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arXiv · 2012.04431

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

Vincent Caudrelier·Matteo Stoppato·Benoit Vicedo
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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 "$H=Tr L^2$".

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BibTeXRIS

Vincent Caudrelier, Matteo Stoppato, Benoit Vicedo. 2020-12-08. On the Zakharov-Mikhailov action: $4$d Chern-Simons origin and covariant Poisson algebra of the Lax connection. https://doi.org/10.1007/s11005-021-01425-x

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