Noncommutative Yang–Mills–Higgs actions from derivation-based differential calculus

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We show that natural noncommutative gauge theory models on R 3 λ can accommodate gauge invariant harmonic terms, thanks to the existence of a relationship between the center of R 3 λ and the components of the gauge invariant 1-form canonical connection. This latter object shows up naturally within the present noncommutative differential calculus. Restricting ourselves to positive actions with covariant coordinates as field variables, a suitable gauge-fixing leads to a family of matrix models with quartic interactions and kinetic operators with compact resolvent. Their perturbative behavior is then studied. We first compute the 2-point and 4-point functions at the one-loop order within a subfamily of these matrix models for which the interactions have a symmetric form. We find that the corresponding contributions are finite. We then extend this result to arbitrary order. We find that the amplitudes of the ribbon diagrams for the models of this subfamily are finite to all orders in perturbation. This result extends finally to any of the models of the whole family of matrix models obtained from the above gauge-fixing. The origin of this result is discussed. Finally, the existence of a particular model related to integrable hierarchies is indicated, for which the partition function is expressible as a product of ratios of determinants.

Section: A Family Of Gauge Invariant Classical Actionsmentioning

confidence: 99%

Section: Jhep12(2015)045mentioning

confidence: 99%

Section: Jhep12(2015)045mentioning

confidence: 99%

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Noncommutative gauge theories on ℝ λ 3 $$ {\mathrm{\mathbb{R}}}_{\uplambda}^3 $$ : perturbatively finite models

Géré

Jurić²,

Wallet

2015

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“…It would be interesting to extend these NCFT to the case of noncommutative gauge theories built from differential calculi which do not belong to the category of usual derivationbased differential calculi [68][69][70]. These latter are known to give rise to gauge theory models whose commutative limit do not coincide with standard gauge theories on R 3 , reflecting the fact that no (analog of) radial dependence can be accommodated in this framework.…”

Section: Jhep07(2017)116mentioning

confidence: 99%

Involutive representations of coordinate algebras and quantum spaces

Jurić

Poulain

Wallet

2017

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We show that su(2) Lie algebras of coordinate operators related to quantum spaces with su(2) noncommutativity can be conveniently represented by SO(3)-covariant poly-differential involutive representations. We show that the quantized plane waves obtained from the quantization map action on the usual exponential functions are determined by polar decomposition of operators combined with constraint stemming from the Wigner theorem for SU(2). Selecting a subfamily of * -representations, we show that the resulting star-product is equivalent to the Kontsevich product for the Poisson manifold dual to the finite dimensional Lie algebra su(2). We discuss the results, indicating a way to extend the construction to any semi-simple non simply connected Lie group and present noncommutative scalar field theories which are free from perturbative UV/IR mixing.

“…The focus will be mostly on formulae and their application to the truncated Heisenberg geometry; for more mathematical recent reviews see [14] or [26]. The gauge potential A and the field strength F are respectively a 1-form and a 2-form,…”

Section: Gauge Fieldsmentioning

confidence: 99%

Gauge fields on noncommutative geometries with curvature

Burić

Grosse

Madore

2010

It was shown recently that the lagrangian of the Grosse-Wulkenhaar model can be written as lagrangian of the scalar field propagating in a curved noncommutative space. In this interpretation, renormalizability of the model is related to the interaction with the background curvature which introduces explicit coordinate dependence in the action. In this paper we construct the U 1 gauge field on the same noncommutative space: since covariant derivatives contain coordinates, the Yang-Mills action is again coordinate dependent. To obtain a two-dimensional model we reduce to a subspace, which results in splitting of the degrees of freedom into a gauge and a scalar. We define the gauge fixing and show the BRST invariance of the quantum action.