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Noncommutative Renormalization
 SÉMINAIRE POINCARÉ X (2007) 1 – 81
, 2007
"... A new version of scale analysis and renormalization theory has been found on the noncommutative Moyal space. It could be useful for physics beyond the standard model or for standard physics in strong external field. The good news is that quantum field theory is better behaved on noncommutative tha ..."
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Cited by 169 (28 self)
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A new version of scale analysis and renormalization theory has been found on the noncommutative Moyal space. It could be useful for physics beyond the standard model or for standard physics in strong external field. The good news is that quantum field theory is better behaved on noncommutative than on ordinary space: indeed it has no Landau ghost. Noncommutativity might therefore be an alternative to supersymmetry. We review this rapidly growing subject.
Noncommutative Induced Gauge Theory
, 2007
"... We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4dimensional Moyal space and compute in position space the oneloop YangMillstype effective theory generated from the integration over the scalar field. We find that the gauge invariant effective action ..."
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Cited by 38 (10 self)
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We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4dimensional Moyal space and compute in position space the oneloop YangMillstype effective theory generated from the integration over the scalar field. We find that the gauge invariant effective action involves, beyond the expected noncommutative version of the pure YangMills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic oscillator term, which for the noncommutative ϕ 4theory on Moyal space ensures renormalisability. The expression of a possible candidate for a renormalisable action for a gauge theory defined on Moyal space is conjectured and discussed.
On the vacuum states for noncommutative gauge theory
, 2008
"... Candidates for renormalisable gauge theory models on Moyal spaces constructed recently have non trivial vacua. We show that these models support vacuum states that are invariant under both global rotations and symplectic isomorphisms which form a global symmetry group for the action. We compute the ..."
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Cited by 27 (8 self)
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Candidates for renormalisable gauge theory models on Moyal spaces constructed recently have non trivial vacua. We show that these models support vacuum states that are invariant under both global rotations and symplectic isomorphisms which form a global symmetry group for the action. We compute the explicit expression in the position space for these vacuum configurations in two and four dimensions.
Renormalization of noncommutative φ 4 4 field theory in x space
 Commun. Math. Phys
"... In this paper we provide a new proof that the GrosseWulkenhaar noncommutative scalar Φ4 4 theory is renormalizable to all orders in perturbation theory, and extend it to more general models with covariant derivatives. Our proof relies solely on a multiscale analysis in x space. We think this proof ..."
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Cited by 24 (9 self)
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In this paper we provide a new proof that the GrosseWulkenhaar noncommutative scalar Φ4 4 theory is renormalizable to all orders in perturbation theory, and extend it to more general models with covariant derivatives. Our proof relies solely on a multiscale analysis in x space. We think this proof is simpler and could be more adapted to the future study of these theories (in particular at the nonperturbative or constructive level). 1
Dimensional regularization and renormalization of noncommutative quantum field theory
, 2008
"... Using the recently introduced parametric representation of noncommutative quantum field theory, we implement here the dimensional regularization and renormalization of the vulcanized Φ ⋆4 4 model on the Moyal space. ..."
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Cited by 24 (6 self)
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Using the recently introduced parametric representation of noncommutative quantum field theory, we implement here the dimensional regularization and renormalization of the vulcanized Φ ⋆4 4 model on the Moyal space.
Noncommutative YangMillsHiggs actions from derivationbased differential calculus
 arXiv:0804.3061, High Energy Physics  Theory (hepth
, 2008
"... Derivations of a noncommutative algebra can be used to construct differential calculi, the socalled derivationbased differential calculi. We apply this framework to a version of the Moyal algebra M. We show that the differential calculus, generated by the maximal subalgebra of the derivation algeb ..."
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Cited by 22 (10 self)
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Derivations of a noncommutative algebra can be used to construct differential calculi, the socalled derivationbased differential calculi. We apply this framework to a version of the Moyal algebra M. We show that the differential calculus, generated by the maximal subalgebra of the derivation algebra of M that can be related to infinitesimal symplectomorphisms, gives rise to a natural construction of YangMillsHiggs models on M and a natural interpretion of the covariant coordinates as Higgs fields. We also compare in detail the main mathematical properties characterizing the present situation to those specific of two other noncommutative geometries, namely the finite dimensional matrix algebra Mn(C) and the algebra of matrix valued functions C ∞ (M) ⊗ Mn(C). The UV/IR mixing problem of the resulting YangMillsHiggs models is also discussed. Work supported by ANR grant NT05343374 “GENOPHY”.
Selfdual noncommutative φ4theory in four dimensions is a nonperturbatively solvable and nontrivial quantum field theory,” arXiv:1205.0465
"... in four dimensions is a nonperturbatively solvable and nontrivial quantum field theory ..."
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Cited by 19 (6 self)
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in four dimensions is a nonperturbatively solvable and nontrivial quantum field theory
Classical solutions of the TEK model and noncommutative instantons in two dimensions
 J. High Energy Phys
"... Abstract: The twisted EguchiKawai (TEK) model provides a nonperturbative definition of noncommutative YangMills theory: the continuum limit is approached at large N by performing suitable double scaling limits, in which nonplanar contributions are no longer suppressed. We consider here the twod ..."
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Cited by 14 (1 self)
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Abstract: The twisted EguchiKawai (TEK) model provides a nonperturbative definition of noncommutative YangMills theory: the continuum limit is approached at large N by performing suitable double scaling limits, in which nonplanar contributions are no longer suppressed. We consider here the twodimensional case, trying to recover within this framework the exact results recently obtained by means of Morita equivalence. We present a rather explicit construction of classical gauge theories on noncommutative toroidal lattice for general topological charges. After discussing the limiting procedures to recover the theory on the noncommutative torus and on the noncommutative plane, we focus our attention on the classical solutions of the related TEK models. We solve the equations of motion and we find the configurations having finite action in the relevant double scaling limits. They can be explicitly described in terms of twisteaters and they exactly correspond to the instanton solutions that are seen to dominate the partition function on the noncommutative torus. Fluxons on the noncommutative plane are recovered as well. We also discuss how the highly nontrivial structure of the exact partition function can emerge from a direct matrix model computation. The quantum consistency of the TEK formulation is eventually checked by computing Wilson loops in a particular limit. Keywords: Noncommutative Gauge Theories, Matrix Models, LargeN limit. Contents
Noncommutative Induced Gauge Theories on Moyal Spaces
, 2007
"... Noncommutative field theories on Moyal spaces can be conveniently handled within a framework of noncommutative geometry. Several renormalisable matter field theories that are now identified are briefly reviewed. The construction of renormalisable gauge theories on these noncommutative Moyal spaces, ..."
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Cited by 13 (4 self)
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Noncommutative field theories on Moyal spaces can be conveniently handled within a framework of noncommutative geometry. Several renormalisable matter field theories that are now identified are briefly reviewed. The construction of renormalisable gauge theories on these noncommutative Moyal spaces, which remains so far a challenging problem, is then closely examined. The computation in 4D of the oneloop effective gauge theory generated from the integration over a scalar field appearing in a renormalisable theory minimally coupled to an external gauge potential is presented. The gauge invariant effective action is found to involve, beyond the expected noncommutative version of the pure YangMills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic term, which for the noncommutative ϕ 4theory on Moyal space ensures renormalisability. A class of possible candidates for renormalisable gauge theory actions defined on Moyal space is presented and discussed.