Multiple M2-branes and the Embedding Tensor

TL;DR

This work shows that the Bagger-Lambert theory for multiple M2-branes emerges from the embedding tensor framework for ${\cal N}=8$ gauge theories in $2+1$ dimensions, with the embedding tensor identified as the totally antisymmetric $f^{abcd}$ defining a 3-algebra. It derives the necessary linear and quadratic constraints on the embedding tensor, demonstrating that consistency of supersymmetry selects the antisymmetric representation and reproduces the Bagger-Lambert couplings; an alternative formulation replaces the fixed tensor by parity-odd scalar fields to render the Chern-Simons term invariant and to permit domain-wall configurations. The paper also contrasts these 3D conformal gaugings with gauged supergravity, showing that supergravity admits additional gaugings due to weaker linear constraints and larger duality groups, and discusses a covariant Theta(x) formulation with 2-form and 3-form Lagrange multipliers that enforces the required constraints. Together, these results illuminate how maximal 3D theories with M2-branes fit into the embedding-tensor paradigm and suggest avenues for broader generalizations and connections to non-Lagrangian frameworks.

Abstract

We show that the Bagger-Lambert theory of multiple M2-branes fits into the general construction of maximally supersymmetric gauge theories using the embedding tensor technique. We apply the embedding tensor technique in order to systematically obtain the consistent gaugings of N=8 superconformal theories in 2+1 dimensions. This leads to the Bagger-Lambert theory, with the embedding tensor playing the role of the four-index antisymmetric tensor defining a ``3-algebra''. We present an alternative formulation of the theory in which the embedding tensor is replaced by a set of unrestricted scalar fields. By taking these scalar fields to be parity-odd, the Chern-Simons term can be made parity-invariant.