Quantized Nambu-Poisson Manifolds and n-Lie Algebras

TL;DR

This paper extends quantization methods to Nambu-Poisson structures, linking them to n-Lie algebras and noncommutative geometries, with applications to quantum brane geometries in M-theory.

Contribution

It introduces a generalized quantization framework translating classical Nambu-Poisson structures into n-Lie algebras, extending Berezin-Toeplitz quantization to new geometries.

Findings

Extended Berezin quantization of spheres and hyperboloids

Interpretation of n-Lie algebras via foliations of noncommutative spaces

Connections to quantum geometry of branes in M-theory

Abstract

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.