A Mathematical Construction of an E6 Grand Unified Theory

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.