Searcharxiv⌕ Search

arXiv subjects

Jean-Christophe Wallet

Publications and source records attributed to Jean-Christophe Wallet.

At least 19 recordsLinked to original sources
arXiv2025

Noncommutative Gauge Theories: Yang-Mills extensions and beyond - An overview

The status of several representative gauge theories on various quantum space-times, mainly focusing on Yang-Mills type extensions together with a few matrix model formulations is overviewed. The common building blocks are derivation based differential calculus possibly twisted and noncommutative analog of the Koszul connection. The star-products related to the quantum space-times are obtained from a combination of harmonic analysis of group algebras combined with Weyl quantization. The remaining problems inherent to gauge theories on Moyal spaces in their two different formulations are outlined. A family of gauge invariant matrix models on $\mathbb{R}^3_λ$, a deformation of $\mathbb{R}^3$ is presented among which a solvable model. The characterization of 11 new quantum Minkowski space-times through their $*$-algebras is given. A gauge theory of Yang-Mills type is constructed on one recently explored of these space-times and compared to its counterpart built on the popular $κ$-Minkowski.

hep-th↗
arXiv2025

Star-products for Lie-algebraic noncommutative Minkowski space-times

Poisson structures of the Poincaré group can be linked to deformations of the Minkowski space-time, classified some time ago by Zakrewski. Based on this classification, various quantum Minkowski space-times with coordinates Lie algebras and specific Poincare Hopf algebras have been exhibited by Mercati and called T-Minkowski space-times. Here we construct the star products and involutions characterizing the $\star$-algebras for a broad family of Lie algebras which includes 11 out of 17 Lie algebras of T-Minkowski spaces. We show that the usual Lebesgue integral defines either a trace or a KMS weight ('twisted trace') depending on whether the Lie group of the coordinates' Lie algebra is unimodular or not. Finally, we give the Poincaré Hopf algebras when they are compatible with our $*$-product. General derivation of such symmetry Hopf algebras are briefly discussed.

hep-th↗
arXiv2025

Gauge Theories on quantum Minkowski spaces: $ρ$ versus $κ$

The $ρ$-Minkowski space-time, a Lie-algebraic deformation of the usual Minkowski space-time is considered. A star-product realization of this quantum space-time together with the characterization of the deformed Poincaré symmetry acting on it are presented. It is shown that appearance of UV/IR mixing is expected already in scalar field theories on $ρ$-Minkowski. Classical and one-loop features of a typical gauge theory on this quantum space-time are presented and critically compared to the situation for $κ$-Minkowski.

hep-th↗
arXiv2024

$κ$-Minkowski as tangent space I: quantum partition of unity

We define a quantum (noncommutative) analogue of locally trivial tangent bundle based on two main elements: the definition of local algebras through quotients of ideals of the global algebra as introduced in [21], and the triviality of the local tangent space as being the $κ$-Minkowski space inspired from [2]. This tangent bundle is explicitly constructed via local coordinate charts. Every local objects are exported to the global algebra through the notion of quantum (noncommutative) partition of unity introduced in this purpose. This partition is also used to export consistently an integral on $κ$-Minkowski to an integral on the global algebra.

math-ph↗
arXiv2024

Gauge theory on $ρ$-Minkowski space-time

We construct a gauge theory model on the 4-dimensional $ρ$-Minkowski space-time, a particular deformation of the Minkowski space-time recently considered. The corresponding star product results from a combination of Weyl quantization map and properties of the convolution algebra of the special Euclidean group. We use noncommutative differential calculi based on twisted derivations together with a twisted notion of noncommutative connection. The twisted derivations pertain to the Hopf algebra of $ρ$-deformed translations, a Hopf subalgebra of the $ρ$-deformed Poincaré algebra which can be viewed as defining the quantum symmetries of the $ρ$-Minkowski space-time. The gauge theory model is left invariant under the action of the $ρ$-deformed Poincaré algebra. The kinetic part of the action is found to coincide with the one of the usual (commutative) electrodynamics.

hep-th↗
arXiv2023

Quantum causality in $κ$-Minkowski and related constraints

We study quantum causal structures in $1+1$ $κ$-Minkowski space-time described by a Lorentzian Spectral Triple whose Dirac operator is built from a natural set of twisted derivations of the $κ$-Poincaré algebra. We show that the Lorentzian Spectral Triple must be twisted to accommodate the twisted nature of the derivations. We exhibit various interesting classes of causal functions, including an analog of the light-cone coordinates. We show in particular that the existence of a causal propagation between two pure states, the quantum analogs of points, can exist provided quantum constraints, linking the momentum and the space coordinate, are satisfied. One of these constraints is a quantum analog of the speed of light limit.

math-ph↗
arXiv2023

Field theories on $ρ$-deformed Minkowski space-time

We study one-loop perturbative properties of scalar field theories on the $ρ$-Minkowski space. The corresponding star-product, together with the involution are characterized from a combination of Weyl quantization and defining properties of the convolution algebra of the Euclidean group linked to the coordinate algebra of the $ρ$-Minkowski space. The natural integration measure linked to the Haar measure of the Euclidean group defines a trace for the star-product. One-loop properties of the 2-point and 4-point functions for families of complex-valued scalar field theories on $ρ$-Minkowski space are examined. For scalar theories with orientable interaction, the 2-point function is found to receive UV quadratically diverging one-loop corrections in 4 dimensions while no IR singularities generating UV/IR mixing appears. These however occur in the one-loop corrections to the 4-point function. As well, one-loop 2-point functions for theories with non-orientable interaction involve such IR singularities. These results are discussed.

hep-th↗
arXiv2022

Gauge theories on quantum spaces

We review the present status of gauge theories built on various quantum space-times described by noncommutative space-times. The mathematical tools and notions underlying their construction are given. Different formulations of gauge theory models on Moyal spaces as well as on quantum spaces whose coordinates form a Lie algebra are covered, with particular emphasis on some explored quantum properties. Recent attempts aiming to include gravity dynamics within a noncommutative framework are also considered.

hep-th↗
arXiv2022

Quantum instability of gauge theories on $κ$-Minkowski space

We consider a gauge theory on the 5-d $κ$-Minkowski which can be viewed as the noncommutative analog of a $U(1)$ gauge theory. We show that the Hermiticity condition obeyed by the gauge potential $A_μ$ is necessarily twisted. Performing a BRST gauge-fixing with a Lorentz-type gauge, we carry out a first exploration of the one loop quantum properties of this gauge theory. We find that the gauge-fixed theory gives rise to a non-vanishing tadpole for the time component of the gauge potential, while there is no non-vanishing tadpole 1-point function for the spatial components of $A_μ$. This signals that the classical vacuum of the theory is not stable against quantum fluctuations. Possible consequences regarding the symmetries of the gauge model and the fate of the tadpole in other gauges of non-covariant type are discussed.

hep-th↗
arXiv2022

Gauge theory models on $κ$-Minkowski space: Results and prospects

Recent results obtained in $κ$-Poincaré invariant gauge theories on $κ$-Minkowski space are reviewed and commented. A Weyl quantization procedure can be applied to convolution algebras to derive a convenient star product. For such a star product, gauge invariant polynomial action functional depending on the curvature exists only in 5 dimensions. The corresponding noncommutative differential calculus and the related connection are twisted together with the BRST structure linked to the gauge invariance. Phenomenological consequences stemming from the existence of one extra dimension are commented. Some consequences of the appearance of a non-vanishing one-loop tadpole upon BRST gauge-fixing are discussed.

hep-th↗
arXiv2022

Algebraic structures in $κ$-Poincaré invariant gauge theories

$κ$-Poincaré invariant gauge theories on $κ$-Minkowski space-time, which are noncommutative analogs of the usual $U(1)$ gauge theory, exist only in five dimensions. These are built from noncommutative twisted connections on a hermitian right module over the algebra coding the $κ$-Minkowski space-time. We show that twisting the action of this algebra on the hermitian module, assumed to be a copy of it, affects neither the value of the above dimension nor the noncommutative gauge group defined as the unitary automorphisms of the module leaving the hermitian structure unchanged. Only the hermiticity condition obeyed by the gauge potential becomes twisted. Similarities between the present framework and algebraic features of twisted spectral triples are exhibited.

hep-th↗
arXiv2022

Quantum causality constraints on kappa-Minkowski space-time

The $κ$-Minkoswki space-time provides a quantum noncommutative-deformation of the usual Minkowski space-time. However, a notion of causality is difficult to be defined in such a space with noncommutative time. In this paper, we define a notion of causality on a (1+1)-dimensional $κ$-Minkoswki space-time using the more general framework of Lorentzian noncommutative geometry. We show that this notion allows specific causal relations, but limited by a general constraint which is a quantum generalization of the traditional speed of light limit.

math-ph↗
arXiv2021

Twisted BRST symmetry in gauge theories on $κ$-Minkowski

Algebraic properties of the BRST symmetry associated to the twisted gauge symmetry occurring in the $κ$-Poincaré invariant gauge theories on the $κ$-Minkowski space are investigated. We find that the BRST operation associated to the gauge invariance of the action functional can be continuously deformed together with its corresponding Leibniz rule, into a nilpotent twisted BRST operation, leading to a twisted BRST symmetry algebra which may be viewed as a noncommutative analog of the usual Yang-Mills BRST algebra.

hep-th↗
arXiv2021

Single Extra Dimension from $κ$-Poincaré and Gauge Invariance

We show that $κ$-Poincaré invariant gauge theories on $κ$-Minkowski space with physically acceptable commutative (low energy) limit must be 5-d. The gauge invariance requirement of the action fixes the dimension of the $κ$-Minkowski space to $d=5$ and selects the unique twisted differential calculus with which the construction can be achieved. We characterize a BRST symmetry related to the 5-d noncommutative gauge invariance though the definition of a nilpotent operation, which is used to construct a gauge-fixed action. We also consider standard scenarios assuming (compactification of) flat extra dimension, for which the 5-d deformation parameter $κ$ can be viewed as the bulk 5-d Planck mass. We study physical properties of the resulting 4-d effective theories. Recent data from collider experiments require $κ\gtrsim\mathcal{O}(10^{13})\ \text{GeV}$. The use of standard test of in-vacuo dispersion relations of Gamma Ray Burst photons increases this lower bound by 4 orders of magnitude. The robustness of this bound is discussed in the light of possible new features of noncommutative causal structures.

hep-th↗
arXiv2020

Gauge theories on $κ$-Minkowski spaces: Twist and modular operators

We discuss the construction of $κ$-Poincaré invariant actions for gauge theories on $κ$-Minkowski spaces. We consider various classes of untwisted and (bi)twisted differential calculi. Starting from a natural class of noncommutative differential calculi based on a particular type of twisted derivations belonging to the algebra of deformed translations, combined with a twisted extension of the notion of connection, we prove an algebraic relation between the various twists and the classical dimension d of the $κ$-Minkowski space(-time) ensuring the gauge invariance of the candidate actions for gauge theories. We show that within a natural differential calculus based on a distinguished set of twisted derivations, d=5 is the unique value for the classical dimension at which the gauge action supports both the gauge invariance and the $κ$-Poincaré invariance. Within standard (untwisted) differential calculi, we show that the full gauge invariance cannot be achieved, although an invariance under a group of transformations constrained by the modular (Tomita) operator stemming from the $κ$-Poincaré invariance still holds.

hep-th↗
arXiv2019

$κ$-Poincaré invariant orientable field theories at 1-loop

We consider a family of $κ$-Poincaré invariant scalar field theories on 4-d $κ$-Minkowski space with quartic orientable interaction, that is for which $ϕ$ and its conjugate $ϕ^†$ alternate in the quartic interaction, and whose kinetic operator is the square of a $U_κ(iso(4))$-equivariant Dirac operator. The formal commutative limit yields the standard complex $ϕ^4$ theory. We find that the 2-point function receives UV linearly diverging 1-loop corrections while it stays free of IR singularities that would signal occurrence of UV/IR mixing. We find that all the 1-loop planar and non-planar contributions to the 4-point function are UV finite, stemming from the existence of the particular estimate for the propagator partly combined with its decay properties at large momenta, implying formally vanishing of the beta-functions at 1-loop so that the coupling constants stay scale-invariant at 1-loop.

hep-th↗
arXiv2017

Involutive representations of coordinate algebras and quantum spaces

We show that $\frak{su}(2)$ Lie algebras of coordinate operators related to quantum spaces with $\frak{su}(2)$ 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 $\mathfrak{su}(2)$. 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.

hep-th↗