Matrix Quantization of Classical Nambu Brackets and Super $p$-Branes

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