Some Noncommutative Multi-instantons from Vortices in Curved Space

TL;DR

This paper constructs explicit noncommutative multi-instanton solutions in four dimensions by reducing the problem to vortex solutions in curved space, using the Fock space approach.

Contribution

It extends Witten's ansatz to noncommutative geometry, providing explicit vortex and multi-instanton solutions in curved space.

Findings

Explicit noncommutative multi-instanton solutions derived

Vortex solutions obtained via Fock space approach

Reduction of 4D instanton problem to 2D vortex equations

Abstract

We construct U(2) noncommutative multi-instanton solutions by extending Witten's ansatz [1] which reduces the problem of cylindrical symmetry in four dimensions to that of a set of Bogomol'nyi equations for an Abelian Higgsmodel in two dimensional curved space. Using the Fock space approach, we give explicit vortex solutions to the Bogomol'nyi equations and, from them, we present multi-instanton solutions.