A Novel Matrix Model for the M5-brane?

We provide a new formal extension of the BFSS matrix model by an additional 5-bracket. Maximal supersymmetry leads us to promote the BFSS 2-bracket structure constants to a dynamical field governed by a Chern-Simons-like kinetic term, as well as a novel potential self-duality relation with respect to the 5-bracket. We show that the full action is invariant under maximal supersymmetry and that the associated supersymmetry algebra closes. This result hinges on a conspiracy of properties of the $SO(9)$ gamma matrices and the 2- and 5-brackets. Compellingly, the resulting model seems to realize some features expected of a theory containing M5-branes and opens up the possibility of further including higher brackets for the M6- and M9-branes.