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38
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.
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”.
Oneloop calculations for a translation invariant noncommutative gauge model
 Eur. Phys. J. C
"... In this paper we discuss oneloop results for the translation invariant noncommutative gauge field model we recently introduced in ref. [1]. This model relies on the addition of some carefully chosen extra terms in the action which mix long and short scales in order to circumvent the infamous UV/IR ..."
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Cited by 18 (12 self)
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In this paper we discuss oneloop results for the translation invariant noncommutative gauge field model we recently introduced in ref. [1]. This model relies on the addition of some carefully chosen extra terms in the action which mix long and short scales in order to circumvent the infamous UV/IR mixing, and were motivated by the renormalizable noncommutative scalar model of Gurau et al. [2].
Spectral action on noncommutative torus
 J. Noncommut. Geom
"... Dedicated to Alain Connes on the occasion of his 60th birthday The spectral action on noncommutative torus is obtained, using a Chamseddine– Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series ..."
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Cited by 18 (9 self)
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Dedicated to Alain Connes on the occasion of his 60th birthday The spectral action on noncommutative torus is obtained, using a Chamseddine– Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series of holomorphic functions are obtained in this context.
On the Problem of Renormalizability in NonCommutative Gauge Field Models — A Critical Review
"... When considering quantum field theories on noncommutative spaces one inevitably encounters the infamous UV/IR mixing problem. So far, only very few renormalizable models exist and all of them describe noncommutative scalar field theories on fourdimensional Euclidean GroenewoldMoyal deformed space ..."
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Cited by 14 (10 self)
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When considering quantum field theories on noncommutative spaces one inevitably encounters the infamous UV/IR mixing problem. So far, only very few renormalizable models exist and all of them describe noncommutative scalar field theories on fourdimensional Euclidean GroenewoldMoyal deformed space, also known as ‘θdeformed space ’ R4 θ. In this work we discuss some major obstacles of constructing a renormalizable noncommutative gauge field model and sketch some possible ways out. 1
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.
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.
Noncommutative εgraded connections
, 2012
"... We introduce the new notion of εgraded associative algebras which takes its root into the notion of commutation factors introduced in the context of Lie algebras [1]. We define and study the associated notion of εderivationbased differential calculus, which generalizes the derivationbased differ ..."
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Cited by 10 (2 self)
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We introduce the new notion of εgraded associative algebras which takes its root into the notion of commutation factors introduced in the context of Lie algebras [1]. We define and study the associated notion of εderivationbased differential calculus, which generalizes the derivationbased differential calculus on associative algebras. A corresponding notion of noncommutative connection is also defined. We illustrate these considerations with various examples of εgraded algebras, in particular some graded matrix algebras and the Moyal algebra. This last example permits also to interpret mathematically a noncommutative gauge field theory.