Machine Learning Line Bundle Cohomologies of Hypersurfaces in Toric Varieties

Different techniques from machine learning are applied to the problem of computing line bundle cohomologies of (hypersurfaces in) toric varieties. While a naive approach of training a neural network to reproduce the cohomologies fails in the general case, by inspecting the underlying functional form of the data we propose a second approach. The cohomologies depend in a piecewise polynomial way on the line bundle charges. We use unsupervised learning to separate the different polynomial phases. The result is an analytic formula for the cohomologies. This can be turned into an algorithm for computing analytic expressions for arbitrary (hypersurfaces in) toric varieties.Comment: 8 pages, 6 figures, 5 tables; typos corrected, reference added, clarifications adde

Computational methods in string theory and applications to the swampland conjectures

The goal of the swampland program is the classification of low energy effective theories which can be consistently coupled to quantum gravity. Due to the vastness of the string landscape most results of the swampland program are still conjectures, yet the web of conjectures is ever growing and many interdependencies between different conjectures are known. A better understanding or even proof of these conjectures would result in restrictions on the allowed effective theories. The aim of this thesis is to develop the necessary mathematical tools to explicitly test the conjectures in a string theory setup. To this end the periods of Calabi-Yau manifolds are computed numerically as well as analytically. Furthermore, tools applicable to general string phenomenological models are discussed, including the computation of target space Calabi-Yau metrics, line bundle cohomologies and Strebel differentials. These periods are used to test two conjectures, the refined swampland distance conjecture as well as the dS conjecture. The first states that an effective field theory is only valid up to a certain value of field excursions. If larger field values are included, the effective description breaks down due to an infinite tower of states becoming exponentially light. The conjecture is tested explicitly by computing the distances in the moduli space of CY manifolds. Challenging this conjecture requires the computation of the periods of different Calabi-Yau spaces. The dS conjecture on the other hand forbids vacua with positive cosmological constant. To test this conjecture, the KKLT construction is examined in detail and some steps of the construction are carried out explicitly. Moreover, the validity of the assumed effective theory in a warped throat is investigated. Besides these traditional approaches more exotic ones are followed, including the construction of dS theories using tachyons as well as modifying the signature of space time.Das Ziel des Swampland Programms ist die Klassifizierung effektiver, zu Quantengravitationstheorien vervollständigbarer Theorien. Aufgrund der enormen Anzahl an möglichen Stringvacua, zusammengefasst in der sogenannten Stringlandschaft, sind die meisten der bisherigen Resultate des Programms Vermutungen. Jedoch existiert ein beständig wachsendes dichtes Netz aus Abhängigkeiten zwischen diesen Vermutungen. Ein besseres Verständnis oder ein Beweis dieser Vermutungen würde die erlaubten Niederenergietheorien einschränken. Das Ziel dieser Arbeit ist deshalb die Entwicklung mathematischer Methoden, die explizite Tests der Swampland Vermutungen in stringtheoretischen Modellen ermöglichen. Insbesondere werden Perioden von Calabi-Yau Mannigfaltigkeiten auf numerischem und analytischem Weg berechnet. Darüber hinaus werden Methoden zur Berechnung von Calabi-Yau Metriken, Linienbündelkohomologien und Strebeldifferentialen behandelt. Diese werden zur Überprüfung zweier Vermutungen eingesetzt, zum Test der Swampland Distanzvermutung sowie zum Test der dS Vermutung. Erstere besagt, dass eine effektive Theorie nur bis zu bestimmten Feldwerten gültig sein kann. Werden diese überschritten werden unendlich viele nicht berücksichtigte Zustände exponentiell leicht und die verwendete effektive Beschreibung bricht zusammen. Diese Vermutung wird durch eine explizite Berechnung von Distanzen zwischen effektiven Theorien in Calabi-Yau Moduliräumen getestet. Die dS Vermutung verbietet hingegen stabile Vacua mit positiver kosmologischer Konstante. Um diese Vermutung zu überprüfen, wird ein Teil der KKLT-Konstruktion explizit durchgeführt. Darüber hinaus wird die Validität der zugrundeliegenden effektiven Theorie in einem warped throat analysiert. Neben diesen traditionellen Herangehensweisen werden exotischere Ansätze für die Konstruktion von dS Räumen untersucht. Dies umfasst Tachyonenkondensation sowie andere Raumzeitsignaturen

The Refined Swampland Distance Conjecture in Calabi-Yau Moduli Spaces

The Swampland Distance Conjecture claims that effective theories derived from a consistent theory of quantum gravity only have a finite range of validity. This will imply drastic consequences for string theory model building. The refined version of this conjecture says that this range is of the order of the naturally built in scale, namely the Planck scale. It is investigated whether the Refined Swampland Distance Conjecture is consistent with proper field distances arising in the well understood moduli spaces of Calabi-Yau compactification. Investigating in particular the non-geometric phases of Kahler moduli spaces of dimension $h^{11}\in\{1,2,101\}$, we always found proper field distances that are smaller than the Planck-length.Comment: 71 pages, 11 figures, v2: refs added, typos correcte

Flux vacua of the mirror octic

We determine all flux vacua with flux numbers $N_{\rm flux}\leq 10$ for a type IIB orientifold-compactification on the mirror-octic three-fold. To achieve this, we develop and apply techniques for performing a complete scan of flux vacua for the whole moduli space - we do not randomly sample fluxes nor do we consider only boundary regions of the moduli space. We compare our findings to results in the literature.Comment: 47 pages, 10 figures, dataset provided as ancillary file; v2: references adde

Analytic Periods via Twisted Symmetric Squares

We study the symmetric square of Picard-Fuchs operators of genus one curves and the thereby induced generalized Clausen identities. This allows the computation of analytic expressions for the periods of all one-parameter K3 manifolds in terms of elliptic integrals. The resulting expressions are globally valid throughout the moduli space and allow the explicit inversion of the mirror map and the exact computation of distances, useful for checks of the Swampland Distance Conjecture. We comment on the generalization to multi-parameter models and provide a two-parameter example.Comment: 36 pages, 1 figure; v2: references added, typos corrected, matches published versio

Beyond Large Complex Structure: Quantized Periods and Boundary Data for One-Modulus Singularities

We study periods in an integral basis near all possible singularities in one-dimensional complex structure moduli spaces of Calabi-Yau threefolds. Near large complex structure points these asymptotic periods are well understood in terms of the topological data of the mirror Calabi-Yau manifold. The aim of this work is to characterize the period data near other boundaries in moduli space such as conifold and K-points. Using results from Hodge theory, we provide the general form of these periods in a quantized three-cycle basis. Based on these periods we compute the prepotential and related physical couplings of the underlying supergravity theory. Moreover, we elucidate the meaning of the model-dependent coefficients that appear in these expressions: these can be identified with certain topological and arithmetic numbers associated to the singular geometry at the moduli space boundary. We illustrate our findings by studying a wide set of examples.Comment: 100 pages, 6 table