140 research outputs found
Induced Gauge Theory on a Noncommutative Space
We consider a scalar theory on canonically deformed Euclidean space
in 4 dimensions with an additional oscillator potential. This model is known to
be renormalisable. An exterior gauge field is coupled in a gauge invariant
manner to the scalar field. We extract the dynamics for the gauge field from
the divergent terms of the 1-loop effective action using a matrix basis and
propose an action for the noncommutative gauge theory, which is a candidate for
a renormalisable model.Comment: Typos corrected, one reference added; eqn. (49) corrected, one
equation number added; 30 page
Heat kernel and number theory on NC-torus
The heat trace asymptotics on the noncommutative torus, where generalized
Laplacians are made out of left and right regular representations, is fully
determined. It turns out that this question is very sensitive to the
number-theoretical aspect of the deformation parameters. The central condition
we use is of a Diophantine type. More generally, the importance of number
theory is made explicit on a few examples. We apply the results to the spectral
action computation and revisit the UV/IR mixing phenomenon for a scalar theory.
Although we find non-local counterterms in the NC theory on \T^4, we
show that this theory can be made renormalizable at least at one loop, and may
be even beyond
Quantum field theory on projective modules
We propose a general formulation of perturbative quantum field theory on
(finitely generated) projective modules over noncommutative algebras. This is
the analogue of scalar field theories with non-trivial topology in the
noncommutative realm. We treat in detail the case of Heisenberg modules over
noncommutative tori and show how these models can be understood as large
rectangular pxq matrix models, in the limit p/q->theta, where theta is a
possibly irrational number. We find out that the modele is highly sensitive to
the number-theoretical aspect of theta and suffers from an UV/IR-mixing. We
give a way to cure the entanglement and prove one-loop renormalizability.Comment: 52 pages, uses feynm
Moyal Planes are Spectral Triples
Axioms for nonunital spectral triples, extending those introduced in the
unital case by Connes, are proposed. As a guide, and for the sake of their
importance in noncommutative quantum field theory, the spaces endowed
with Moyal products are intensively investigated. Some physical applications,
such as the construction of noncommutative Wick monomials and the computation
of the Connes--Lott functional action, are given for these noncommutative
hyperplanes.Comment: Latex, 54 pages. Version 3 with Moyal-Wick section update
Position-dependent noncommutative products: classical construction and field theory
We look in Euclidean for associative star products realizing the
commutation relation , where the
noncommutativity parameters depend on the position
coordinates . We do this by adopting Rieffel's deformation theory
(originally formulated for constant and which includes the Moyal
product as a particular case) and find that, for a topology ,
there is only one class of such products which are associative. It corresponds
to a noncommutativity matrix whose canonical form has components
and ,
with an arbitrary positive smooth bounded function. In Minkowski
space-time, this describes a position-dependent space-like or magnetic
noncommutativity. We show how to generalize our construction to
arbitrary dimensions and use it to find traveling noncommutative lumps
generalizing noncommutative solitons discussed in the literature. Next we
consider Euclidean field theory on such a noncommutative
background. Using a zeta-like regulator, the covariant perturbation method and
working in configuration space, we explicitly compute the UV singularities. We
find that, while the two-point UV divergences are non-local, the four-point UV
divergences are local, in accordance with recent results for constant .Comment: 1+22 pages, no figure
Heat kernel, effective action and anomalies in noncommutative theories
Being motivated by physical applications (as the phi^4 model) we calculate
the heat kernel coefficients for generalised Laplacians on the Moyal plane
containing both left and right multiplications. We found both star-local and
star-nonlocal terms. By using these results we calculate the large mass and
strong noncommutativity expansion of the effective action and of the vacuum
energy. We also study the axial anomaly in the models with gauge fields acting
on fermions from the left and from the right.Comment: 21 pages, v2: references adde
Index theory for locally compact noncommutative geometries
Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, we prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in our algebra
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