January 26, 2024
One of the challenges of heterotic compactification on a Calabi-Yau threefold is to determine the physical $(\mathbf{27})^3$ Yukawa couplings of the resulting four-dimensional $\mathcal{N}=1$ theory. In general, the calculation necessitates knowledge of the Ricci-flat metric. However, in the standard embedding, which references the tangent bundle, we can compute normalized Yukawa couplings from the Weil-Petersson metric on the moduli space of complex structure deformations of the Calabi-Yau manifold. In various examples (the Fermat quintic, the intersection of two cubics in $\mathbb{P}^5$, and the Tian-Yau manifold), we calculate the normalized Yukawa couplings for $(2,1)$-forms using the Weil-Petersson metric obtained from the Kodaira-Spencer map. In cases where $h^{1,1}=1$, this is compared to a complementary calculation based on performing period integrals. A third expression for the normalized Yukawa couplings is obtained from a machine learned approximate Ricci-flat metric making use of explicit harmonic representatives. The excellent agreement between the different approaches opens the door to precision string phenomenology.
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April 3, 2022
This thesis is concerned with heterotic E8 x E8 string models that can produce quasirealistic N = 1 supersymmetric extensions of the Standard Model in the low-energy limit. We start rather generally by deriving the four-dimensional spectrum and Lagrangian terms from the ten-dimensional theory, through a process of compactification over six-dimensional Calabi-Yau manifolds, upon which holomorphic poly-stable vector bundles are defined. We then specialise to a class of heteroti...
February 2, 2024
We present a numerical computation, based on neural network techniques, of the physical Yukawa couplings in a heterotic string theory compactification on a smooth Calabi-Yau threefold with non-standard embedding. The model belongs to a large class of heterotic line bundle models that have previously been identified and whose low-energy spectrum precisely matches that of the MSSM plus fields uncharged under the Standard Model group. The relevant quantities for the calculation,...
July 18, 2024
Calabi--Yau compactifications of the $E_8\times E_8$ heterotic string provide a promising route to recovering the four-dimensional particle physics described by the Standard Model. While the topology of the Calabi--Yau space determines the overall matter content in the low-energy effective field theory, further details of the compactification geometry are needed to calculate the normalized physical couplings and masses of elementary particles. In this work, we present numeric...
December 16, 2015
We develop techniques, based on differential geometry, to compute holomorphic Yukawa couplings for heterotic line bundle models on Calabi-Yau manifolds defined as complete intersections in projective spaces. It is shown explicitly how these techniques relate to algebraic methods for computing holomorphic Yukawa couplings. We apply our methods to various examples and evaluate the holomorphic Yukawa couplings explicitly as functions of the complex structure moduli. It is shown ...
February 21, 2024
We study the modular symmetry in heterotic string theory on Calabi-Yau threefolds. In particular, we examine whether moduli-dependent holomorphic Yukawa couplings are described by modular forms in the context of heterotic string theory with standard embedding. We find that $SL(2,\mathbb{Z})$ modular symmetry emerges in asymptotic regions of the Calabi-Yau moduli space. The instanton-corrected holomorphic Yukawa couplings are then given by modular forms under $SL(2,\mathbb{Z})...
July 12, 2016
We develop methods to compute holomorphic Yukawa couplings for heterotic compactifications on complete intersection Calabi-Yau manifolds, generalising results of an earlier paper for Calabi-Yau hypersurfaces. Our methods are based on constructing the required bundle-valued forms explicitly and evaluating the relevant integrals over the projective ambient space. We also show how our approach relates to an earlier, algebraic one to calculate the holomorphic Yukawa couplings. A ...
April 15, 2009
We present a practical, algebraic method for efficiently calculating the Yukawa couplings of a large class of heterotic compactifications on Calabi-Yau three-folds with non-standard embeddings. Our methodology covers all of, though is not restricted to, the recently classified positive monads over favourable complete intersection Calabi-Yau three-folds. Since the algorithm is based on manipulating polynomials it can be easily implemented on a computer. This makes the automate...
January 29, 2018
We propose an analytic method to calculate the matter field K\"ahler metric in heterotic compactifications on smooth Calabi-Yau three-folds with Abelian internal gauge fields. The matter field K\"ahler metric determines the normalisations of the ${\cal N}=1$ chiral superfields, which enter the computation of the physical Yukawa couplings. We first derive the general formula for this K\"ahler metric by a dimensional reduction of the relevant supergravity theory and find that i...
August 29, 2018
This thesis contributes with a number of topics to the subject of string compactifications. In the first half of the work, I discuss the Hodge plot of Calabi-Yau threefolds realised as hypersurfaces in toric varieties. The intricate structure of this plot is explained by the existence of certain webs of elliptic-K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along K3 slices. Any two half-polytopes over a given slice can be combined...
March 10, 2015
Yau proved an existence theorem for Ricci-flat K\"ahler metrics in the 1970's, but we still have no closed form expressions for them. Nevertheless there are several ways to get approximate expressions, both numerical and analytical. We survey some of this work and explain how it can be used to obtain physical predictions from superstring theory.