Quantized Nambu-Poisson Manifolds and n-Lie Algebras

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.Comment: 43 pages, minor corrections, presentation improved, references adde

2D fuzzy Anti-de Sitter space from matrix models

We study the fuzzy hyperboloids AdS^2 and dS^2 as brane solutions in matrix models. The unitary representations of SO(2,1) required for quantum field theory are identified, and explicit formulae for their realization in terms of fuzzy wavefunctions are given. In a second part, we study the (A)dS^2 brane geometry and its dynamics, as governed by a suitable matrix model. In particular, we show that trace of the energy-momentum tensor of matter induces transversal perturbations of the brane and of the Ricci scalar. This leads to a linearized form of Henneaux-Teitelboim-type gravity, illustrating the mechanism of emergent gravity in matrix models.Comment: 25 page

A canonical rank-three tensor model with a scaling constraint

A rank-three tensor model in canonical formalism has recently been proposed. The model describes consistent local-time evolutions of fuzzy spaces through a set of first-class constraints which form an on-shell closed algebra with structure functions. In fact, the algebra provides an algebraically consistent discretization of the Dirac-DeWitt constraint algebra in the canonical formalism of general relativity. However, the configuration space of this model contains obvious degeneracies of representing identical fuzzy spaces. In this paper, to delete the degeneracies, another first-class constraint representing a scaling symmetry is added to propose a new canonical rank-three tensor model. A consequence is that, while classical solutions of the previous model have typically runaway or vanishing behaviors, the new model has a compact configuration space and its classical solutions asymptotically approach either fixed points or cyclic orbits in time evolution. Among others, fixed points contain configurations with group symmetries, and may represent stationary symmetric fuzzy spaces. Another consequence on the uniqueness of the local Hamiltonian constraint is also discussed, and a minimal canonical tensor model, which is unique, is given.Comment: 11 pages, minor corrections: typos corrected, references added, a discussion added in the final sectio

Beyond fuzzy spheres

We study polynomial deformations of the fuzzy sphere, specifically given by the cubic or the Higgs algebra. We derive the Higgs algebra by quantizing the Poisson structure on a surface in $\mathbb{R}^3$. We find that several surfaces, differing by constants, are described by the Higgs algebra at the fuzzy level. Some of these surfaces have a singularity and we overcome this by quantizing this manifold using coherent states for this nonlinear algebra. This is seen in the measure constructed from these coherent states. We also find the star product for this non-commutative algebra as a first step in constructing field theories on such fuzzy spaces.Comment: 9 pages, 3 Figures, Minor changes in the abstract have been made. The manuscript has been modified for better clarity. A reference has also been adde

Use of idempotent functions in the aggregation of different filters for noise removal

The majority of existing denoising algorithms obtain good results for a specific noise model, and when it is known previously. Nonetheless, there is a lack in denoising algorithms that can deal with any unknown noisy images. Therefore, in this paper, we study the use of aggregation functions for denoising purposes, where the noise model is not necessary known in advance; and how these functions affect the visual and quantitative results of the resultant images

On supermatrix models, Poisson geometry and noncommutative supersymmetric gauge theories

We construct a new supermatrix model which represents a manifestly supersymmetric noncommutative regularisation of the $UOSp(2\vert 1)$ supersymmetric Schwinger model on the supersphere. Our construction is much simpler than those already existing in the literature and it was found by using Poisson geometry in a substantial way.Comment: 29 pages, we enlarge Section 3.3 by adding a comparison with older results on the subject of the component expansion