标题： 三维混合BF拉格朗日1-形式：Hitchin可积系统的变分公式 显示英文标题

标题： The 3d mixed BF Lagrangian 1-form: a variational formulation of Hitchin's integrable system

摘要： 我们引入了规范拉格朗日$1$-形式的概念，将拉格朗日$1$-形式的概念扩展到规范理论的框架中。 这个一般形式主义被应用于在光滑主丛 $G$-丛 $\mathcal{P}$在紧黎曼曲面 $C$上全纯结构空间的余切丛上的自然几何拉格朗日量 $1$-形式，无论该黎曼曲面具有任意亏格 $g$，是否有标记点，以规范光滑丛自同构的对称群 $\mathcal{P}$。 所得构造产生了一个多形式版本的$3$维混合 BF 作用量，包含所谓的类型 A 和 B 缺陷，为在$C$上 Hitchen 的完全可积系统提供了变分公式。 通过转换到全纯局部平凡化并部分满足经典条件，我们得到了一个统一的作用量，用于描述 Hitchen 系统的 Lax 方程族，该作用量基于亚纯 Lax 矩阵。 对于带有标记点的亏格$0$和$1$的情况进行了更详细的处理，分别给出了有理 Gaudin 层次结构和椭圆 Gaudin 层次结构的显式拉格朗日$1$-形式，其中椭圆自旋 Calogero-Moser 层次结构作为特例出现。 显示英文摘要

摘要： We introduce the concept of gauged Lagrangian $1$-forms, extending the notion of Lagrangian $1$-forms to the setting of gauge theories. This general formalism is applied to a natural geometric Lagrangian $1$-form on the cotangent bundle of the space of holomorphic structures on a smooth principal $G$-bundle $\mathcal{P}$ over a compact Riemann surface $C$ of arbitrary genus $g$, with or without marked points, in order to gauge the symmetry group of smooth bundle automorphisms of $\mathcal{P}$. The 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.