi associées. Nous adoptons une approche infinitésimale pour donner un critère de trivialité de l'application d'Abel-Jacobi d'une famille lagrangienne, et utilisons ce critère pour donner une réponse négative à la question précédente, ajoutant aux subtilités d'une conjecture de Voisin. Nous explorons également comment la maximalité de la variation des structures de Hodge sur la cohomologie de degré 1 de la famille lagrangienne implique la trivialité de l'application d'Abel-Jacobi. La deuxième partie de la thèse, à paraître dans International Mathematics Research Notices [C. Bai, On some birational invariants of hyper-Kähler manifolds, ArXiv: 2210.12455, to appear in International Mathematics Research Notices, 2024] et présentée dans le chapitre 3, étudie le degré d'irrationalité, la gonalité fibrante et le genre fibrant des variétés hyper-kählériennes projectives. Nous commençons par donner une légère amélioration d'un résultat de Voisin sur la borne inférieure du degré d'irrationalité des variétés hyper-kählériennes générales de Mumford-Tate. Nous étudions ensuite la relation entre les trois invariants birationnels susmentionnés pour les surfaces K3 projectives de nombre de Picard 1, rajoutant la compréhension sur une conjecture de Bastianelli, De Poi, Ein, Lazarsfeld, Ullery sur le comportement asymptotique du degré d'irrationalité des surfaces K3 projectives très générales. La troisième partie de la thèse, présentée dans le chapitre 4, étudie les applications de Voisin de dimension supérieure sur les variétés de Calabi-Yau strictes. Voisin a construit des applications auto-rationnelles de variétés de Calabi-Yau obtenues comme des variétés de r-plans dans des hypersurfaces cubiques de dimension adéquate. Cette application a été largement étudiée dans le cas r=1, qui est le cas de Beauville-Donagi. Dans les cas de dimensions supérieures, nous étudions d'abord l'action de l'application de Voisin sur les formes holomorphes. Nous démontrons ensuite la conjecture de Bloch généralisée pour l'action des applications de Voisin sur les groupes de Chow dans le cas de r=2. Enfin, via l'étude de l'application de Voisin, nous apportons des éléments de preuve à une conjecture de Voisin sur l'existence d'un 0-cycle spécial sur les variétés de Calabi-Yau strictes
This thesis is devoted to the study of algebraic cycles in projective hyper-Kähler manifolds and strict Calabi-Yau manifolds. It contributes to the understanding of Beauville's and Voisin's conjectures on the Chow rings of projective hyper-Kähler manifolds and strict Calabi-Yau manifolds. It also studies some birational invariants of projective hyper-Kähler manifolds.The first part of the thesis, appeared in Mathematische Zeitschrift [C. Bai, On Abel-Jacobi maps of Lagrangian families, Math. Z. 304, 34 (2023)] and presented in Chapter 2, studies whether the Lagrangian subvarieties in a hyper-Kähler manifold sharing the same cohomological class have the same Chow class as well. We study the notion of Lagrangian families and its associated Abel-Jacobi maps. We take an infinitesimal approach to give a criterion for the triviality of the Abel-Jacobi map of a Lagrangian family, and use this criterion to give a negative answer to the above question, adding to the subtleties of a conjecture of Voisin. We also explore how the maximality of the variation of the Hodge structures on the degree 1 cohomology the Lagrangian family implies the triviality of the Abel-Jacobi map. The second part of the thesis, to appear in International Mathematics Research Notices [C. Bai, On some birational invariants of hyper-Kähler manifolds, ArXiv: 2210.12455, to appear in International Mathematics Research Notices, 2024] and presented in Chapter 3, studies the degree of irrationality, the fibering gonality and the fibering genus of projective hyper-Kähler manifolds, with emphasis on the K3 surfaces case, en mettant l'accent sur le cas des surfaces K3. We first give a slight improvement of a result of Voisin on the lower bound of the degree of irrationality of Mumford-Tate general hyper-Kähler manifolds. We then study the relation of the above three birational invariants for projective K3 surfaces of Picard number 1, adding the understandinf of a conjecture of Bastianelli, De Poi, Ein, Lazarsfeld, Ullery on the asymptotic behavior of the degree of irrationality of very general projective K3 surfaces. The third part of the thesis, presented in Chapter 4, studies the higher dimensional Voisin maps on strict Calabi-Yau manifolds. Voisin constructed self-rational maps of Calabi-Yau manifolds obtained as varieties of r-planes in cubic hypersurfaces of adequate dimension. This map has been thoroughly studied in the case r=1, which is the Beauville-Donagi case. For higher dimensional cases, we first study the action of the Voisin map on the holomorphic forms. We then prove the generalized Bloch conjecture for the action of the Voisin maps on Chow groups for the case of r=2. Finally, via the study of the Voisin map, we provide evidence for a conjecture of Voisin on the existence of a special 0-cycle on strict Calabi-Yau manifolds

In this thesis we explore various aspects of Calabi-Yau (CY) manifolds and com- pactifications of the heterotic string over them. At first we focus on classifying symmetries and computing Hodge numbers of smooth CY quotients. Being non- simply connected, these quotients are an integral part of CY compactifications of the heterotic string, aimed at producing realistic string vacua. Discrete symmetries of such spaces that are generically present in the moduli space, are phenomenologically important since they may appear as symmetries of the associated low energy theory. We classify such symmetries for the class of smooth Complete Intersection CY (CICY) quotients, resulting in a large number of regular and R-symmetry examples. Our results strongly suggest that generic, non-freely acting symmetries for CY quotients arise relatively frequently. A large number of string derived Standard Models (SM) were recently obtained over this class of CY manifolds indicating that our results could be phenomenologically important. We also specialise to certain loci in the moduli space of a quintic quotient to produce highly symmetric CY quotients. Our computations thus far are the first steps towards constructing a sizeable class of highly symmetric smooth CY quotients. Knowledge of the topological properties of the internal space is vital in determining the suitability of the space for realistic string compactifications. Employing the tools of polynomial deformation and counting of invariant Kähler classes, we compute the Hodge numbers of a large number of smooth CICY quotients. These were later verified by independent cohomology computations. We go on to develop the machinery to understand the geometry of CY manifolds embedded as hypersurfaces in a product of del Pezzo surfaces. This led to an interesting account of the quotient space geometry, enabling the computation of Hodge numbers of such CY quotients. Until recently only a handful of CY compactifications were known that yielded low energy theories with desirable MSSM features. The recent construction of rank 5 line bundle sums over smooth CY quotients has led to several SU(5) GUTs with the exact MSSM spectrum. We derive semi-analytic results on the finiteness of the number of such line bundle models, and study the relationship between the volume of the CY and the number of line bundle models over them. We also imply a possible correlation between the observed number of generations and the value of the gauge coupling constants of the corresponding GUTs. String compactifications with underlying SO(10) GUTs are theoretically attractive especially since the discovery that neutrinos have non-zero mass. With this in mind, we construct tens of thousands of rank 4 stable line bundle sums over smooth CY quotients leading to SO(10) GUTs.

This thesis is concerned with Calabi-Yau threefolds and vector bundles upon them, which are the basic mathematical objects at the centre of smooth supersymmetric compactifications of heterotic string theory. We begin by explaining how these objects arise in physics, and give a brief review of the techniques of algebraic geometry which are used to construct and study them. We then turn to studying multiply-connected Calabi-Yau threefolds, which are of particular importance for realistic string compactifications. We construct a large number of new examples via free group actions on complete intersection Calabi-Yau manifolds (CICY's). For special values of the parameters, these group actions develop fixed points, and we show that, on the quotient spaces, this leads to a particular class of singularities, which are quotients of the conifold. We demonstrate that, in many cases at least, such a singularity can be resolved to yield another smooth Calabi-Yau threefold, with different Hodge numbers and fundamental group. This is a new example of the interconnectedness of the moduli spaces of distinct Calabi-Yau threefolds. In the second part of the thesis we turn to a study of two new `three-generation' manifolds, constructed as quotients of a particular CICY, which can also be represented as a hypersurface in dP6 x dP6, where dP6 is the del Pezzo surface of degree six. After describing the geometry of this manifold, and especially its non-Abelian quotient, in detail, we show how to construct on the quotient manifolds vector bundles which lead to four-dimensional heterotic models with the standard model gauge group and three generations of particles. The example described in detail has the spectrum of the minimal supersymmetric standard model plus a single vector-like pair of colour triplets.

In this thesis, we apply the methods of numerical differential geometry to several different problems in heterotic Calabi-Yau compactification. We review algorithms for computing both the Ricci-flat metric on Calabi-Yau manifolds and Hermitian Yang-Mills connections on poly-stable holomorphic vector bundles over those spaces. We apply the numerical techniques for obtaining Ricci-flat metrics to study hierarchies of curvature scales over Calabi-Yau manifolds as a function of their complex structure moduli. The work we present successfully finds known large curvature regions on these manifolds, and provides useful information about curvature variation at general points in moduli space. This research is important in determining the validity of the low energy effective theories used in the description of Calabi-Yau compactifications. The numerical techniques for obtaining Hermitian Yang-Mills connections are applied in two different fashions in this thesis. First, we demonstrate that they can be successfully used to numerically determine the stability of vector bundles with qualitatively different features to those that have appeared in the literature to date. Second, we use these methods to further develop some calculations of holomorphic Chern-Simons invariant contributions to the heterotic superpotential that have recently appeared in the literature. A complete understanding of these quantities requires explicit knowledge of the Hermitian Yang-Mills connections involved. This feature makes such investigations prohibitively hard to pursue analytically, and a natural target for numerical techniques.
Doctor of Philosophy
String theory is one of the most promising attempts to unify gravity with the other three fundamental interactions (electromagnetic, weak and strong) of nature. It is believed to give a self-consistent theory of quantum gravity, which, at low energy, could contain all of the physics that we known, from the Standard Model of particle physics to cosmology. String theories are often defined in nine spatial dimensions. To obtain a theory with three spatial dimensions one needs to hide, or ``compactify," six of the dimensions on a compact space which is small enough to have remained unobserved by our experiments. Unfortunately, the geometries of these spaces, called Calabi-Yau manifolds, and additional structures associated to them, called holomorphic vector bundles, turns out to be extremely complex. The equations determining the exact solutions of string theory for these quantities are highly non-linear partial differential equations (PDE's) which are simply impossible to solve analytically with currently known techniques. Nevertheless, knowledge of these solutions is critical in understanding much of the detailed physics that these theories imply. For example, to compute how the particles seen in three dimensions would interact with each other in a string theoretic model, the explicit form of these solutions would be required. Fortunately, numerical methods do exist for finding approximate solutions to the PDE's of interest. In this thesis we implement these algorithmic techniques and use them to study a variety of physical questions associated to the attempt to link string theory to the physics observed in our experiments.

This thesis contributes with a number of topics to the subject of string compactifications, especially in the instance of the E₈ × E₈ heterotic string theory compactified on smooth Calabi-Yau threefolds. In the first half of the work, I discuss the Hodge plot associated with Calabi-Yau threefolds that are hypersurfaces in toric varieties. The intricate structure of this plot is explained by the existence of certain webs of elliptic-K3 fibrations, whose mirror images are also elliptic-K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along slices corresponding to the K3 fiber. Any two half-polytopes over a given slice can be combined into a reflexive polytope. This fact, together with a remarkable relation on the additivity of Hodge numbers, give to the Hodge plot the appearance of a fractal. Moving on, I discuss a different type of web of manifolds, by looking at smooth Z₃-quotients of Calabi-Yau three-folds realised as complete intersections in products of projective spaces. Non-simply connected Calabi-Yau three-folds provide an essential ingredient in heterotic string compactifications. Such manifolds are rare in the classical constructions, but they can be obtained as quotients of homotopically trivial Calabi-Yau three-folds by free actions of finite groups. Many of these quotients are connected by conifold transitions. In the second half of the work, I explore an algorithmic approach to constructing E₈ × E₈ heterotic compactifications using holomorphic and poly-stable sums of line bundles over complete intersection Calabi-Yau three-folds that admit freely acting discrete symmetries. Such Abelian bundles lead to N = 1 supersymmetric GUT theories with gauge group SU(5) × U(4) and matter fields in the 10, ⁻10, ⁻5, 5 and 1 representations of SU(5). The extra U(1) symmetries are generically Green-Schwarz anomalous and, as such, they survive in the low energy theory only as global symmetries. These, in turn, constrain the low energy theory and in many cases forbid the existence of undesired operators, such as dimension four or five proton decay operators. The line bundle construction allows for a systematic computer search resulting in a plethora of models with the exact matter spectrum of the Minimally Supersymmetric Standard Model, one or more pairs of Higgs doublets and no exotic fields charged under the Standard Model group. In the last part of the thesis I focus on the case study of a Calabi-Yau hypersurface embedded in a product of four CP1 spaces, referred to as the tetraquadric manifold. I address the question of the finiteness of the class of consistent and physically viable line bundle models constructed on this manifold. Line bundle sums are part of a moduli space of non-Abelian bundles and they provide an accessible window into this moduli space. I explore the moduli space of heterotic compactifications on the tetraquadric hypersurface around a locus where the vector bundle splits as a direct sum of line bundles, using the monad construction. The monad construction provides a description of poly-stable S(U(4) × U(1))–bundles leading to GUT models with the correct field content in order to induce standard-like models. These deformations represent a class of consistent non-Abelian models that has co-dimension one in Kähler moduli space.

In dieser Arbeit werden Aspekte der N = 1 Dualität zwischen dem Heterotischen String (der auf einer komplex dreidimensionalen Calabi-Yau Mannigfaltigkeit mit einem Vektorbündel kompaktifiziert wird) und der F-Theorie (die auf einer komplex vierdimensionalen Calabi-Yau Mannigfaltigkeit kompaktifiziert wird) diskutiert. Zu Beginn wird eine allgemeine Beschreibung der Stringdualitäten gegeben. Die Berech- nungen der notwendigen Calabi-Yau Mannigfaltigkeiten- und Vektorbündeldaten, welche Charakteristische Klassen und Bündelmoduli involvieren, werden im Detail durchgeführt. Die acht- bzw. sechsdimensionale Dualität zwischen dem Heterotis- chen String und der F-Theorie wird diskutiert. Im Anschluß erfolgt ein Vergleich der vierdimensionalen Spektren (dies involviert den Vergleich von N = 1 chiralen Multipletts) und ein Vergleich der Anomaliebedingungen (welche zu konsistenten vierdimensionalen Het/F-Theorie Kompaktifizierungen führen). Weiterhin werden vierdimensionale N = 1 Het/F-Theorie Beispiele konstruiert, insbesondere wird eine Klasse von elliptisch gefaserten Calabi-Yau's über del Pezzoflächen betrachtet.
We discuss aspects of N = 1 duality between the heterotic string compactified on a Calabi-Yau threefold with a vector bundle and F-theory on a Calabi-Yau fourfold. After a description of string duality intended for the non-specialist the framework and the constraints for heterotic/F-theory compactifications are presented. The computations of the necessary Calabi-Yau manifold and vector bundle data, involving characteristic classes and bundle moduli, are given in detail. The eight- and six- dimensional dualities are reviewed. The matching of the spectrum of chiral multiplets and of the number of heterotic five-branes respectively F-theory three-branes, needed for anomaly cancellation in four-dimensional vacua, is pointed out. Several examples of four-dimensional dual pairs are constructed where on both sides the geometry of the involved manifolds relies on del Pezzo surfaces.

We introduce slow-roll inflation in string theory on both a conceptual level and a detailed one. In order to do this we first briefly review important concepts of inflation and string theory. We then reconstruct models of string inflation in the so-called Racetrack scenario for two different cases where the difference being the number of Kähler moduli used as inflaton. Furthermore, we briefly relate our results to the more recent discussion on whether AdS/dS solutions actually exist in string theory. In this instance our results seem to indicate that uplifting is a crucial component to obtain AdS/dS solutions.

This article discusses the link between matrix models and string theory, giving emphasis on topological string theory and the Dijkgraaf–Vafa correspondence, along with applications of this correspondence and its generalizations to supersymmetric gauge theory, enumerative geometry, and mirror symmetry. The article first provides an overview of strings and matrices, noting that the correspondence between matrix models and string theory makes it possible to solve both non-critical strings and topological strings. It then describes some basic aspects of topological strings on Calabi-Yau manifolds and states the Dijkgraaf–Vafa correspondence, focusing on how it is connected to string dualities and how it can be used to compute superpotentials in certain supersymmetric gauge theories. In addition, it shows how the correspondence extends to toric manifolds and leads to a matrix model approach to enumerative geometry. Finally, it reviews matrix quantum mechanics and its applications in superstring theory.