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

TL;DR

This paper derives a 2D integrable field theory from 4D Chern-Simons theory, establishing its covariant Poisson structure and Hamiltonian formulation, thus connecting higher-dimensional gauge theory with integrable models.

Contribution

It introduces the covariant Poisson bracket structure for the Zakharov-Shabat Lax connection and derives the covariant Hamiltonian formula, a novel approach in integrable systems.

Findings

Derived 2D Zakharov-Mikhailov action from 4D Chern-Simons theory

Established covariant Poisson bracket $r$-matrix structure of the Lax connection

Presented a covariant Hamiltonian formula analogous to $H=Tr L^2$

Abstract

We derive the $2$d Zakharov-Mikhailov action from $4$d Chern-Simons theory. This $2$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 $2$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 term of the Lax connection which is the covariant analogue of the well-known formula "$H=Tr L^2$".