Lie dialgebras, gauge theory, and Lagrangian multiforms for integrable models

Lagrangian multiforms provide a variational framework for describing integrable hierarchies. This thesis presents two approaches for systematically constructing Lagrangian one-forms, which cover the case of finite-dimensional integrable hierarchies, thus addressing one of the central open problems in the theory of Lagrangian multiforms. The first approach, based on the theory of Lie dialgebras, incorporates into Lagrangian one-forms the notion of the classical $r$-matrix and produces Lagrangian one-forms living on coadjoint orbits. We prove an important structural result relating the closure relation for Lagrangian one-forms to the Poisson involutivity of Hamiltonians and the double zero on Euler-Lagrange equations. In the second approach, we extend the notion of Lagrangian one-forms to the setting of gauge theories and derive a variational formulation of the Hitchin system associated with a compact Riemann surface of arbitrary genus. We show that this description corresponds to a Lagrangian one-form for classical $3$d holomorphic-topological BF theory coupled with so-called type A and type B defects. Notably, this establishes an explicit connection between $3$d holomorphic-topological BF theory and the Hitchin system at the classical level. Further, we derive a unifying action for a hierarchy of Lax equations describing the Hitchin system in terms of meromorphic Lax matrices. As applications of the two approaches, we also obtain explicit Lagrangian one-forms for the hierarchies of various well-known integrable models.

💡 Research Summary

This thesis addresses a central open problem in the theory of Lagrangian multiforms: the systematic construction of Lagrangian one‑forms for finite‑dimensional integrable hierarchies. Two complementary approaches are developed, each providing a general framework that yields explicit Lagrangian one‑forms for a wide class of well‑known integrable models.

1. Lie‑dialgebra approach.
The first part builds on the theory of Lie dialgebras, algebraic structures equipped with two compatible Lie brackets. By incorporating a classical r‑matrix that satisfies the (modified) Classical Yang‑Baxter Equation, the author constructs geometric Lagrangian one‑forms living on coadjoint orbits of a Lie algebra. A key structural theorem proves that the closure condition dℒ₁=0 for the Lagrangian one‑form is equivalent to the involutivity of the associated Hamiltonians (i.e. {H_i, H_j}=0) and to the appearance of a double zero on the Euler–Lagrange equations. This result establishes a direct variational link between the multiform closure relation and the Poisson‑commuting conserved quantities that define integrability in the Liouville sense.

Using this machinery, explicit Lagrangian one‑forms are derived for:

• The open Toda chain (both standard and skew‑symmetric r‑matrix versions);
• The rational Gaudin model;
• Cyclotomic Gaudin models, including their realizations as periodic Toda chains, the discrete self‑trapping (DST) model, and a coupled Toda‑DST system.

The construction shows how seemingly different models can be unified under a single algebraic framework, with the closure relation guaranteeing the compatibility of all flows in the hierarchy.

2. Gauge‑theoretic approach.
The second part extends the notion of Lagrangian one‑forms to gauge theories. The author considers three‑dimensional holomorphic‑topological BF theory and introduces two types of defects (type A and type B) localized at marked points on a compact Riemann surface Σ of arbitrary genus. By performing a symplectic reduction on the moment‑map level set μ⁻¹(0)/G, a Lagrangian one‑form is obtained that reproduces the classical Hitchin integrable system on Σ. This establishes an explicit classical correspondence between 3d holomorphic‑topological BF theory with defects and the Hitchin system.

A unifying action functional is then derived for a hierarchy of Lax equations describing the Hitchin system in terms of meromorphic Lax matrices L(z). The author works out the genus‑zero (rational) and genus‑one (elliptic) cases in detail, producing Lagrangian one‑forms for:

• The rational Gaudin hierarchy;
• The elliptic Gaudin hierarchy;
• The elliptic spin Calogero–Moser model as a special sub‑case.

3. Applications and examples.
Each of the above models is presented with full algebraic setup, Lax representation, and explicit Lagrangian one‑form. The thesis verifies the closure relation and Poisson involutivity for every case, confirming that the constructed multiforms indeed encode integrability. Notably, the cyclotomic Gaudin framework allows the coupling of the periodic Toda chain with the DST model, producing a novel integrable hierarchy whose Lagrangian one‑form is written down explicitly.

4. Conclusions and outlook.
The work demonstrates that:

• Lie‑dialgebra techniques provide a geometric, orbit‑based method to generate Lagrangian multiforms for any integrable system admitting a classical r‑matrix;
• Gauge‑theoretic constructions link topological field theories with classical integrable systems, opening pathways to incorporate defects, higher‑genus curves, and possibly quantum deformations.

Future directions suggested include quantization of the constructed multiforms, extensions to infinite‑dimensional field theories, and exploration of connections with supersymmetric gauge theories and the geometric Langlands program. Overall, the thesis delivers a robust, unified variational framework that significantly advances the applicability of Lagrangian multiforms in the study of integrable models.