Matrix regularization of classical Nambu brackets and super p-branes

Abstract

Abstract We present an explicit matrix algebra regularization of the algebra of volume-preserving diffeomorphisms of the n-torus. That is, we approximate the corresponding classical Nambu brackets using $$ \mathfrak{sl}\left({N}^{\left\lceil \frac{n}{2}\right\rceil },\mathrm{\mathbb{C}}\right) $$ sl N n 2 ℂ -matrices equipped with the finite bracket given by the completely anti-symmetrized matrix product, such that the classical brackets are retrieved in the N → ∞ limit. We then apply this approximation to the super 4-brane in 9 dimensions and give a regularized action in analogy with the matrix regularization of the supermembrane. This action exhibits a reduced gauge symmetry that we discuss from the viewpoint of L∞-algebras in a slight generalization to the construction of Lie 2-algebras from Bagger-Lambert 3-algebras.