8,945 research outputs found
Convergence of Fuzzy Tori and Quantum Tori for the quantum Gromov-Hausdorff Propinquity: an explicit approach
Quantum tori are limits of finite dimensional C*-algebras for the quantum
Gromov-Hausdorff propinquity, a metric defined by the author as a strengthening
of Rieffel's quantum Gromov-Hausdorff designed to retain the C*-algebraic
structure. In this paper, we propose a proof of the continuity of the family of
quantum and fuzzy tori which relies on explicit representations of the
C*-algebras rather than on more abstract arguments, in a manner which takes
full advantage of the notion of bridge defining the quantum propinquity.Comment: 41 Pages. This paper is the second half of ArXiv:1302.4058v2. The
latter paper has been divided in two halves for publications purposes, with
the first half now the current version of 1302.4058, which has been accepted
in Trans. Amer. Math. Soc. This second half is now a stand-alone paper, with
a brief summary of 1302.4058 and a new introductio
Non-constant Non-commutativity in 2d Field Theories and a New Look at Fuzzy Monopoles
We write down scalar field theory and gauge theory on two-dimensional
noncommutative spaces with nonvanishing curvature and non-constant
non-commutativity. Usual dynamics results upon taking the limit of
going to i) a commutative manifold having nonvanishing curvature
and ii) the noncommutative plane. Our procedure does not require introducing
singular algebraic maps or frame fields. Rather, we exploit the K\"ahler
structure in the limit i) and identify the symplectic two-form with the volume
two-form. As an example, we take to be the stereographically
projected fuzzy sphere, and find magnetic monopole solutions to the
noncommutative Maxwell equations. Although the magnetic charges are conserved,
the classical theory does not require that they be quantized. The
noncommutative gauge field strength transforms in the usual manner, but the
same is not, in general, true for the associated potentials. We develop a
perturbation scheme to obtain the expression for gauge transformations about
limits i) and ii). We also obtain the lowest order Seiberg-Witten map to write
down corrections to the commutative field equations and show that solutions to
Maxwell theory on are stable under inclusion of lowest order
noncommutative corrections. The results are applied to the example of
noncommutative AdS.Comment: 27 page
The lowest modes around Gaussian solutions of tensor models and the general relativity
In the previous paper, the number distribution of the low-lying spectra
around Gaussian solutions representing various dimensional fuzzy tori of a
tensor model was numerically shown to be in accordance with the general
relativity on tori. In this paper, I perform more detailed numerical analysis
of the properties of the modes for two-dimensional fuzzy tori, and obtain
conclusive evidences for the agreement. Under a proposed correspondence between
the rank-three tensor in tensor models and the metric tensor in the general
relativity, conclusive agreement is obtained between the profiles of the
low-lying modes in a tensor model and the metric modes transverse to the
general coordinate transformation. Moreover, the low-lying modes are shown to
be well on a massless trajectory with quartic momentum dependence in the tensor
model. This is in agreement with that the lowest momentum dependence of metric
fluctuations in the general relativity will come from the R^2-term, since the
R-term is topological in two dimensions. These evidences support the idea that
the low-lying low-momentum dynamics around the Gaussian solutions of tensor
models is described by the general relativity. I also propose a renormalization
procedure for tensor models. A classical application of the procedure makes the
patterns of the low-lying spectra drastically clearer, and suggests also the
existence of massive trajectories.Comment: 31 pages, 8 figures, Added references, minor corrections, a
misleading figure replace
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
Spectral geometry with a cut-off: topological and metric aspects
Inspired by regularization in quantum field theory, we study topological and
metric properties of spaces in which a cut-off is introduced. We work in the
framework of noncommutative geometry, and focus on Connes distance associated
to a spectral triple (A, H, D). A high momentum (short distance) cut-off is
implemented by the action of a projection P on the Dirac operator D and/or on
the algebra A. This action induces two new distances. We individuate conditions
making them equivalent to the original distance. We also study the
Gromov-Hausdorff limit of the set of truncated states, first for compact
quantum metric spaces in the sense of Rieffel, then for arbitrary spectral
triples. To this aim, we introduce a notion of "state with finite moment of
order 1" for noncommutative algebras. We then focus on the commutative case,
and show that the cut-off induces a minimal length between points, which is
infinite if P has finite rank. When P is a spectral projection of , we work
out an approximation of points by non-pure states that are at finite distance
from each other. On the circle, such approximations are given by Fejer
probability distributions. Finally we apply the results to Moyal plane and the
fuzzy sphere, obtained as Berezin quantization of the plane and the sphere
respectively.Comment: Reference added. Minor corrections. Published version. 38 pages, 2
figures. Journal of Geometry and Physics 201
Fuzzy de Sitter Space from kappa-Minkowski Space in Matrix Basis
We consider the Lie group generated by the Lie algebra
of -Minkowski space. Imposing the invariance of the metric under the
pull-back of diffeomorphisms induced by right translations in the group, we
show that a unique right invariant metric is associated with
. This metric coincides with the metric of de Sitter
space-time. We analyze the structure of unitary representations of the group
relevant for the realization of the non-commutative
-Minkowski space by embedding into -dimensional Heisenberg
algebra. Using a suitable set of generalized coherent states, we select the
particular Hilbert space and realize the non-commutative -Minkowski
space as an algebra of the Hilbert-Schmidt operators. We define dequantization
map and fuzzy variant of the Laplace-Beltrami operator such that dequantization
map relates fuzzy eigenvectors with the eigenfunctions of the Laplace-Beltrami
operator on the half of de Sitter space-time.Comment: 21 pages, v3 differs from version published in Fortschritte der
Physik by a note and references added and adjuste
Symmetry, Gravity and Noncommutativity
We review some aspects of the implementation of spacetime symmetries in
noncommutative field theories, emphasizing their origin in string theory and
how they may be used to construct theories of gravitation. The geometry of
canonical noncommutative gauge transformations is analysed in detail and it is
shown how noncommutative Yang-Mills theory can be related to a gravity theory.
The construction of twisted spacetime symmetries and their role in constructing
a noncommutative extension of general relativity is described. We also analyse
certain generic features of noncommutative gauge theories on D-branes in curved
spaces, treating several explicit examples of superstring backgrounds.Comment: 52 pages; Invited review article to be published in Classical and
Quantum Gravity; v2: references adde
On noncommutative spherically symmetric spaces
Two families of noncommutative extensions are given of a general space-time
metric with spherical symmetry, both based on the matrix truncation of the
functions on the sphere of symmetry. The first family uses the truncation to
foliate space as an infinite set of spheres, is of dimension four and
necessarily time-dependent; the second can be time-dependent or static, is of
dimension five and uses the truncation to foliate the internal space.Comment: 22 page
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