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

Abstract We derive the 2 d 2d Zakharov–Mikhailov action from 4 d 4d Chern–Simons theory. This 2 d 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 2 d 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={\, Tr\, }L² H=TrL2 ”.