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Register a new userMatrix Regularization of Classical Nambu Brackets and Super $p$-Branes
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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.
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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 a truncation of the topological p-brane theory previously studied by the authors. The resulting topological p-algebra quantum mechanics is discussed in detail and the relation with the M2-M5 system is pointed out in the p=3 case, providing a geometrical argument for the emergence of the 3-algebra structure in the Bagger-Lambert-Gustavsson theory
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 transformations that connect different thermodynamics metrics to each other. Using our bracket approach, we obtain interesting exact relations between the Hessian matrix with any number of parameters and specific heat capacities. Finally, we employ this approach to investigate some thermodynamic properties of the Meyers-Perry black holes with three spins.
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 $mathcal{epsilon}(4|N; mathbb{C})$ for N=1,2. Further we present our results: N=1 and N=2 supersymmetric D=4 Poincare and Euclidean r-matrices obtained by using D= 4 Poincare r-matrices provided by Zakrzewski [1]. For N=2 we shall consider the general superalgebras with two central charges.
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