Selfdual strings and loop space Nahm equations

TL;DR

This work advocates a loop-space formulation as the natural setting for selfdual strings on M5-branes and for the underlying membrane dynamics. It develops loop-space coordinates and a pair of SU(2) Bogomolnyi/Nahm structures that together encode the $SO(4)$ selfdual-string equations and their ADHMN-like construction, while deriving Nahm equations from a loop-space membrane theory with a central extension. The analysis connects the loop-space Nahm data to two decoupled Nahm systems and shows a consistent reduction to 4D Yang–Mills theory, clarifying how membrane physics can reproduce known gauge-theory limits. A key result is the argument that finite-dimensional Lie algebras cannot realize the membrane's algebraic structure, motivating the loop-algebra (and hence loop-space) framework as essential for a nontrivial interacting membrane theory, with connections to the Basu–Harvey construction. Overall, the paper links loop-space coordinates, Bogomolnyi equations, and Nahm-type constructions to a broader picture of M2–M5 dynamics and their reductions.

Abstract

We give two independent arguments why the classical membrane fields should be loops. The first argument comes from how we may construct selfdual strings in the M5 brane from a loop space version of the Nahm equations. The second argument is that there appears to be no infinite set of finite-dimensional Lie algebras (such as $su(N)$ for any $N$) that satisfies the algebraic structure of the membrane theory.