A recent series of works characterized the Standard Model (SM) gauge group G_SM as the subgroup of SO(9) that, in the octonionic model of the latter, preserves the split $\mathbb{O}=\mathbb{C}\oplus {\mathbb{C}}^3$ of the space of octonions into a copy of the complex plane plus the rest. This description, however, proceeded via the exceptional Jordan algebras $J_3(\mathbb{O}),J_2(\mathbb{O})$ and, in this sense, remained indirect. One of the goals of this paper is to provide as explicit a description as possible and also to clarify the underlying geometry. The other goal is to emphasize the role played by different complex structures in the spaces $\mathbb{O}$ and ${\mathbb{O}}^2$. We provide a new characterization of G_SM: The group G_SM is the subgroup of Spin(9) that commutes with a certain complex structure J_R in the space ${\mathbb{O}}^2$ of Spin(9) spinors. The complex structure J_R is parameterized by a choice of a unit imaginary octonion. This characterization of G_SM is essentially octonionic in the sense that J_R is restrictive because octonions are non-associative. The quaternionic analog of J_R is the complex structure in the space ${\mathbb{H}}^2$ of Spin(5) spinors that commutes with all Spin(5) transformations.

中文翻译：

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标准模型量规组的SO（9）表征

最近的一系列工作将标准模型（SM）量规组G _SM表征为SO（9）的子组，在后者的渗透压模型中保留了分裂。$\mathbb{Ø}=\mathbb{C}\oplus {\mathbb{C}}^3$的八度空间复制到复杂平面的副本中，再加上其余部分。但是，这种描述是通过特殊的约旦代数进行的${Ĵ}_3（\mathbb{Ø}），{Ĵ}_{\mathrm{2个}}（\mathbb{Ø}）$从这个意义上讲，仍然是间接的。本文的目标之一是提供尽可能明确的描述，并阐明基本的几何形状。另一个目标是强调空间中不同复杂结构的作用$\mathbb{Ø}$ 和 ${\mathbb{Ø}}^{\mathrm{2个}}$。我们提供了一个新的表征ģ _SM：组G ^ _SM是自旋的亚组（9）具有一定的复杂结构，其通勤Ĵ _ř在该空间中${\mathbb{Ø}}^{\mathrm{2个}}$Spin（9）个spinors的数量。通过选择单位虚张度来参数化复杂结构J _R。从J _R是限制性的意义上来说，G _SM的这种表征本质上是张调的，因为是不缔合的。J _R的四元离子类似物是空间中的复杂结构${\mathbb{H}}^{\mathrm{2个}}$ 与所有Spin（5）转换换向的Spin（5）旋转子的数量。