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

TL;DR

This paper develops a matrix algebra approach to quantize Nambu brackets on the n-torus and applies it to regularize super 4-branes in 9 dimensions, revealing a novel gauge symmetry structure.

Contribution

It introduces an explicit matrix algebra quantization of classical Nambu brackets and applies it to super p-branes, connecting to Lie 2-algebras and Bagger--Lambert structures.

Findings

Matrix quantization recovers classical Nambu brackets in the large N limit.

Regularized super 4-brane action with reduced gauge symmetry.

Connection between gauge symmetry and L-infinity algebra structures.

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 $N\rightarrow \infty$ 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.