On relating multiple M2 and D2-branes

TL;DR

This work proposes to study multiple M2-branes at the level of field equations by using a four-index structure constant $f^{ABC}{}_D$ that satisfies the fundamental identity but is not totally antisymmetric. By selecting a non-antisymmetric $f^{ABC}{}_D$ with $f^{phi a b}{}_c=f^{ab}{}_c$ and applying the Mukhi–Papageorgakis Higgs mechanism, the authors recover the infrared limit of $2+1$D SYM for D2-branes with a general semi-simple gauge group, to leading order in $g_{YM}^{-1}$, while revealing a free $ ext{U}(1)$ center-of-mass multiplet that naturally arises from the M2 description. The results suggest a consistent M2–D2 connection at leading order despite the absence of a Lagrangian, and they highlight the tight constraints imposed by the fundamental identity on allowable structure constants. The uniqueness of the embedding and the possible higher-order corrections remain important directions for future work in understanding M-theory dynamics.

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.