On Noncommutative Merons and Instantons

TL;DR

This paper explores noncommutative Yang-Mills and self-dual Yang-Mills equations, constructing solutions that generalize known instantons and merons, and introduces new ansatzes reducing these equations to difference equations.

Contribution

It introduces novel ansatzes for noncommutative YM equations, reducing them to difference equations and constructing explicit solutions, including noncommutative generalizations of instantons and merons.

Findings

Constructed noncommutative generalizations of known SDYM solutions.

Reduced noncommutative YM equations to difference equations on matrices.

Showed non-existence of noncommutative meron solutions with certain ansatz.

Abstract

The Yang-Mills (YM) and self-dual Yang-Mills (SDYM) equations on the noncommutative Euclidean four-dimensional space are considered. We introduce an ansatz for a gauge potential reducing the noncommutative SDYM equations to a difference form of the Nahm equations. By constructing solutions to the difference Nahm equations, we obtain solutions of the noncommutative SDYM equations. They are noncommutative generalizations of the known solutions to the SDYM equations such as the Minkowski solution, the one-instanton solution and others. Using the noncommutative deformation of the Corrigan-Fairlie-'t Hooft-Wilzek ansatz, we reduce the noncommutative YM equations to equations on a scalar field which have meron solutions in the commutative limit and show that they have no such solutions in the noncommutative case. To overcome this difficulty, another ansatz reducing the noncommutative YM equations to a system of difference equations on matrix-valued functions is used. For self-dual configurations this system is reduced to the difference Nahm equations.