Results 1  10
of
27
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 ..."
Abstract

Cited by 22 (10 self)
 Add to MetaCart
(Show Context)
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”.
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 ..."
Abstract

Cited by 12 (3 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
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.
Examples of derivationbased differential calculi related to noncommutative gauge theories
 INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS
, 2008
"... ..."
Gauge theories in noncommutative geometry
, 2011
"... In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and gauge transformations. Two different approaches to noncommut ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
In this review we present some of the fundamental mathematical structures which permit to define noncommutative gauge field theories. In particular, we emphasize the theory of noncommutative connections, with the notions of curvatures and gauge transformations. Two different approaches to noncommutative geometry are covered: the one based on derivations and the one based on spectral triples. Examples of noncommutative gauge field theories are given to illustrate the constructions and to display some of the common features.
On the Origin of the Harmonic Term in Noncommutative Quantum Field Theory
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2010
"... The harmonic term in the scalar field theory on the Moyal space removes the UVIR mixing, so that the theory is renormalizable to all orders. In this paper, we review the three principal interpretations of this harmonic term: the Langmann–Szabo duality, the superalgebraic approach and the noncommuta ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
The harmonic term in the scalar field theory on the Moyal space removes the UVIR mixing, so that the theory is renormalizable to all orders. In this paper, we review the three principal interpretations of this harmonic term: the Langmann–Szabo duality, the superalgebraic approach and the noncommutative scalar curvature interpretation. Then, we show some deep relationship between these interpretations.
Matrix Bases for Star Products: a Review?
, 2014
"... Abstract. We review the matrix bases for a family of noncommutative? products based on a Weyl map. These products include the Moyal product, as well as the Wick–Voros products and other translation invariant ones. We also review the derivation of Lie algebra type star products, with adapted matrix b ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. We review the matrix bases for a family of noncommutative? products based on a Weyl map. These products include the Moyal product, as well as the Wick–Voros products and other translation invariant ones. We also review the derivation of Lie algebra type star products, with adapted matrix bases. We discuss the uses of these matrix bases for field theory, fuzzy spaces and emergent gravity. Key words: noncommutative geometry; star products; matrix models 2010 Mathematics Subject Classification: 58Bxx; 40C05; 46L65 1
LYCEN 200805 UWThPh200806
, 2008
"... Translationinvariant models for noncommutative gauge fields ..."