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

Vincent Caudrelier, Derek Harland, Anup Anand Singh, Benoit Vicedo

TL;DR

This work develops a variational formulation for Hitchin’s completely integrable system by merging Lagrangian multiforms with 3d mixed holomorphic–topological BF theory. By gauging the natural Lagrangian 1-form on the cotangent bundle of holomorphic structures and performing a symplectic reduction, the authors obtain a multiform BF action that encodes the Hitchin hierarchy, including marked-point extensions via type A and B defects. They derive an explicit unifying Lagrangian 1-form for the Hitchin system, show its Euler–Lagrange equations reproduce the Lax form, and provide concrete genus-zero and genus-one examples: the rational Gaudin hierarchy and the elliptic Gaudin/elliptic spin Calogero–Moser hierarchies. The results illuminate a direct gauge-theoretic origin for Hitchin integrable systems and open avenues toward higher-dimensional generalisations and path-integral quantisation of integrable hierarchies.

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 $\mathcal{P}$ 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.

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

TL;DR

This work develops a variational formulation for Hitchin’s completely integrable system by merging Lagrangian multiforms with 3d mixed holomorphic–topological BF theory. By gauging the natural Lagrangian 1-form on the cotangent bundle of holomorphic structures and performing a symplectic reduction, the authors obtain a multiform BF action that encodes the Hitchin hierarchy, including marked-point extensions via type A and B defects. They derive an explicit unifying Lagrangian 1-form for the Hitchin system, show its Euler–Lagrange equations reproduce the Lax form, and provide concrete genus-zero and genus-one examples: the rational Gaudin hierarchy and the elliptic Gaudin/elliptic spin Calogero–Moser hierarchies. The results illuminate a direct gauge-theoretic origin for Hitchin integrable systems and open avenues toward higher-dimensional generalisations and path-integral quantisation of integrable hierarchies.

Abstract

We introduce the concept of gauged Lagrangian -forms, extending the notion of Lagrangian -forms to the setting of gauge theories. This general formalism is applied to a natural geometric Lagrangian -form on the cotangent bundle of the space of holomorphic structures on a smooth principal -bundle over a compact Riemann surface of arbitrary genus , with or without marked points, in order to gauge the symmetry group of smooth bundle automorphisms of . The resulting construction yields a multiform version of the d mixed BF action with so-called type A and B defects, providing a variational formulation of Hitchin's completely integrable system over . 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 and with marked points are treated in greater detail, producing explicit Lagrangian -forms for the rational Gaudin hierarchy and the elliptic Gaudin hierarchy, respectively, with the elliptic spin Calogero-Moser hierarchy arising as a special subcase.

Paper Structure

This paper contains 32 sections, 11 theorems, 256 equations, 1 figure.

Table of Contents

  1. Introduction and summary of main results
  2. Acknowledgements.
  3. Gauged univariational principle and symmetry reduction
  4. Lagrangian 1-forms, univariational principle and symmetry
  5. Univariational principle for Lagrangian $1$-forms
  6. Group actions and symmetry
  7. Gauged Lagrangian 1-form and gauged univariational principle
  8. Gauging a symmetry in a Lagrangian $1$-form
  9. Gauged univariational principle
  10. 3d mixed BF theory and Hitchin's system
  11. Geometric setup for the Hitchin system
  12. Holomorphic structures on a principal $G$-bundle
  13. Cotangent bundle $T^\ast \mathcal{M}$ and action of $\mathcal{G}$
  14. Lagrangian 1-form on TM
  15. Lagrangian 1-form for the Hitchin system on mu G
  16. ...and 17 more sections

Key Result

Proposition 2.1

The action ungauged action is invariant under the infinitesimal action of $G$ on $T^\ast M \times \mathbb{R}^n$ generated by action coords if and only if each $H_i$ is invariant under this infinitesimal group action, i.e. The Noether charges associated with this global $G$ symmetry are then given by $\mu_a(p,q) = - p_\nu X_a^\nu(q)$.

Figures (1)

  • Figure 1: The three levels of Hitchin's integrable system and their corresponding actions.

Theorems & Definitions (21)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Theorem 2.4
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • ...and 11 more