28 research outputs found
The action functional for Moyal planes
Modulo some natural generalizations to noncompact spaces, we show in this
letter that Moyal planes are nonunital spectral triples in the sense of Connes.
The action functional of these triples is computed, and we obtain the expected
result, ie the noncommutative Yang-Mills action associated with the Moyal
product. In particular, we show that Moyal gauge theory naturally fit into the
rigorous framework of noncommutative geometry.Comment: latex, 10 page
Dixmier traces on noncompact isospectral deformations
We extend the isospectral deformations of Connes, Landi and Dubois-Violette
to the case of Riemannian spin manifolds carrying a proper action of the
noncompact abelian group . Under deformation by a torus action, a standard
formula relates Dixmier traces of measurable operators to integrals of
functions on the manifold. We show that this relation persists for actions of
, under mild restrictions on the geometry of the manifold which guarantee
the Dixmier traceability of those operators.Comment: 30 pages, no figures; several minor improvements, to appear in J.
Funct. Ana
Spectral geometry of the Moyal plane with harmonic propagation
We construct a `non-unital spectral triple of finite volume' out of the Moyal
product and a differential square root of the harmonic oscillator Hamiltonian.
We find that the spectral dimension of this triple is d but the KO-dimension is
2d. We add another Connes-Lott copy and compute the spectral action of the
corresponding U(1)-Yang-Mills-Higgs model. We find that the `covariant
coordinate' involving the gauge field combines with the Higgs field to a
unified potential, yielding a deep unification of discrete and continuous parts
of the geometry.Comment: 37 page
Fourier analysis on the affine group, quantization and noncompact Connes geometries
We find the Stratonovich-Weyl quantizer for the nonunimodular affine group of
the line. A noncommutative product of functions on the half-plane, underlying a
noncompact spectral triple in the sense of Connes, is obtained from it. The
corresponding Wigner functions reproduce the time-frequency distributions of
signal processing. The same construction leads to scalar Fourier
transformations on the affine group, simplifying and extending the Fourier
transformation proposed by Kirillov.Comment: 37 pages, Latex, uses TikZ package to draw 3 figures. Two new
subsections, main results unchange
The spectral action for Moyal planes
Extending a result of D.V. Vassilevich, we obtain the asymptotic expansion
for the trace of a "spatially" regularized heat operator associated with a
generalized Laplacian defined with integral Moyal products. The Moyal
hyperplanes corresponding to any skewsymmetric matrix being spectral
triples, the spectral action introduced in noncommutative geometry by A.
Chamseddine and A. Connes is computed. This result generalizes the Connes-Lott
action previously computed by Gayral for symplectic .Comment: 20 pages, no figure, few improvment
Heat-Kernel Approach to UV/IR Mixing on Isospectral Deformation Manifolds
30 pages, no figure, version 2We work out the general features of perturbative field theory on noncommutative manifolds defined by isospectral deformation. These (in general curved) `quantum spaces\', generalizing Moyal planes and noncommutative tori, are constructed using Rieffel\'s theory of deformation quantization for action of . Our framework, incorporating background field methods and tools of QFT in curved spaces, allows to deal both with compact and non-compact spaces, as well as with periodic or not deformations, essentially in the same way. We compute the quantum effective action up to one loop for a scalar theory, showing the different UV/IR mixing phenomena for different kinds of isospectral deformations. The presence and behavior of the non-planar parts of the Green functions is understood simply in terms of off-diagonal heat kernel contributions. For periodic deformations, a Diophantine condition on the noncommutativity parameters is found to play a role in the analytical nature of the non-planar part of the one-loop reduced effective action. Existence of fixed points for the action may give rise to a new kind of UV/IR mixing