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. As applications of this approach, we obtain explicit Lagrangian one-forms for the hierarchies of the open Toda chain and the non-cyclotomic and cyclotomic rational Gaudin models, as well as the periodic Toda chain and the discrete self-trapping model as realisations of the cyclotomic Gaudin model. The versatility of this approach is further demonstrated by coupling the periodic Toda chain with the discrete self-trapping model and obtaining a Lagrangian one-form for the corresponding hierarchy.

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 3d holomorphic-topological BF theory coupled with so-called type A and type B defects. Notably, this establishes an explicit connection between 3d holomorphic-topological BF theory and the Hitchin system at the classical level. Further, 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 zero and one are treated in greater detail, leading to explicit Lagrangian one-forms for the hierarchies of the rational and the elliptic Gaudin models, respectively, and of the elliptic spin Calogero-Moser model as a special subcase of the latter.