Three Generations and a Trio of Trialities
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915248290660352 |
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| author | Furey, N. Hughes, M. J. |
| author_facet | Furey, N. Hughes, M. J. |
| contents | We identify the Standard Model's $\mathfrak{su}(3)\oplus \mathfrak{su}(2)\oplus \mathfrak{u}(1)$ internal symmetries within the triality symmetries $\mathfrak{tri}(\mathbb{C}) \oplus \mathfrak{tri}(\mathbb{H}) \oplus \mathfrak{tri}(\mathbb{O})$. From here, the corresponding Standard Model group action is applied to the triality triple $\left( Ψ_+, Ψ_-,V\right)$ for $Ψ_+, Ψ_-, V \in \mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$. Together, $Ψ_+$ and $Ψ_-$ provide the correct irreducible representations for two generations. Owing to a certain Cartan Factorization, which we define, $V$ provides the irreducible representations for a third generation. Said more explicitly in another way, division algebraic multiplication merges a third generation of spinor representations into a set of scalar bosons. This set of scalar bosons includes the familiar Standard Model Higgs representation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_17948 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Three Generations and a Trio of Trialities Furey, N. Hughes, M. J. High Energy Physics - Phenomenology High Energy Physics - Theory We identify the Standard Model's $\mathfrak{su}(3)\oplus \mathfrak{su}(2)\oplus \mathfrak{u}(1)$ internal symmetries within the triality symmetries $\mathfrak{tri}(\mathbb{C}) \oplus \mathfrak{tri}(\mathbb{H}) \oplus \mathfrak{tri}(\mathbb{O})$. From here, the corresponding Standard Model group action is applied to the triality triple $\left( Ψ_+, Ψ_-,V\right)$ for $Ψ_+, Ψ_-, V \in \mathbb{C}\otimes\mathbb{H}\otimes\mathbb{O}$. Together, $Ψ_+$ and $Ψ_-$ provide the correct irreducible representations for two generations. Owing to a certain Cartan Factorization, which we define, $V$ provides the irreducible representations for a third generation. Said more explicitly in another way, division algebraic multiplication merges a third generation of spinor representations into a set of scalar bosons. This set of scalar bosons includes the familiar Standard Model Higgs representation. |
| title | Three Generations and a Trio of Trialities |
| topic | High Energy Physics - Phenomenology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2409.17948 |