Selfdual strings and loop space Nahm equations

TL;DR

This paper argues that classical membrane fields should be loops, using loop space Nahm equations to construct selfdual strings and highlighting the absence of finite-dimensional Lie algebras fitting membrane theory.

Contribution

It provides two independent theoretical arguments supporting the loop nature of membrane fields and introduces a loop space version of the Nahm equations.

Findings

Selfdual strings can be constructed from loop space Nahm equations.

Finite-dimensional Lie algebras are insufficient for membrane algebraic structures.

Classical membrane fields are better described as loops.

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.