[Paper Review] Heterotic and M-theory Compactifications for String Phenomenology

Lara B. Anderson|ArXiv.org|Aug 27, 2008

Black Holes and Theoretical PhysicsPhysics and Astronomy129 references17 citations

TL;DR

This thesis develops two complementary approaches to string phenomenology: M-theory compactifications on $G_2$ orbifolds with $b{C}^2/\mathbb{Z}_N$ singularities and heterotic compactifications using monad bundles on Calabi-Yau manifolds. It constructs a supersymmetric effective theory by coupling 11D supergravity to a 7D Yang-Mills theory on the fixed plane, derives the Kähler potential, gauge-kinetic function, and superpotential, and establishes consistency with smooth $G_2$ spaces via Higgsing. For heterotic models, it classifies positive monad bundles on complete intersection Calabi-Yau threefolds, proves stability, computes the particle spectrum (with no anti-generations), and develops new cohomology techniques, yielding ~7,000 finite models with moduli-dependent spectra.

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.

Motivation & Objective

• To construct a consistent, supersymmetric effective four-dimensional $N=1$ theory from M-theory compactification on $G_2$ orbifolds with $b{C}^2/\mathbb{Z}_N$ singularities.
• To incorporate gauge fields at singular loci by coupling 11D supergravity to a 7D Yang-Mills theory on the fixed plane.
• To derive the Kähler potential, gauge-kinetic function, and superpotential for the resulting four-dimensional theory.
• To generalize the construction to heterotic $E_8\times E_8$ compactifications on Calabi-Yau manifolds using monad bundles.
• To classify stable positive monad bundles on complete intersection Calabi-Yau threefolds, compute their particle spectra, and develop efficient cohomology techniques.

Proposed method

• Construct M-theory on $b{C}^2/\mathbb{Z}_N$ by coupling 11D supergravity to a 7D Yang-Mills theory on the orbifold fixed plane.
• Derive the effective 4D $N=1$ supergravity action, including the Kähler potential, gauge-kinetic function, and superpotential, to leading non-trivial order in the Newton constant.
• Implement blowing-up of the orbifold via a Higgs mechanism along D-flat directions, showing consistency with smooth $G_2$ compactifications.
• Develop an algorithmic framework for $E_8\times E_8$ heterotic compactifications using monad bundles on cyclic and complete intersection Calabi-Yau threefolds.
• Prove stability of positive monad bundles using a new method for $SU(n)$ bundles on CICYs and compute cohomology of line bundles via new techniques.
• Apply the heterotic anomaly condition to constrain the class of models, yielding a finite set of ~7,000 stable, phenomenologically viable models.

Experimental results

Research questions

• RQ1 How can M-theory compactification on $G_2$ orbifolds with $b{C}^2/\mathbb{Z}_N$ singularities be consistently coupled to gauge fields at the fixed plane to yield a supersymmetric 4D $N=1$ effective theory?
• RQ2 What is the explicit form of the Kähler potential, gauge-kinetic function, and superpotential in the resulting 4D effective theory?
• RQ3 How does the Higgsing mechanism along D-flat directions reproduce the geometry of smooth $G_2$ spaces in the context of M-theory compactifications?
• RQ4 What is the complete particle spectrum of $E_8\times E_8$ heterotic compactifications on complete intersection Calabi-Yau threefolds using positive monad bundles?
• RQ5 Can a finite and computable class of phenomenologically viable models be constructed by imposing the heterotic anomaly condition on positive monad bundles?

Key findings

• The effective 4D $N=1$ theory derived from M-theory compactification on $G_2$ orbifolds with $b{C}^2/\mathbb{Z}_N$ singularities is supersymmetric to leading non-trivial order in the 11D Newton constant.
• The Kähler potential, gauge-kinetic function, and superpotential are explicitly computed, providing a complete description of the low-energy effective action.
• Blowing-up the orbifold singularity via a Higgs mechanism along D-flat directions reproduces the physics of smooth $G_2$ spaces, confirming consistency with known results.
• The class of positive monad bundles satisfying the heterotic anomaly condition on complete intersection Calabi-Yau threefolds is finite, with approximately 7,000 such models.
• The particle spectrum of these models contains no anti-generations and is manifestly moduli-dependent, consistent with $N=1$ supersymmetry.
• A new method for proving slope-stability of $SU(n)$ vector bundles on CICYs is developed, enabling the classification of stable bundles and efficient cohomology computations.