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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 $mathfrak{sl}(N^{lceiltfrac{n}{2}rceil},mathbb{C})$-matrices equipped with the finite bracket given by the completely anti-symmetrized matrix product, such that the classical brackets are retrieved in the $Nrightarrow infty$ limit. We then apply this approximation to the super $4$-brane in $9$ dimensions and give a regularized action in analogy with the matrix regularization of the supermembrane. This action exhibits a reduced gauge symmetry that we discuss from the viewpoint of $L_infty$-algebras in a slight generalization to the construction of Lie $2$-algebras from Bagger-Lambert $3$-algebras.
Inspired by the recent advances in multiple M2-brane theory, we consider the generalizations of Nahm equations for arbitrary p-algebras. We construct the topological p-algebra quantum mechanics associated to them and we show that this can be obtained
As an extension to our earlier work cite{Mirza2}, we employ the Nambu brackets to prove that the divergences of heat capacities correspond to their counterparts in thermodynamic geometry. We also obtain a simple representation for the conformal trans
It is shown that the algebra of diffeomorphism-invariant charges of the Nambu-Goto string cannot be quantized in the framework of canonical quantization. The argument is shown to be independent of the dimension of the underlying Minkowski space.
We review the AKSZ construction as applied to the topological open membranes and Poisson sigma models. We describe a generalization to open topological p-branes and Nambu-Poisson sigma models.
We provide the classification of real forms of complex D=4 Euclidean algebra $mathcal{epsilon}(4; mathbb{C}) = mathfrak{o}(4;mathbb{C})) ltimes mathbf{T}_{mathbb{C}}^4$ as well as (pseudo)real forms of complex D=4 Euclidean superalgebras $mathcal{eps