On the Fairlie's Moyal formulation of M(atrix)- theory

TL;DR

This paper reexamines membrane equations in M-theory using the Moyal formulation, relating them to $SU( olinebreak{}( olinebreak{} ext{infinity})$ Yang-Mills equations, and provides explicit solutions including known special cases.

Contribution

It introduces a simplified form of Nahm's equations in the Moyal framework and establishes their systematic relation to $SU( olinebreak{}( olinebreak{} ext{infinity})$ Yang-Mills equations, offering explicit membrane solutions.

Findings

Reformulation of Nahm's equations in a simple form

Relation between membrane equations and $SU( olinebreak{}( olinebreak{} ext{infinity})$ Yang-Mills equations

Explicit membrane solutions including known special cases

Abstract

Starting from the Moyal formulation of M-theory in the large N-limit, we propose to reexamine the associated membrane equations of motion in 10 dimensions formulated in terms of Poisson bracket. Among the results obtained, we rewrite the coupled first order Nahm's equations into a simple form leading in turn to their systematic relation with $SU(\infty)$ Yang Mills equations of motion. The former are interpreted as the vanishing condition of some conserved currents which we propose. We develop also an algebraic analysis in which an ansatz is considered and find an explicit form for the membrane solution of our problem. Typical solutions known in literature can also emerge as special cases of the proposed solution