Quantized Nambu-Poisson Manifolds and n-Lie Algebras

Details Quantized Nambu-Poisson Manifolds and n-Lie Algebras Quantized Nambu-Poisson Manifolds and n-Lie Algebras

2010-01-19

arxiv.org/abs/1001.3275v2

J.Math.Phys.51:122303,2010

Physics

High Energy Physics

High Energy Physics - Theory

43 pages, minor corrections, presentation improved, references added

Scientific paper

10.1063/1.3503773

We investigate the geometric interpretation of quantized Nambu-Poisson structures in terms of noncommutative geometries. We describe an extension of the usual axioms of quantization in which classical Nambu-Poisson structures are translated to n-Lie algebras at quantum level. We demonstrate that this generalized procedure matches an extension of Berezin-Toeplitz quantization yielding quantized spheres, hyperboloids, and superspheres. The extended Berezin quantization of spheres is closely related to a deformation quantization of n-Lie algebras, as well as the approach based on harmonic analysis. We find an interpretation of Nambu-Heisenberg n-Lie algebras in terms of foliations of R^n by fuzzy spheres, fuzzy hyperboloids, and noncommutative hyperplanes. Some applications to the quantum geometry of branes in M-theory are also briefly discussed.

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