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 <i>G</i>-bundle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>P</mml:mi></mml:math> over a compact Riemann surface <i>C</i> of arbitrary genus <i>g</i>, with or without marked points, in order to gauge the symmetry group of smooth bundle automorphisms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>P</mml:mi></mml:math> . 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 <i>C</i>. By passing to holomorphic local trivialisations and going partially on-shell, we obtain a <i>unifying action</i> 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.