On the Weyl-Wigner-Moyal Description of SU$(\infty)$ Nahm Equations

TL;DR

This paper explores the translation of SU(∞) Nahm equations from the reduced Self-dual Yang-Mills theory into the Weyl-Wigner-Moyal formalism, revealing connections to hyper-Kähler metrics and BPS monopoles.

Contribution

It introduces a novel application of the Weyl-Wigner-Moyal formalism to the Nahm equations, linking gauge theory and gravitational solutions.

Findings

Establishes a correspondence between BPS monopoles and hyper-Kähler metrics.

Demonstrates the applicability of the Weyl-Wigner-Moyal formalism to integrable gauge theories.

Abstract

We show how the reduced Self-dual Yang-Mills theory described by the Nahm equations can be carried over to the Weyl-Wigner-Moyal formalism employed recently in Self-dual gravity. Evidence of the existence of correspondence between BPS magnetic monopoles and space-time hyper-K\"ahler metrics is given.