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Derivations of the Moyal Algebra and Noncommutative Gauge Theories
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2009
"... The differential calculus based on the derivations of an associative algebra underlies most of the noncommutative field theories considered so far. We review the essential properties of this framework and the main features of noncommutative connections in the case of non graded associative unital a ..."
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Cited by 12 (3 self)
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The differential calculus based on the derivations of an associative algebra underlies most of the noncommutative field theories considered so far. We review the essential properties of this framework and the main features of noncommutative connections in the case of non graded associative unital algebras with involution. We extend this framework to the case of Z2graded unital involutive algebras. We show, in the case of the Moyal algebra or some related Z2graded version of it, that the derivation based differential calculus is a suitable framework to construct Yang–Mills–Higgs type models on Moyal (or related) algebras, the covariant coordinates having in particular a natural interpretation as Higgs fields. We also exhibit, in one situation, a link between the renormalisable NC ϕ4model with harmonic term and a gauge theory model. Some possible consequences of this are briefly discussed.
VignesTourneret F., Quantum field theory on the degenerate Moyal space
"... We prove that the selfinteracting scalar field on the fourdimensional degenerate Moyal plane is renormalisable to all orders when adding a suitable counterterm to the Lagrangean. Despite the apparent simplicity of the model, it raises several non trivial questions. Our result is a first step towar ..."
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We prove that the selfinteracting scalar field on the fourdimensional degenerate Moyal plane is renormalisable to all orders when adding a suitable counterterm to the Lagrangean. Despite the apparent simplicity of the model, it raises several non trivial questions. Our result is a first step towards the definition of renormalisable quantum field theories on a noncommutative Minkowski space. 1
Non Commutative Field Theory on Rank One Symmetric Spaces,” arXiv:0806.4255 [hepth
"... Quantum field theory has been shown recently renormalizable on flat Moyal space and better behaved than on ordinary spacetime. Some models at least should be completely finite, even beyond perturbation theory. In this paper a first step is taken to extend such theories to nonflat backgrounds such ..."
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Quantum field theory has been shown recently renormalizable on flat Moyal space and better behaved than on ordinary spacetime. Some models at least should be completely finite, even beyond perturbation theory. In this paper a first step is taken to extend such theories to nonflat backgrounds such as solvable symmetric spaces. 1
Minimalist translationinvariant noncommutative scalar field theory, Preprint arXiv:0803.1035
, 2008
"... We prove the perturbative renormalisability of a minimalist translationinvariant noncommutative scalar field theory. The result is based on a careful analysis of the uv/ir mixing of the noncommutative φ ⋆4 4 model on the Moyal space R4 Θ. 1 ..."
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We prove the perturbative renormalisability of a minimalist translationinvariant noncommutative scalar field theory. The result is based on a careful analysis of the uv/ir mixing of the noncommutative φ ⋆4 4 model on the Moyal space R4 Θ. 1
Construction of 2dimensional GrosseWulkenhaar model
, 2011
"... In this paper we construct the noncommutative GrosseWulkenhaar model on 2dimensional Moyal plane with the method of loop vertex expansion. We treat renormalization with this new tool, adapt Nelson’s argument and prove Borel summability of the perturbation series. This is the first noncommutative ..."
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In this paper we construct the noncommutative GrosseWulkenhaar model on 2dimensional Moyal plane with the method of loop vertex expansion. We treat renormalization with this new tool, adapt Nelson’s argument and prove Borel summability of the perturbation series. This is the first noncommutative quantum field theory model to be built in a nonperturbative sense.
Instantons, fluxons and open gauge string theory
, 2004
"... Preprint typeset in JHEP style HYPER VERSION ..."
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UWThPh200616 Exact renormalization of a noncommutative φ 3 model in 6 dimensions
, 2007
"... The noncommutative selfdual φ 3 model in 6 dimensions is quantized and essentially solved, by mapping it to the Kontsevich model. The model is shown to be renormalizable and asymptotically free, and solvable genus by genus. It requires both wavefunction and coupling constant renormalization. The exa ..."
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The noncommutative selfdual φ 3 model in 6 dimensions is quantized and essentially solved, by mapping it to the Kontsevich model. The model is shown to be renormalizable and asymptotically free, and solvable genus by genus. It requires both wavefunction and coupling constant renormalization. The exact (“allorder”) renormalization of the bare parameters is determined explicitly, which turns out to depend on the genus 0 sector only. The running coupling constant is also computed exactly, which decreases more rapidly than predicted by the oneloop beta function. A phase transition to an unstable phase is found. 1
Renormalization of Orientable NonCommutative Complex Φ 6 3 Model
, 2008
"... In this paper we prove that the GrosseWulkenhaar type noncommutative orientable complex scalar ϕ6 3 theory, with two noncommutative coordinates and the third one commuting with the other two, is renormalizable to all orders in perturbation theory. Our proof relies on a multiscale analysis in x sp ..."
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In this paper we prove that the GrosseWulkenhaar type noncommutative orientable complex scalar ϕ6 3 theory, with two noncommutative coordinates and the third one commuting with the other two, is renormalizable to all orders in perturbation theory. Our proof relies on a multiscale analysis in x space. 1
Duality and Braiding in Twisted Quantum Field Theory,” arXiv:0711.1525 [hepth
"... We reexamine various issues surrounding the definition of twisted quantum field theories on flat noncommutative spaces. We propose an interpretation based on nonlocal commutative field redefinitions which clarifies previously observed properties such as the formal equivalence of Green’s functions i ..."
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We reexamine various issues surrounding the definition of twisted quantum field theories on flat noncommutative spaces. We propose an interpretation based on nonlocal commutative field redefinitions which clarifies previously observed properties such as the formal equivalence of Green’s functions in the noncommutative and commutative theories, causality, and the absence of UV/IR mixing. We use these fields to define the functional integral formulation of twisted quantum field theory. We exploit techniques from braided tensor algebra to argue that the twisted Fock space states of these free fields obey conventional statistics. We support our claims with a detailed analysis of the modifications induced in the presence of background magnetic fields, which induces additional twists by magnetic translation operators and alters the effective noncommutative geometry seen by the twisted quantum fields. When two such field theories are dual to one another, we demonstrate that only our braided physical states Twisted quantum field theory is a modification of the traditional approach to noncommutative field theory [19, 42] aimed at restoring the symmetries of spacetime which are broken by noncommutativity.