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arXiv 2012.044312020-12hep-thmath-phmath.MPnlin.SI

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

Vincent Caudrelier, Matteo Stoppato, Benoit Vicedo

备注：: 16 pages. Final authors' version with added references and explanations
发表信息：: Lett. Math. Phys. 111, 82 (2021)
DOI：: 10.1007/s11005-021-01425-x

英文摘要：

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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