On the Zakharov–Mikhailov action: 4d Chern–Simons origin and covariant Poisson algebra of the Lax connection

We derive the 2d Zakharov–Mikhailov action from 4d Chern–Simons theory. This 2d action is known to produce as equations of motion the flatness condition of a large class of Lax connections of Zakharov–Shabat type, which includes an ultralocal variant of the principal chiral model as a special case. At the 2d level, we determine for the first time the covariant Poisson bracket r-matrix structure of the Zakharov–Shabat Lax connection, which is of rational type. The flatness condition is then derived as a covariant Hamilton equation. We obtain a remarkable formula for the covariant Hamiltonian in terms of the Lax connection which is the covariant analogue of the well-known formula “H=TrL2”.