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

References in corpus (18)

• Gauge Symmetry and Supersymmetry of Multiple M2-Branes
• Algebraic structures on parallel M2-branes
• Fractional M2-branes
• Viscosity, Black Holes, and Quantum Field Theory
• M2-branes, 3-Lie Algebras and Plucker relations
• Multiple M2-branes and Generalized 3-Lie algebras
• Bagger-Lambert Theory for General Lie Algebras
• M-theory branes and their interactions
• Multiple Membranes from Gauged Supergravity
• Toric Calabi-Yau four-folds dual to Chern-Simons-matter theories
• Ternary Virasoro - Witt Algebra
• On the structure of k-Lie algebras
• Light-cone M5 and multiple M2-branes
• Truncated Nambu-Poisson Bracket and Entropy Formula for Multiple Membranes
• M-Theory Superalgebra From Multiple Membranes
• A note on Quantum Aspects of Multiple Membranes
• Relaxed Three-Algebras: Their Matrix Representations and Implications for Multi M2-brane Theory
• Octonions, G_2 and generalized Lie 3-algebras

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