Equivalence of Projections as Gauge Equivalence on Noncommutative Space

TL;DR

This paper introduces a framework for understanding the equivalence of projections as gauge transformations on noncommutative space, with applications to noncommutative instantons and resolutions of singularities.

Contribution

It proposes a novel framework for gauge equivalence of projections on noncommutative space and applies it to analyze U(2) instantons, revealing noncommutative resolutions of singular gauge transformations.

Findings

Zero winding number configuration is gauge equivalent to noncommutative BPST instanton.

Framework clarifies gauge transformations as noncommutative resolutions of singularities.

Application to ADHM construction enhances understanding of instantons on noncommutative space.

Abstract

Projections play crucial roles in the ADHM construction on noncommutative $\R^4$. In this article a framework for the description of equivalence relations between projections is proposed. We treat the equivalence of projections as ``gauge equivalence'' on noncommutative space. We find an interesting application of this framework to the study of U(2) instanton on noncommutative $\R^4$: A zero winding number configuration with a hole at the origin is ``gauge equivalent'' to the noncommutative analog of the BPST instanton. Thus the ``gauge transformation'' in this case can be understood as a noncommutative resolution of the singular gauge transformation in ordinary $\R^4$.