Abelian Toda field theories on the noncommutative plane

TL;DR

This paper extends abelian Toda field theories to the noncommutative plane using a zero-curvature approach, deriving new equations, actions, and connections to higher-dimensional gauge theories, with explicit examples and validation of the Ward conjecture.

Contribution

It introduces a noncommutative generalization of abelian Toda theories via zero-curvature conditions and derives their actions, equations, and conserved currents, including explicit models and links to Yang-Mills equations.

Findings

Constructed noncommutative Toda field theories with conserved currents.

Derived explicit actions for noncommutative Toda models.

Connected noncommutative Toda equations to dimensional reduction of Yang-Mills.

Abstract

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