Abelian Toda field theories on the noncommutative plane

摘要: Generalizations of GL(n) abelian Toda and $\widetilde{GL}(n)$ affine field theories to the noncommutative plane are constructed. Our proposal relies on extension a zero-curvature condition satisfied by algebra-valued gauge potentials dependent fields. This can be expressed as Leznov-Saveliev equations which make possible define generalizations systems second order differential equations, with an infinite chain conserved currents. The actions corresponding these also provided. special cases GL(2) Liouville $\widetilde{GL}(2)$ sinh/sine-Gordon explicitly studied. It is shown that from (anti-)self-dual Yang-Mills in four dimensions it obtain dimensional reduction motion two-dimensional models fact supports validity version Ward conjecture. relation our previous versions some specific reported literature presented well.