On the ADHM construction of noncommutative U(2) k-instanton

TL;DR

This paper reformulates the ADHM construction within noncommutative geometry to explicitly compute the U(2) instanton number, linking algebraic and topological aspects in noncommutative space.

Contribution

It introduces a new formulation of the ADHM construction using noncommutative algebra elements, enabling explicit calculation of instanton numbers in noncommutative space.

Findings

Explicit calculation of U(2) instanton number in noncommutative space.

Connection between algebraic projector trace and topological winding number.

Reformulation facilitates analysis of instantons in noncommutative geometry.

Abstract

The basic objects of the ADHM construction are reformulated in terms of elements of the $A_{\theta}(R^4)$ algebra of the noncommutative $R_{\theta}^4$ space. This new formulation of the ADHM construction makes possible the explicit calculus of the U(2) instanton number which is shown to be the product of a trace of finite rank projector of the Fock representation space of the algebra $A_{\theta}(R^4)$ times a noncommutative version of the winding number.