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

## TL;DR

This paper introduces a new mathematical framework for Hitchin's integrable system using gauged Lagrangian 1-forms in gauge theories.

## Contribution

The paper introduces gauged Lagrangian 1-forms and applies them to Hitchin's system, yielding a variational formulation with defects.

## Key findings

- A multiform 3d mixed BF action with defects is derived for Hitchin's integrable system.
- Explicit Lagrangian 1-forms are constructed for the rational and elliptic Gaudin hierarchies.
- The elliptic spin Calogero–Moser hierarchy is shown as a special subcase.

## Abstract

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 \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {P}$$\end{document}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.