On the Zakharov–Mikhailov action: $$4\hbox {d}$$ Chern–Simons origin and covariant Poisson algebra of the Lax connection

We derive the $$2\hbox {d}$$ 2 d Zakharov–Mikhailov action from $$4\hbox {d}$$ 4 d Chern–Simons theory. This $$2\hbox {d}$$ 2 d 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 $$2\hbox {d}$$ 2 d 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 “$$H={{\,\mathrm{Tr}\,}}L^2$$ H = Tr L 2 ”.