Supersymmetric Yang-Mills Theory From Lorentzian Three-Algebras

TL;DR

This work extends the BF membrane model based on a Lorentzian three-algebra by adding a supersymmetric Faddeev–Popov ghost sector, producing a BRST-invariant action that eliminates negative-norm states. For a trivial vacuum, the theory is BRST-equivalent to a trivial theory, while a nonzero vev for $X_+^I$ yields a reformulation of 2+1D maximally supersymmetric Yang–Mills with formal $SO(8)$ invariance. The authors show how to construct gauge-invariant, BRST-invariant operators forming full $SO(8)$ multiplets, providing an $SO(8)$ covariant framework to discuss chiral primaries and their holographic connections to M-theory on $AdS_4 imes S^7$ via a D2-brane route. This approach clarifies unitarity properties, offers a new lens on the M2-brane worldvolume theory, and suggests a path to compute M2-related observables from D2 dynamics, albeit without a direct IR fixed-point realization in this formulation.

Abstract

We show that by adding a supersymmetric Faddeev-Popov ghost sector to the recently constructed Bagger-Lambert theory based on a Lorentzian three algebra, we obtain an action with a BRST symmetry that can be used to demonstrate the absence of negative norm states in the physical Hilbert space. We show that the combined theory, expanded about its trivial vacuum, is BRST equivalent to a trivial theory, while the theory with a vev for one of the scalars associated with a null direction in the three-algebra is equivalent to a reformulation of maximally supersymmetric 2+1 dimensional Yang-Mills theory in which there a formal SO(8) superconformal invariance.