Non-Commutative Instantons and the Seiberg-Witten Map

TL;DR

This paper introduces a broad class of non-commutative instanton solutions in higher-dimensional Yang-Mills theories and analyzes how the Seiberg-Witten map translates these solutions into commutative variables, revealing their singular nature.

Contribution

It provides a new ansatz for instanton solutions in non-commutative Yang-Mills theories and simplifies the Seiberg-Witten map's action on these solutions.

Findings

Discovery of a large class of instanton solutions including shift operator and Nekrasov-Schwarz instantons.

Simplified form of the Seiberg-Witten map in operator language.

Instantons become singular when described in commutative variables.

Abstract

We present several results concerning non-commutative instantons and the Seiberg-Witten map. Using a simple ansatz we find a large new class of instanton solutions in arbitrary even dimensional non-commutative Yang-Mills theory. These include the two dimensional ``shift operator'' solutions and the four dimensional Nekrasov-Schwarz instantons as special cases. We also study how the Seiberg-Witten map acts on these instanton solutions. The infinitesimal Seiberg-Witten map is shown to take a very simple form in operator language, and this result is used to give a commutative description of non-commutative instantons. The instanton is found to be singular in commutative variables.