Nambu-Lie 3-Algebras on Fuzzy 3-Manifolds

TL;DR

This paper explores Nambu-Poisson 3-algebras on three-dimensional manifolds, demonstrating their equivalence to volume-preserving diffeomorphisms and proposing a quantization method that preserves classical properties.

Contribution

It establishes a connection between Nambu-Poisson 3-algebras and volume-preserving diffeomorphisms on 3-manifolds and introduces a quantization scheme for these algebras.

Findings

Nambu-Poisson algebra is identical to $SDiff({\

The fundamental identity corresponds to the $SDiff({\

A quantization prescription that reproduces classical limits.

Abstract

We consider Nambu-Poisson 3-algebras on three dimensional manifolds $ {\cal M}_{3} $, such as the Euclidean 3-space $R^{3}$, the 3-sphere $S^{3}$ as well as the 3-torus $T^{3}$. We demonstrate that in the Clebsch-Monge gauge, the Lie algebra of volume preserving diffeomorphisms $SDiff({\cal M}_{3})$ is identical to the Nambu-Poisson algebra on ${\cal M}_{3}$. Moreover the fundamental identity for the Nambu 3-bracket is just the commutation relation of $ SDiff({\cal M}_{3})$. We propose a quantization prescription for the Nambu-Poisson algebra which provides us with the correct classical limit. As such it possesses all of the expected classical properties constituting, in effect, a concrete representation of Nambu-Lie 3-algebras.