A Mathematical Construction of an E6 Grand Unified Theory

TL;DR

This paper provides a rigorous mathematical construction of an E6 grand unified theory, highlighting its subgroup structures, anomaly-free representations, and symmetry breaking mechanisms, with explicit checks and alternative embedding approaches.

Contribution

It offers a detailed mathematical framework for E6 GUT, including explicit verification of kernel actions and a novel approach to symmetry breaking via Lie algebra embeddings.

Findings

Verified trivial action of Z4 kernel on fermions

Explicitly embedded su(5) into so(10) for symmetry breaking

Discussed phenomenological implications of E6 GUT

Abstract

Of the five exceptional groups, $\mathrm{E}_6$ is considered the most attractive for unification due to the following reasons: (i) it contains both $\mathrm{Spin} (10) \times \mathrm{U}(1)$ and $\mathrm{SU} (3) \times \mathrm{SU}(3) \times \mathrm{SU}(3)$ as maximal subgroups, each of which admit embeddings of the Standard Model; (ii) uniquely among the exceptional groups, it admits complex representations; in particular, its 27 dimensional fundamental representation accommodates one generation of left-handed fermions under the usual charge assignments; (iii) all of its representations are anomaly-free. In this master's thesis, written in the spirit of Baez and Huerta's "The Algebra of Grand Unified Theories", we rigorously show how an $\mathrm{E}_6$ grand unified theory is mathematically constructed. Our modest contribution to the literature includes an explicit check that that $\mathbb{Z}_4$ kernel of the homomorphism $\mathrm{Spin} (10) \times \mathrm{U}(1) \to \mathrm{E}_6$ acts trivially on every fermion; we also formulate symmetry breaking, in particular the symmetry breaking of the exotic $\mathrm{E}_6$ fermions under $\mathrm{Spin} (10) \to \mathrm{SU}(5)$, using a different approach than the usual Dynkin diagrams: we explicitly embedded $\mathfrak{su}(5) \hookrightarrow \mathfrak{so}(10) \cong \mathfrak{spin} (10)$ and solve the related eigenvalue problem. Phenomenological aspects of grand unified theories are also discussed.