Symbolic approximations to Ricci-flat metrics via extrinsic symmetries of Calabi–Yau hypersurfaces

Abstract Ever since Yau's non-constructive existence proof of Ricci-flat metrics on Calabi–Yau manifolds, finding their explicit construction remains a major obstacle to development of both string theory and algebraic geometry. Recent computational approaches employ machine learning to create novel neural representations for approximating these metrics, offering high accuracy but limited interpretability. In this paper, we analyse machine learning approximations to flat metrics of Fermat Calabi–Yau n-folds and some of their one-parameter deformations in three dimensions in order to discover their new properties. We formalise cases in which the flat metric has more symmetries than the underlying manifold, and prove that these symmetries imply that the flat metric admits a surprisingly compact representation for certain choices of complex structure moduli. We show that such symmetries uniquely determine the flat metric on certain loci, for which we present an analytic form. We also incorporate our theoretical results into neural networks to reduce Ricci curvature for multiple Calabi–Yau manifolds compared to previous machine learning approaches. We conclude with distilling the ML models to obtain for the first time closed form expressions for Kähler metrics with near-zero scalar curvature.</jats:p

On the class S origin of spindle solutions

We analyse the backreacted geometry corresponding to a stack of M5-branes wrapped on a spindle, with a view towards precision tests of the dual N = 1 superconformal field theory. We carefully study the singular loci of the uplifted geometry and show that these correspond to C3/Zn conical singularities. Therefore, these solutions present one of the first explicit realisations of honest locally N = 1 preserving punctures in class S. Additionally we study the symmetries and anomalies of the dual field theory through anomaly inflow and compute a variety of holographic observables including dimensions of BPS operators. This work paves the way for advancements in the study and identification of the precise dual field theories

The Sinkhorn algorithm, parabolic optimal transport and geometric Monge–Amp\ue8re equations

We show that the discrete Sinkhorn algorithm—as applied in the setting of Optimal Transport on a compact manifold—converges to the solution of a fully non-linear parabolic PDE of Monge–Amp\ue8re type, in a large-scale limit. The latter evolution equation has previously appeared in different contexts (e.g. on the torus it can be be identified with the Ricci flow). This leads to algorithmic approximations of the potential of the Optimal Transport map, as well as the Optimal Transport distance, with explicit bounds on the arithmetic complexity of the construction and the approximation errors. As applications we obtain explicit schemes of nearly linear complexity, at each iteration, for optimal transport on the torus and the two-sphere, as well as the far-field antenna problem. Connections to Quasi-Monte Carlo methods are exploited

dHYM connections coupled to a variable Kähler metric

The deformed Hermitian Yang-Mills (dHYM) equation is a special Lagrangian type condition in complex geometry. It requires the complex analogue of the Lagrangian phase, defined for Chern connections on holomorphic line bundles using a background Kahler metric, to be constant. In this paper we introduce and study dHYM equations with variable Kahler metric. These are coupled equations involving both the Lagrangian phase and the radius function, at the same time. They are obtained by using the extended gauge group to couple the moment map interpretation of dHYM connections, due to Collins-Yau and mirror to Thomas’ moment map for special Lagrangians, to the Donaldson-Fujiki picture of scalar curvature as a moment map. As a consequence one expects that solutions should satisfy a mixture of K-stability and Bridgeland-type stability. In special limits, or in special cases, we recover the Kahler-Yang-Mills system of Alvarez-Consul, Garcia-Fernandez and Garcıa-Prada, and ´the coupled Kahler-Einstein equations of Hultgren-Witt Nystrom. After establishing several general results we focus on the equations and their large/small radius limits on abelian varieties, with a source term, and on ruled surfaces, allowing solutions to develop conical singularities

Navigating the string landscape with machine learning techniques

This thesis explores the possibility of using heuristic search algorithms from data science, namely genetic algorithms and reinforcement learning, to navigate the string landscape to uncover phenomenologically interesting constructions. Specifically, we apply these algorithms to construct holomorphic slope-stable vector bundles over Calabi-Yau three-folds. These vector bundles lead to the particle spectrum of the minimally supersymmetric standard model (plus uncharged moduli), via compactifications of the E8 × E8 heterotic string. We explore two types of vector bundles: sums of line bundles, which have been extensively explored in existing literature and thus serve as a benchmark for the effectiveness of the algorithms, and monad bundles, where only one quasi-realistic model was previously known. For both environments, these search algorithms were able to discover many models, while exploring as little as 10−19 of the total space. As an example of these methods in a simpler context, we also explored Froggatt-Nielsen models of quark masses

Aspects of Mirror Symmetry for Non-Fano Toric Varieties

Correlation functions in the topological A-model can be computed with open Gromov-Witten invariants whereas the mirror topological B-model admits a simpler description in terms of period integrals. We first review preliminary material in Chapter 1. Then in Chapter 2 we use tropical geometry to construct the Duistermaat-Heckman measure for non-Fano toric varieties. This allows one to compute the asymptotic terms of period integrals. In Chapter 3 we solve the generalized Picard-Fuchs system for the Hirzebruch surfaces, and hence compute periods to all orders of the mirror complex structure moduli. Near the large complex structure limit point, we use toric degenerations, scattering diagrams, and Landau-Ginzburg models from the Gross-Siebert mirror symmetry program to compute open Gromov-Witten invariants. The results of Chapter 3 extend the result of Gräfnitz-Ruddat-Zaslow to non-Fano toric varieties. Namely, the proper Landau-Ginzburg superpotential is the open mirror map even in the non-Fano setting. We explicitly compute invariants for the Hirzebruch surfaces and observe novel features such as internal scattering and negative kinks in the scattering diagram

Challenging swampland conjectures in exotic corners of the landscape

Effektive Feld-Theorie erlaubt uns die Physik unserer Welt auf unterschiedlichen Energie-Niveaus zu beschreiben. Umgekehrt muss eine effektive Theorie, mit der wir unsere Welt auf alltäglichen Skalen darstellen, bei hohen Energien mit der Quantengravitation vereinigt werden. Dies ist nicht uneingeschränkt möglich. Deshalb möchte das Swampland-Programm die Voraussetzungen unter welchen eine effektive Theorie mit der Gravitation gekoppelt werden kann eruieren. Diese Bedingungen werden “Swampland Conjectures” genannt. Sie geben klare Vorhersagen und begrenzen die effektiven Theorien, welche über das Standardmodell hinaus in Betracht gezogen werden sollten. Zum Beispiel soll ein langlebiges de-Sitter (dS) Universum schlicht unmöglich sein. In Anti-de-Sitter (AdS) ist die kosmologische Konstante an die Massen eines Turms von Teilchen gekoppelt, sodass der Energiebereich in welchem die effektive Theorie valide ist im Grenzfall des flachen Raumes verschwindet. Allerdings sind Swampland Conjectures generell nicht mathematisch beweisbar. Nur wenige konnten bislang rigoros bewiesen werden, und auch diese nur in speziellen Bereichen der String Theorie. Die meisten Swampland Conjectures beziehen Inspiration und Evidenz von der String Theorie und Schwarzen Löchern. String Theorie, eine natürliche Theorie der Quantengravitation, hat eine große Zahl an Vakua mit vielen gut verstandenen effektiven Theorien. Dies stellt eine riesige Datenmenge dar, mit welcher Swampland Conjectures überprüft werden können. Unglücklicherweise sind die am besten verstandenen Vakua nicht unbedingt eine repräsentative Menge. Mit ähnlichen Konfigurationen von BPS D-Branen und p-Form Flüssen sowie perturbativen Beiträgen besteht die reelle Gefahr, dass manche Swampland Conjectures nur ein Produkt der selektiven Datenauswahl sind. Die Arbeit, welche in dieser Dissertation präsentiert wird, testet die dS und AdS Swampland Conjectures in bislang weniger gut untersuchten Bereichen der String Theorie. Mit non-BPS Branen und Exotischen String Theorien versuchen wir Hindernisse, welche dS Vakua verhindern, zu umgehen. Immer wenn wir einen Schritt näher an dS kommen, tauchen stattdessen neue Probleme auf. Demzufolge wird die dS Swampland Conjecture auch in diesen exotischen Bereichen des String Landscape bestätigt. Schließlich untersuchen wir nicht-perturbative Beiträge und erkennen, dass die AdS Swampland Conjectures um log-terme ergänzt werden müssen. Zusammengefasst bestätigen wir die Swampland Conjectures bis auf log Korrekturen auch in exotischen Bereichen der String Theorie.Effective field theories are the way physics describes the world at different energies. Conversely, this means that any effective theory of our universe should couple to quantum gravity at high energies. Realizing that this is not always possible, the swampland program tries to delineate the conditions under which a low energy theory can be consistently completed with gravity in the UV. These conditions are called swampland conjectures. They give real predictions and bounds on the effective theories we should consider in beyond the standard model physics. For instance, it is conjectured that a long-lived de Sitter (dS) vacuum is simply impossible. In Anti-de Sitter (AdS), the magnitude of the cosmological constant is related to the mass of a tower of states, so that the energy cutoff of the effective theory goes to zero as we approach flat space. However, as the name “conjecture” already implies, these statements are not in general mathematically proven. Only a very few conjectures have been rigorously proven, and even then only in special subsectors of string theory. The bulk of the swampland conjectures takes inspiration and evidence from string theory and black hole physics. String theory as a natural theory of quantum gravity has a huge number of vacua, with many well characterized effective theories. This provides an enormous data set that swampland conjectures can be tested on. Unfortunately, the best understood vacua of string theory are not necessarily a representative set. With similar setups, only using standard D-branes, p-form fluxes and perturbative contributions, there is a danger that the swampland conjectures are a product of the lamppost effect. This is the motivation for the work presented in this thesis. We investigate the dS and AdS swampland conjectures in less explored regimes of string theory, in order to escape the lamppost. We introduce non-BPS branes and consider exotic string theories to try and get around various obstructions to finding dS vacua. Always we observe that while we do manage to circumvent obstructions, new problems appear. This confirms the no-dS conjecture also in exotic corners of the string landscape. Finally we consider non-perturbative contributions and find that here the AdS swampland conjectures have to be corrected by log-terms. In summary, we find that even in strange and new corners of string theory, and up to quantum log-corrections, the swampland conjectures still hold

On the Global Topology of Moduli Spaces of Riemannian Metrics with Holonomy Sp⁡(n)\operatorname{Sp}(n)

We discuss aspects of the global topology of moduli spaces of hyperkähler metrics. If the second Betti number is larger than $4$, we show that each connected component of these moduli spaces is not contractible. Moreover, in certain cases, we show that the components are simply connected and determine the second rational homotopy group. By that, we prove that the rank of the second homotopy group is bounded from below by the number of orbits of MBM-classes in the integral cohomology. \\ An explicit description of the moduli space of these hyperkähler metrics in terms of Torelli theorems will be given. We also provide such a description for the moduli space of Einstein metrics on the Enriques manifold. For the Enriques manifold, we also give an example of a desingularization process similar to the Kummer construction of Ricci-flat metrics on a Kummer $K3$ surface.\\ We will use these theorems to provide topological statements for moduli spaces of Ricci-flat and Einstein metrics in any dimension larger than $3$. For a compact simply connected manifold $N$ we show that the moduli space of Ricci flat metrics on $N\times T^k$ splits homeomorphically into a product of the moduli space of Ricci flat metrics on $N$ and the moduli of sectional curvature flat metrics on the torus $T^k$

Light hyperweak new gauge bosons from kinetic mixing in string models

String theory is at the moment our best candidate for a unified quantum theory of gravity, aiming to reconcile all the known (and unknown) interactions with gravity as well as provide insights for currently mysterious phenomena that the Standard Model and the modern Cosmology are not able to explain. In fact, it is believed that most of the problems associated to the Standard Model can indeed be resolved in string theory. Supersymmetry is supposed to be an elegant solution to the Hierarchy problem (even though more and more stringent bounds in this direction are being placed by the fact that we have been unable to experimentally find supersymmetry yet), while all the axions that compactifications bring into play can be used to resolve the strong CP problem as well as provide good candidates for Dark Matter. Inflationary models can also be constructed in string theory, providing, then, the most diffused solution to the Horizon problem. This work, in particular, is formulated in type IIB string theory compactified on an orientifolded Calabi-Yau three-fold in LARGE Volume Scenario (LVS) and focuses on the stabilisation of all the moduli in play compatible with the construction of a hidden gauge sector whose gauge boson kinetically mixes to the visible sector U(1), acquiring a mass via a completely stringy process resulting in the St{\"u}ckelberg mechanism. The "compatibility" regards the fact that certain experimental bounds should be respected combined with recent data extrapolated by Coherent Elastic Neutrino-Nucleus Scattering (CE$\nu$NS) events at the Spallation Neutron Source at Oak Ridge National Laboratory. We are going to see that in this context we will be able to fix all the moduli as well as present a brane and fluxes set-up reproducing the correct mass and coupling of the hidden gauge boson. We also get a TeV scale supersymmetry, since the gravitino in this model will be of order O(TeV), with an uplifted vacuum to reproduce a de Sitter universe as well