Algebraic structures on parallel M2-branes

TL;DR

The paper addresses constructing a non-abelian, SUSY-preserving worldvolume theory for parallel M2-branes by postulating a two-set algebra ${\cal A},{\cal B}$ with three products and exploring abelian reductions from $\mathcal{N}=1$ 10D super Yang–Mills to $1+2$D. It develops a gamma-matrix–based finite-dimensional realization to illustrate consistency and derives SUSY closure conditions, including a Chern–Simons–like gauge sector and a loop-space–inspired generalization. It further proposes an infinite-dimensional loop-space formulation with a two-bracket $<\phi,\varphi> = \int ds [\phi(s)\dot{\varphi}(s) - \varphi(s)\dot{\phi}(s)]$ and discusses a reduction to D2-branes that may emerge from summing over loops, potentially connecting M2- and D2-brane dynamics. Overall, the work offers a novel, non-Lie-algebra approach to M2-brane physics and outlines a pathway toward a loop-based realization that could underpin M-theory brane interactions.

Abstract

In the course of closing supersymmetry on parallel M2 branes up to a gauge transformation, following the suggestion in hep-th/0611108 of incorporating a gauge field which only has topological degrees of freedom, we are led to assume a certain algebraic structure for the low energy theory supposedly living on parallel M2 branes.