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Heterotic and M-theory Compactifications for String Phenomenology

Lara B. Anderson

TL;DR

This thesis advances string phenomenology along two complementary tracks. First, it develops a concrete M-theory framework for compactifications on G2 spaces with co-dimension four ADE singularities, coupling 11D supergravity to seven-dimensional SU(N) gauge theories on orbifold fixed planes, and derives the four-dimensional effective theory including Kähler potentials, gauge kinetic terms, and superpotentials; it also analyzes blow-up via Higgs/D-flat directions and explores flux/Wilson-line effects. Second, it presents an algorithmic heterotic program based on monad bundles on cyclic Calabi–Yau and complete intersection Calabi–Yau manifolds, proving finiteness of positive monads under anomaly constraints, computing full particle spectra, and establishing practical cohomology techniques (Koszul/Leray/MBBW) to enable large-scale scans (~thousands of bundles on thousands of CYs). The work demonstrates precise matches between singular and blown-up M-theory descriptions, derives Gukov-type superpotentials for matter sectors, and shows how flux and Wilson lines shape 4D physics; in the heterotic arena, it delivers a finite, stable set of monad vacua with three-family prospects and no anti-generations, catalyzing data-driven searches for realistic vacua across CY landscapes. Collectively, the results establish a computational pipeline linking high-dimensional M-theory geometry to phenomenologically viable 4D theories and provide a scalable framework for exploring heterotic vacua with realistic gauge groups and matter content. The techniques and findings offer a path toward systematic, large-scale construction and evaluation of string-derived models with potential experimental relevance.

Abstract

In this thesis, we explore two approaches to string phenomenology. In the first half of the work, we investigate M-theory compactifications on spaces with co-dimension four, orbifold singularities. We construct M-theory on C^2/Z_N by coupling 11-dimensional supergravity to a seven-dimensional Yang-Mills theory located on the orbifold fixed-plane. The resulting action is supersymmetric to leading non-trivial order in the 11-dim Newton constant. We thereby reduce M-theory on a G2 orbifold with C^2/Z_N singularities, explicitly incorporating the additional gauge fields at the singularities. We derive the Kahler potential, gauge-kinetic function and superpotential for the resulting N=1 four-dimensional theory. Blowing-up of the orbifold is described by a Higgs effect and the results are consistent with the corresponding ones obtained for smooth G2 spaces. Further, we consider flux and Wilson lines on singular loci of the G2 space, and discuss the relation to N=4 SYM theory. In the second half, we develop an algorithmic framework for E8 x E8 heterotic compactifications with monad bundles. We begin by considering cyclic Calabi-Yau manifolds where we classify positive monad bundles, prove stability, and compute the complete particle spectrum for all bundles. Next, we generalize the construction to bundles on complete intersection Calabi-Yau manifolds. We show that the class of positive monad bundles, subject to the heterotic anomaly condition, is finite (~7000 models). We compute the particle spectrum for these models and develop new techniques for computing the cohomology of line bundles. There are no anti-generations of particles and the spectrum is manifestly moduli-dependent. We further study the slope-stability of positive monad bundles and develop a new method for proving stability of SU(n) vector bundles.

Heterotic and M-theory Compactifications for String Phenomenology

TL;DR

This thesis advances string phenomenology along two complementary tracks. First, it develops a concrete M-theory framework for compactifications on G2 spaces with co-dimension four ADE singularities, coupling 11D supergravity to seven-dimensional SU(N) gauge theories on orbifold fixed planes, and derives the four-dimensional effective theory including Kähler potentials, gauge kinetic terms, and superpotentials; it also analyzes blow-up via Higgs/D-flat directions and explores flux/Wilson-line effects. Second, it presents an algorithmic heterotic program based on monad bundles on cyclic Calabi–Yau and complete intersection Calabi–Yau manifolds, proving finiteness of positive monads under anomaly constraints, computing full particle spectra, and establishing practical cohomology techniques (Koszul/Leray/MBBW) to enable large-scale scans (~thousands of bundles on thousands of CYs). The work demonstrates precise matches between singular and blown-up M-theory descriptions, derives Gukov-type superpotentials for matter sectors, and shows how flux and Wilson lines shape 4D physics; in the heterotic arena, it delivers a finite, stable set of monad vacua with three-family prospects and no anti-generations, catalyzing data-driven searches for realistic vacua across CY landscapes. Collectively, the results establish a computational pipeline linking high-dimensional M-theory geometry to phenomenologically viable 4D theories and provide a scalable framework for exploring heterotic vacua with realistic gauge groups and matter content. The techniques and findings offer a path toward systematic, large-scale construction and evaluation of string-derived models with potential experimental relevance.

Abstract

In this thesis, we explore two approaches to string phenomenology. In the first half of the work, we investigate M-theory compactifications on spaces with co-dimension four, orbifold singularities. We construct M-theory on C^2/Z_N by coupling 11-dimensional supergravity to a seven-dimensional Yang-Mills theory located on the orbifold fixed-plane. The resulting action is supersymmetric to leading non-trivial order in the 11-dim Newton constant. We thereby reduce M-theory on a G2 orbifold with C^2/Z_N singularities, explicitly incorporating the additional gauge fields at the singularities. We derive the Kahler potential, gauge-kinetic function and superpotential for the resulting N=1 four-dimensional theory. Blowing-up of the orbifold is described by a Higgs effect and the results are consistent with the corresponding ones obtained for smooth G2 spaces. Further, we consider flux and Wilson lines on singular loci of the G2 space, and discuss the relation to N=4 SYM theory. In the second half, we develop an algorithmic framework for E8 x E8 heterotic compactifications with monad bundles. We begin by considering cyclic Calabi-Yau manifolds where we classify positive monad bundles, prove stability, and compute the complete particle spectrum for all bundles. Next, we generalize the construction to bundles on complete intersection Calabi-Yau manifolds. We show that the class of positive monad bundles, subject to the heterotic anomaly condition, is finite (~7000 models). We compute the particle spectrum for these models and develop new techniques for computing the cohomology of line bundles. There are no anti-generations of particles and the spectrum is manifestly moduli-dependent. We further study the slope-stability of positive monad bundles and develop a new method for proving stability of SU(n) vector bundles.

Paper Structure

This paper contains 150 sections, 584 equations, 3 figures, 18 tables.

Table of Contents

  1. Introduction
  2. The Problem of String Phenomenology
  3. An Overview
  4. M-theory on manifolds of $G_2$ holonomy
  5. Introduction
  6. Manifolds of $G_2$ holonomy
  7. An example $G_2$ manifold
  8. M-theory compactified on a smooth $G_2$-manifold
  9. M-theory on singular $G_2$-spaces
  10. Arguments from String dualities
  11. Co-dimension $4$ Singularities
  12. Co-dimension $7$ singularities
  13. Review of Hořava-Witten Theory
  14. M-theory on the orbifold $\mathbb{C}^{2}/\mathbb{Z}_{N}$
  15. Introduction
  16. ...and 135 more sections

Figures (3)

  • Figure 1: (a) Histogram for the index, ${\rm ind}(V)$, of the 7118 positive monads found over 36 favourable CICYs: the horizontal axis is ${\rm ind}(V)$ and the vertical, the number of bundles; (b) the same data set, but only taking those monads which have ${\rm ind}(V) = 3k$ for some positive integer $k$ and such that $k$ divides the Euler number of the corresponding CICY.
  • Figure 2: (a) Histogram for the index, ${\rm ind}(V)$, of the positive monads, 5680 of rank 3 (in red), 1334 of rank 4 (in blue), and 104 of rank 5 (in gray), found over 36 favourable CICYs: the horizontal axis is ${\rm ind}(V)$ and the vertical, the number of bundles; (b) the same data set, but only taking those monads which have ${\rm ind}(V) = 3k$ for some positive integer $k$ and such that $k$ divides the Euler number of the corresponding CICY.
  • Figure 3: The Kähler cone and two potentially de-stabilizing line bundles $l_1$ and $l_2$.