论文标题
在Zakharov-Mikhailov行动中：$ 4 $ D CHERN-SIMONS ORIGIN和lax Connection的协变量泊松代数
On the Zakharov-Mikhailov action: $4$d Chern-Simons origin and covariant Poisson algebra of the Lax connection
论文作者
论文摘要
我们从$ 4 $ d Chern-Simons理论中得出$ 2 $ D Zakharov-Mikhailov Action。这种$ 2 $ D的动作被称为运动方程，这是Zakharov-Shabat类型的大量松弛连接的平坦条件，其中包括主要手性模型的超局部变体作为特殊情况。在$ 2 $ d的水平上，我们首次确定Zakharov-Shabat Lax连接的协变量Poisson barket $ r $ -Matrix结构是合理的类型。然后将平坦条件推导为协变量的汉密尔顿方程。我们在宽松连接的术语中获得了协方差汉密尔顿的非凡公式，这是众所周知的公式“ $ h = tr l^2 $”的协变量。
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 "$H=Tr L^2$".
