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

Area of Science:

• Mathematical Physics
• Gauge Theory
• Differential Geometry

Background:

• Lagrangian 1-forms are fundamental in classical mechanics.
• Gauge theories describe fundamental forces using symmetry principles.
• Hitchin's system is a completely integrable system in mathematical physics.

Purpose of the Study:

• To extend Lagrangian 1-forms to gauge theories.
• To apply this formalism to construct a variational principle for Hitchin's system.
• To derive a unifying action for Lax equations describing the Hitchin system.

Main Methods:

• Introduction of gauged Lagrangian 1-forms.
• Application to the cotangent bundle of holomorphic structures on principal G-bundles.
• Construction of a 3D mixed BF action with defects.
• Derivation of a unifying action for Lax equations via holomorphic trivializations and partial on-shell reduction.

Main Results:

• A general formalism for gauged Lagrangian 1-forms is established.
• A multiform 3D mixed BF action with defects is constructed, providing a variational formulation of Hitchin's system.
• A unifying action for a hierarchy of Lax equations describing the Hitchin system is obtained.
• Explicit Lagrangian 1-forms for rational and elliptic Gaudin hierarchies are derived.

Conclusions:

• The developed formalism provides a powerful framework for studying integrable systems within gauge theory.
• The unifying action offers a new perspective on the Hitchin system and its associated hierarchies.
• This work connects concepts from differential geometry, gauge theory, and integrable systems.