On relating multiple M2 and D2-branes

TL;DR

This paper explores alternative algebraic structures for multiple M2-branes by relaxing Lagrangian constraints, leading to new insights into their relation with D2-branes and gauge theories.

Contribution

It introduces non-antisymmetric structure constants satisfying the fundamental identity, providing a novel approach to M2-brane theories without requiring a Lagrangian formulation.

Findings

Reproduces D2-brane SYM theory to leading order

Provides new perspective on M2-brane center of mass

Suggests alternative algebraic structures for M2-branes

Abstract

Due to the difficulties of finding superconformal Lagrangian theories for multiple M2-branes, we will in this paper instead focus on the field equations. By relaxing the requirement of a Lagrangian formulation we can explore the possibility of having structure constants $f^{ABC}{}_D$ satisfying the fundamental identity but which are not totally antisymmetric. We exemplify this discussion by making use of an explicit choice of a non-antisymmetric $f^{ABC}{}_D$ constructed from the Lie algebra structure constants $f^{ab}{}_c$ of an arbitrary gauge group. Although this choice of $f^{ABC}{}_D$ does not admit an obvious Lagrangian description, it does reproduce the correct SYM theory for a stack of $N$ D2-branes to leading order in $g_{YM}^{-1}$ upon reduction and, moreover, it sheds new light on the centre of mass coordinates for multiple M2-branes.