Comments on the Bagger-Lambert theory and multiple M2-branes

TL;DR

This work scrutinizes the Bagger–Lambert construction for multiple M2-branes, focusing on the explicit $SO(4)$ example. It shows that the theory can be recast as an $SU(2) imes SU(2)$ gauge theory with bifundamental matter, and that parity invariance is achieved via opposite-sign Chern–Simons terms together with an exchange of the two gauge fields. The analysis reveals a moduli-space mismatch: the vanishing-potential sector is $(R^8 imes R^8)/O(2)$, not aligning with the expected M2-brane moduli space $(R^8)^N/S_N$, and it highlights a difficulty in constructing the full set of superconformal primary operators within this framework because only even $SO(8)$ representations appear. These findings suggest the $SO(4)$ Bagger–Lambert theory may be related but is not yet the correct worldvolume theory for generic M2-branes, indicating the need for additional algebraic structure in three-algebras to realize the desired spectrum. The work therefore narrows the gap between the Bagger–Lambert proposal and the M2-brane worldvolume theory, pointing to avenues for refining the algebraic framework or identifying new examples with the appropriate operator content.

Abstract

We study the SO(8) superconformal theory proposed recently by Bagger and Lambert as a possible worldvolume theory for multiple M2-branes. For their explicit example with gauge group SO(4), we rewrite the theory (originally formulated in terms of a three-algebra) as an ordinary SU(2) \times SU(2) gauge theory with bifundamental matter. In this description, the parity invariance of the theory, required for a proper description of M2-branes, is clarified. We describe the subspace of scalar field configurations on which the potential vanishes, and note that this does not coincide with the moduli space for a stack of M2-branes. Finally, we point out a difficulty in constructing the required set of superconformal primary operators which should be present in the correct theory describing multiple M2-branes.