Modave Lectures on Classical Integrability in $2d$ Field Theories

These lecture notes are based on a blackboard course given at the XVII Modave Summer School in Mathematical Physics held from 13 -- 17 September 2021 in Brussels (Belgium), and aimed at Ph.D. students in High Energy Theoretical Physics. We start with introducing classical integrability in finite-dimensional systems to set the stage for our main purpose: introducing two-dimensional classical field theories which are integrable. We focus on their zero-curvature formulation through the so-called Lax connection, which ensures the existence of an infinite tower of conserved charges. We then move on to their Poisson bracket structure, known as the Sklyanin or Maillet structure, which ensures complete classical integrability. All the concepts that we encounter will be illustrated with the integrable Principal Chiral Model, which is the canonical sigma-model that appears (or its generalisations) on the worldsheet of many string backgrounds. Along the way we briefly comment on the properties of integrability at the quantum level.