Classical 2d Integrable Systems and Gauge Theories

We compare constructions of 2d integrable models through two gauge field theories. The first one is the 4d Chern-Simons (4d-CS) theory proposed by Costello and Yamazaki. The second one is the Affine Higgs bundles (AHB). It is 2d generalization of the Hitchin integrable systems. We consider in more detail the latter construction and illustrate it by considering the Landau-Lifshitz model.