标题： 粒子理论的代数路线图，第三部分：交叉点 显示英文标题

标题： An Algebraic Roadmap of Particle Theories, Part III: Intersections

摘要： 本文中，我们绕过了在文献[1]中确立的详细对称性破缺路径。相反，通过一个单一的代数约束，直接建立了从Spin(10)模型到标准模型的途径。然而，这个单一约束可以重新配置为要求三个$\mathfrak{so}(10)$操作在固定多向量费米子空间上一致。这种$\mathfrak{so}(10) \mapsto \mathfrak{su}(3)_{C} \oplus \mathfrak{su}(2)_{L} \oplus{u}(1)_{Y}$破缺（来自三方交集）在某些方面反映了八元数三重性上下文中$\mathfrak{so}(8) \mapsto \mathfrak{g}_2$破缺（同样来自三方交集）。通过将这一结果扩展到包含四元数和复数，我们发现五方交集会破缺$\mathfrak{so}(10) \mapsto \mathfrak{su}(3)_{C} \oplus \mathfrak{u}(1)_{Q}$。这些是标准模型在希格斯机制后的未破缺规范对称性。 显示英文摘要

摘要： In this article, we bypass the detailed symmetry breaking pathways established in [1]. Instead, a direct route from the Spin(10) model to the Standard Model is enabled via a single algebraic constraint. This single constraint, however, may be reconfigured as a requirement that three $\mathfrak{so}(10)$ actions coincide on a fixed space of multi-vector fermions. This $\mathfrak{so}(10) \mapsto \mathfrak{su}(3)_{C} \oplus \mathfrak{su}(2)_{L} \oplus{u}(1)_{Y}$ breaking (from a three-way intersection) mirrors, in certain ways, the $\mathfrak{so}(8) \mapsto \mathfrak{g}_2$ breaking (from a three-way intersection) in the context of octonionic triality. By extending this result to include quaternions and complex numbers, we find that a five-way intersection breaks $\mathfrak{so}(10) \mapsto \mathfrak{su}(3)_{C} \oplus \mathfrak{u}(1)_{Q}$. These are the Standard Model's unbroken gauge symmetries, post-Higgs-mechanism.