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.