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

Abstract

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 $\mathfrak{sl}(N^{\lceil \frac{n}{2}\rceil },\mathbb{C})$-matrices equipped with the finite bracket given by the completely anti-symmetrized matrix product, such that the classical brackets are retrieved in the $N\to \infty$ 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 $L_{\infty }$-algebras in a slight generalization to the construction of Lie $2$-algebras from Bagger--Lambert $3$-algebras.

36 pages. Updated title and refined presentation. Published in JHEP

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Cited by in corpus (3)

• Supersymmetric M2-Brane Matrix Model with Restricted Volume-Preserving Deformations: Lorentz Covariance and BPS Spectrum
• Generalized Heisenberg Dynamics Revisited
• A Lorentz Covariant Matrix Model for Bosonic M2-Branes: Nambu Brackets and Restricted Volume-Preserving Deformations