Abstract
AbstractWe 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
”.
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
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