The 3d Mixed BF Lagrangian 1-Form: A Variational Formulation of Hitchin's Integrable System

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 $P$ 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 $P$ . The resulting construction yields a multiform version of the 3d 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.