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Classical integrability in 2D and asymptotic symmetries

These lecture notes are a contribution to the proceedings of the school "Geometric, Algebraic and Topological Methods for Quantum Field Theory", held in Villa de Leyva, Colombia, from 31st of July to 9th of August 2023. Its intention is to put together several basic tools of classical integrability and contrast them with those available in the formulation of asymptotic symmetries and the definition of canonical charges in gauge theories. We consider as a working example the Chern-Simons theory in 3D dimensions, motivated by its various applications in condensed matter physics, gravity, and black hole physics. We review basic aspects of the canonical formulation, symplectic geometry, Liouville integrability, and Lax Pairs. We define the Hamiltonian formulation of the Chern-Simons action and the canonical generators of the gauge symmetries, which are surface integrals that subject to non-trivial boundary conditions, realize transformations that do change the physical state, namely large (or improper gauge transformations). We propose asymptotic conditions that realize an infinite set of abelian conserved charges associated with integral models. We review two different cases: the Korteweg-de Vries equation for its connection with the Virasoro algebra and fluid dynamics, and the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, as it embeds an infinite class of non-linear notable integrable evolution equations. We propose a concrete example for gravity in 3D with $Λ<0$, where we find a near-horizon asymptotic dynamics. We finalize offering some insights on the initial value problem, its connection with integrable systems and flat connections. We study some properties of the Monodromy matrix and recover the infinite KdV charges from the trace invariants extracted from the Monodromy evolution equation that can be written in a Lax form.