AI 中文总结
研究规范理论几何优先与对称优先表述的不等价性，通过分析生成结构限制、主丛与物质丛关系及自然函子性质，揭示二者在理论数量、结构确定及函子特性上的差异。
AI 中文摘要
本文认为规范理论的几何优先和对称优先表述不等价，在三个方面存在差异。一是几何优先表述中，当生成结构限于有限张量数据时，允许的理论更少，如具有加法结构群\(\mathbb{R}\)的带电理论无此类表示。二是即使对称优先理论有几何优先表述，其主丛和物质丛也不能确定生成规范群的结构化向量丛。三是从几何优先生成对象到带物质的主丛的自然函子是忠实的，但既不是本质满射也不是满的。本质满射在标准模型规范群的对角商等情况不成立，满性在实定向纤维\(\mathbb{R}^{2m}\)时不成立。
英文摘要
This paper argues that the geometry-first and symmetry-first formulations of gauge theory are not equivalent. They differ in three respects. First, the geometry-first formulation---in which gauge groups arise as automorphism groups of structured fundamental vector bundles---admits fewer theories when its generating structures are restricted to finite tensorial data. Charged theories with additive structure group $\mathbb{R}$ have no such presentation. Second, even when a symmetry-first theory has a geometry-first presentation, its principal bundle and matter bundles do not determine which structured vector bundles generated the gauge group. Third, the natural functor from geometry-first generating objects to principal bundles with matter is faithful but neither essentially surjective nor full. Essential surjectivity fails for the diagonal quotient of the Standard Model gauge group on what I call `the classical menu', and for the additive-$\mathbb{R}$ examples on every finite tensorial menu. Fullness fails for a real oriented fibre $\mathbb{R}^{2m}$: the principal-bundle category admits the outer automorphism of $SO(2m)$ induced by an improper orthogonal map, but no structure-preserving fibre map induces it.
Comments27 pages
