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 \( \mathfrak{sl}\left({N}^{\left\lceil \frac{n}{2}\right\rceil },\mathrm{\mathbb{C}}\right) \)-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.

我们提出了n环面的体积保持微分同胚代数的显式矩阵代数正则化。也就是说，我们使用\( \mathfrak{sl}\left({N}^{\left\lceil \frac{n}{2}\right\rceil },\mathrm{\mathbb{ C}}\right) \) -矩阵配备了由完全反对称矩阵乘积给出的有限括号，使得经典括号在N → ∞限制中检索。然后，我们将这种近似应用于 9 维的超 4 膜，并给出与超膜的矩阵正则化类似的正则化动作。这个动作展示了我们从L _∞的角度讨论的规范对称性降低-代数对从 Bagger-Lambert 3-代数构造 Lie 2-代数的轻微推广。