3 维混合 BF 范数 1 形式：Hitchin 可积系统的变分表述

我们提出了准规范 $1$ 形式的概念，扩展了在规范理论中 $1$ 形式的范数概念。该一般形式被应用于在任意基数 $g$（可带或不带标记点）的紧黏 Riemann 表面 $C$ 上的光滑主 $G$-束 $\mathcal{P}$ 上自然的几何 $1$ 形式，以规范化光滑束自同构的对称性群。 resulting construction yields a multiform version of the $3$d mixed BF action with so-called type A and B defects, providing a variational formulation of Hitchin's completely integrable system over $C$. By passing to holomorphic local trivialisations and going partially on-shell, we obtain a unifying action for a hierarchy of Lax equations describing the Hitchin system in terms of meromorphic Lax matrices. The cases of genus $0$ and $1$ with marked points are treated in greater detail, producing explicit Lagrangian $1$-forms for the rational Gaudin hierarchy and the elliptic Gaudin hierarchy, respectively, with the elliptic spin Calogero-Moser hierarchy arising as a special subcase.