307 research outputs found
Curved Noncommutative Tori as Leibniz Quantum Compact Metric Spaces
We prove that curved noncommutative tori, introduced by Dabrowski and Sitarz,
are Leibniz quantum compact metric spaces and that they form a continuous
family over the group of invertible matrices with entries in the commutant of
the quantum tori in the regular representation, when this group is endowed with
a natural length function.Comment: 16 Pages, v3: accepted in Journal of Math. Physic
Connes distance by examples: Homothetic spectral metric spaces
We study metric properties stemming from the Connes spectral distance on
three types of non compact noncommutative spaces which have received attention
recently from various viewpoints in the physics literature. These are the
noncommutative Moyal plane, a family of harmonic Moyal spectral triples for
which the Dirac operator squares to the harmonic oscillator Hamiltonian and a
family of spectral triples with Dirac operator related to the Landau operator.
We show that these triples are homothetic spectral metric spaces, having an
infinite number of distinct pathwise connected components. The homothetic
factors linking the distances are related to determinants of effective Clifford
metrics. We obtain as a by product new examples of explicit spectral distance
formulas. The results are discussed.Comment: 23 pages. Misprints corrected, references updated, one remark added
at the end of the section 3. To appear in Review in Mathematical Physic
The beat of a fuzzy drum: fuzzy Bessel functions for the disc
The fuzzy disc is a matrix approximation of the functions on a disc which
preserves rotational symmetry. In this paper we introduce a basis for the
algebra of functions on the fuzzy disc in terms of the eigenfunctions of a
properly defined fuzzy Laplacian. In the commutative limit they tend to the
eigenfunctions of the ordinary Laplacian on the disc, i.e. Bessel functions of
the first kind, thus deserving the name of fuzzy Bessel functions.Comment: 30 pages, 8 figure
An Obstruction to Quantization of the Sphere
In the standard example of strict deformation quantization of the symplectic
sphere , the set of allowed values of the quantization parameter
is not connected; indeed, it is almost discrete. Li recently constructed a
class of examples (including ) in which can take any value in an
interval, but these examples are badly behaved. Here, I identify a natural
additional axiom for strict deformation quantization and prove that it implies
that the parameter set for quantizing is never connected.Comment: 23 page. v2: changed sign conventio
The Moyal Sphere
We construct a family of constant curvature metrics on the Moyal plane and
compute the Gauss-Bonnet term for each of them. They arise from the conformal
rescaling of the metric in the orthonormal frame approach. We find a particular
solution, which corresponds to the Fubini-Study metric and which equips the
Moyal algebra with the geometry of a noncommutative sphere.Comment: 16 pages, 3 figure
On the role of twisted statistics in the noncommutative degenerate electron gas
We consider the problem of a degenerate electron gas in the background of a
uniformly distributed positive charge, ensuring overall neutrality of the
system, in the presence of non-commutativity. In contrast to previous
calculations that did not include twisted statistics, we find corrections to
the ground state energy already at first order in perturbation theory when the
twisted statistics is taken into account. These corrections arise since the
interaction energy is sensitive to two particle correlations, which are
modified for twisted anti-commutation relations
BRST quantization of quasi-symplectic manifolds and beyond
We consider a class of \textit{factorizable} Poisson brackets which includes
almost all reasonable Poisson structures. A particular case of the factorizable
brackets are those associated with symplectic Lie algebroids. The BRST theory
is applied to describe the geometry underlying these brackets as well as to
develop a deformation quantization procedure in this particular case. This can
be viewed as an extension of the Fedosov deformation quantization to a wide
class of \textit{irregular} Poisson structures. In a more general case, the
factorizable Poisson brackets are shown to be closely connected with the notion
of -algebroid. A simple description is suggested for the geometry underlying
the factorizable Poisson brackets basing on construction of an odd Poisson
algebra bundle equipped with an abelian connection. It is shown that the
zero-curvature condition for this connection generates all the structure
relations for the -algebroid as well as a generalization of the Yang-Baxter
equation for the symplectic structure.Comment: Journal version, references and comments added, style improve
Unstable solitons on noncommutative tori and D-branes
We describe a class of exact solutions of super Yang-Mills theory on
even-dimensional noncommutative tori. These solutions generalize the solitons
on a noncommutative plane introduced in hep-th/0009142 that are conjectured to
describe unstable D2p-D0 systems. We show that the spectrum of quadratic
fluctuations around our solutions correctly reproduces the string spectrum of
the D2p-D0 system in the Seiberg-Witten decoupling limit. In particular the
fluctuations correctly reproduce the 0-0 string winding modes. For p=1 and p=2
we match the differences between the soliton energy and the energy of an
appropriate SYM BPS state with the binding energies of D2-D0 and D4-D0 systems.
We also give an example of a soliton that we conjecture describes branes of
intermediate dimension on a torus such as a D2-D4 system on a four-torus.Comment: 22 pages, Latex; v.2: references adde
Noncommutative spacetime symmetries: Twist versus covariance
We prove that the Moyal product is covariant under linear affine spacetime
transformations. From the covariance law, by introducing an -space
where the spacetime coordinates and the noncommutativity matrix components are
on the same footing, we obtain a noncommutative representation of the affine
algebra, its generators being differential operators in -space. As
a particular case, the Weyl Lie algebra is studied and known results for Weyl
invariant noncommutative field theories are rederived in a nutshell. We also
show that this covariance cannot be extended to spacetime transformations
generated by differential operators whose coefficients are polynomials of order
larger than one. We compare our approach with the twist-deformed enveloping
algebra description of spacetime transformations.Comment: 19 pages in revtex, references adde
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