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.

arXiv commentMaster's thesis, 81 pages

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