Nambu-Lie 3-Algebras on Fuzzy 3-Manifolds

Abstract

We consider Nambu-Poisson 3-algebras on three dimensional manifolds ${\mathcal{M}}_3$, 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 $SDiff({\mathcal{M}}_3)$ is identical to the Nambu-Poisson algebra on ${\mathcal{M}}_3$. Moreover the fundamental identity for the Nambu 3-bracket is just the commutation relation of $SDiff({\mathcal{M}}_3)$. 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.

44 pages, v2:typos corrected, references added

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