Date of Award

Fall 2025

Project Type

Dissertation

Program or Major

Physics

Degree Name

Doctor of Philosophy

First Advisor

Per Berglund

Second Advisor

James Halverson

Third Advisor

Vishnu Jejjala

Abstract

Low-energy couplings of certain superstring theories are related to the masses of the particles. One of the most promising superstring theories is $\mathcal{N}=1$ supersymmetric $E_8\times E_8$ heterotic string theory. In a certain embedding of the standard model into $E_8\times E_8$ gauge group of the heterotic string, the particle spectrum consists of $(\mathbf{27})^3$, $(\overline{\mathbf{27}})^3$, $(\mathbf{1})^3$ and $\mathbf{1}\cdot \mathbf{27}\cdot\overline{\mathbf{27}}$ couplings. While the $(\overline{\mathbf{27}})^3$ couplings are quasi-topological, the $(\mathbf{27})^3$ explicitly depend on the Ricci-flat metric on Calabi-Yau.

In this dissertation, we explore recently developed techniques for calculating the normalized Yukawa couplings on various Calabi-Yau manifolds. In Chapter 2, we describe a novel architecture, called ``Spectral Neural Networks", which offers state-of-the-art approximations to the Ricci-flat metric on Calabi-Yau manifolds. In particular, we investigate the behavior of characteristic forms associated to the metrics on the Cefal\'u family of quartic twofolds and the Dwork family of quintic threefolds. We observe that the numerical stability of various techniques is significantly influenced by the choice of neural network architecture. We further demonstrate that the spectral neural networks offer better accuracy, whilst also remaining numericaly stable when manifolds are singular or near-singular. In Chapter 3, we focus on calculating the normalized Yukawa couplings for $(2,1)$-forms on various Calabi-Yau manifolds, such as: the Fermat quintic, the intersection on two cubics in $\mathbb{P}^5$, and the Tian-Yau manifold. In the cases where $h^{2,1}=1$, this is compared to a complementary calculation based on performing period integrals. In the case of the Tian-Yau manifold, which defines a model with three generations and has $h^{2,1}>1$, we provide a first-of-its-kind calculation of the normalized Yukawa couplings, thereby calculating the normalization constants. Finally, we complement this method with a more general approach which involves calculation of harmonic representatives. In particular, we present an equivariant neural network architecture which approximates the harmonic representatives of cohomology classes in $H^1(T_X)$. We observe that the results produced using machine-learning techniques are also in excellent agreement with the expected values. The precise agreement between different approaches opens the door to precision string phenomenology. As part of this work, we have developed a Python library called cymyc, which streamlines calculation of the Calabi-Yau metric and the Yukawa couplings on arbitrary Calabi-Yau manifolds that are realized as complete intersections and provides a framework for studying the differential geometric properties, such as curvature. As a next step, several possible generalizations and extensions of this work are proposed. Finally, in Chapter 4, we describe a Reinforcement-Learning based algorithm for scanning the landscape for Calabi-Yau manifolds which admit properties necessary for reproducing the Standard Model.

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