On the Zakharov–Mikhailov action: $$4\hbox {d}$$ Chern–Simons origin and covariant Poisson algebra of the Lax connection
Vincent Caudrelier, Matteo Stoppato, Benoît Vicedo
Abstract
Abstract We derive the $$2\hbox {d}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>2</mml:mn><mml:mtext>d</mml:mtext></mml:mrow></mml:math> Zakharov–Mikhailov action from $$4\hbox {d}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>4</mml:mn><mml:mtext>d</mml:mtext></mml:mrow></mml:math> Chern–Simons theory. This $$2\hbox {d}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>2</mml:mn><mml:mtext>d</mml:mtext></mml:mrow></mml:math> 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 $$2\hbox {d}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>2</mml:mn><mml:mtext>d</mml:mtext></mml:mrow></mml:math> 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 “ $$H={{\,\mathrm{Tr}\,}}L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>H</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mspace/><mml:mi>Tr</mml:mi><mml:mspace/></mml:mrow><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:math> ”.