On the Problem of Multiple M2 Branes

TL;DR

The paper proposes a simplified three-dimensional BL-type framework for an arbitrary number $N$ of M2 branes in $d=11$, built from a degenerate $U(N)$ Nambu bracket and a non-antisymmetric $f^{abcd}$, with fields in the adjoint of $SU(N)$ plus gauge-singlet scalars that can trigger a transition to D2 branes in $d=10$. It develops a nonlinear gauged ${\cal N}=8$ action incorporating auxiliary fields and a potential, analyzes the linear limit and the M2\to D2 transition, and outlines how a consistent ${\cal N}=8$ SUSY invariance can be maintained in the simplified setting through targeted cancellations. The work places the simplified model within the broader BL construction, showing how it can be embedded into the BL framework and how the standard M2/D2 reduction emerges when a singlet acquires a vev, albeit at the cost of discarding the full non-adjoint gauge-field structure. Overall, the paper offers a down-to-earth description of multiple M2 branes at arbitrary $N$ that connects to the BL program and clarifies the role of antisymmetry constraints in the underlying 3-algebra.

Abstract

A simplified version of 3d BL theory is considered, which allows any number N of M2 branes in d=11. The underlying 3-algebra structure is provided by degenerate U(N) Nambu bracket [X,Y,Z] = tr(X) [Y,Z] + tr(Y) [Z,X] + tr(Z) [X,Y], the corresponding f^{abcd} is not totally antisymmetric and extended supersymmetry of the action remains to be checked. All the fields, including auxiliary non-propagating gauge fields, are in adjoint representation of SU(N) and the only remnant of 3-algebra structure is an octuple of gauge singlets, acquiring vacuum expectation value in transition to D2 branes in d=10.